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Cleve Moler is the author of the first MATLAB, one of the founders of MathWorks, and is currently Chief Mathematician at the company. He writes here about MATLAB, scientific computing and interesting mathematics.
Updated: 1 day 9 hours ago

Fun With The Pascal Triangle

Mon, 02/19/2018 - 22:30
The Wikipedia article on Pascal's Triangle has hundreds of properties of the triangle and there are dozens of other Web pages devoted to it. Here are a few facts that I find most interesting. Contents Blaise Pascal Blaise Pascal (1623-1662) was a 17th century French mathematician, physicist, inventor and theologian. His Traité du triangle arithmétique (Treatise on Arithmetical Triangle) was published posthumously in 1665. But this was not the first publication about the triangle. Various versions appear in Indian, Chinese, Persian, Italian and other manuscripts centuries before Pascal. Binomial Coefficients The binomial coefficient usually denoted by ${n} \choose {k}$ is the number of ways of picking $k$ unordered outcomes from $n$ possibilities. These coefficients appear in the expansion of the binomial $(x+1)^n$. For example, when $n = 7$ syms x n = 7; x7 = expand((x+1)^n) x7 = x^7 + 7*x^6 + 21*x^5 + 35*x^4 + 35*x^3 + 21*x^2 + 7*x + 1 Formally, the binomial coefficients are given by $${{n} \choose {k}} = \frac {n!} {k! (n-k)!}$$ But premature floating point overflow of the factorials makes this an unsatisfactory basis for computation. A better way employs the recursion $$ {{n} \choose {k}} = {{n-1} \choose {k}} + {{n-1} \choose {k-1}}$$ This is used by the MATLAB function nchoosek(n,k). Pascal Matrices MATLAB offers two Pascal matrices. One is symmetric, positive definite and has the binomial coefficients on the antidiagonals. P = pascal(7) P = 1 1 1 1 1 1 1 1 2 3 4 5 6 7 1 3 6 10 15 21 28 1 4 10 20 35 56 84 1 5 15 35 70 126 210 1 6 21 56 126 252 462 1 7 28 84 210 462 924 The other is lower triangular, with the binomial coefficients in the rows. (We will see why the even numbered columns have minus signs in a moment.) L = pascal(7,1) L = 1 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 -2 1 0 0 0 0 1 -3 3 -1 0 0 0 1 -4 6 -4 1 0 0 1 -5 10 -10 5 -1 0 1 -6 15 -20 15 -6 1 The individual elements are P(i,j) = P(j,i) = nchoosek(i+j-2,j-1) And (temporarily ignoring the minus signs) for i $\ge$ j L(i,j) = nchoosek(i-1,j-1) The first fun fact is that L is the (lower) Cholesky factor of P. L = chol(P)' L = 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 2 1 0 0 0 0 1 3 3 1 0 0 0 1 4 6 4 1 0 0 1 5 10 10 5 1 0 1 6 15 20 15 6 1 So we can reconstruct P from L. P = L*L' P = 1 1 1 1 1 1 1 1 2 3 4 5 6 7 1 3 6 10 15 21 28 1 4 10 20 35 56 84 1 5 15 35 70 126 210 1 6 21 56 126 252 462 1 7 28 84 210 462 924 Pascal Triangle The traditional Pascal triangle is obtained by rotating P clockwise 45 degrees, or by sliding the rows of L to the right in half increments. Each element of the resulting triangle is the sum of the two above it. triprint(L) 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Square Root of Identity When the even numbered columns of L are given minus signs the matrix becomes a square root of the identity. L = pascal(n,1) L_squared = L^2 L = 1 0 0 0 0 0 0 1 -1 0 0 0 0 0 1 -2 1 0 0 0 0 1 -3 3 -1 0 0 0 1 -4 6 -4 1 0 0 1 -5 10 -10 5 -1 0 1 -6 15 -20 15 -6 1 L_squared = 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 Here is an exercise for you. What is sqrt(eye(n))? Why isn't it L? Cube Root of Identity When I first saw this, I was amazed. Rotate L counterclockwise. The result is a cube root of the identity. X = rot90(L,-1) X_cubed = X^3 X = 1 1 1 1 1 1 1 -6 -5 -4 -3 -2 -1 0 15 10 6 3 1 0 0 -20 -10 -4 -1 0 0 0 15 5 1 0 0 0 0 -6 -1 0 0 0 0 0 1 0 0 0 0 0 0 X_cubed = 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 Sierpinski Which binomial coefficients are odd? It's a fledgling fractal. odd = @(x) mod(x,2)==1; n = 56; L = abs(pascal(n,1)); spy(odd(L)) title('odd(L)') Fibonacci The sums of the antidiagonals of L are the Fibonacci numbers. n = 12; A = fliplr(abs(pascal(n,1))) for k = 1:n F(k) = sum(diag(A,n-k)); end F A = 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 2 1 0 0 0 0 0 0 0 0 1 3 3 1 0 0 0 0 0 0 0 1 4 6 4 1 0 0 0 0 0 0 1 5 10 10 5 1 0 0 0 0 0 1 6 15 20 15 6 1 0 0 0 0 1 7 21 35 35 21 7 1 0 0 0 1 8 28 56 70 56 28 8 1 0 0 1 9 36 84 126 126 84 36 9 1 0 1 10 45 120 210 252 210 120 45 10 1 1 11 55 165 330 462 462 330 165 55 11 1 F = 1 1 2 3 5 8 13 21 34 55 89 144 pi The elements in the third column of lower triangular Pascal matrix are the triangle numbers. The n-th triangle number is the number of bowling pins in the n-th row of an array of bowling pins. $$t_n = {{n+1} \choose {2}}$$ L = pascal(12,1); t = L(3:end,3)' t = 1 3 6 10 15 21 28 36 45 55 Here's an unusual series relating the triangle numbers to $\pi$. The signs go + + - - + + - - . pi - 2 = 1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 - 1/36 + 1/45 + 1/55 - ... type pi_pascal function pie = pi_pascal(n) tk = 1; s = 1; for k = 2:n tk = tk + k; if mod(k+1,4) > 1 s = s + 1/tk; else s = s - 1/tk; end end pie = 2 + s; Ten million terms gives $\pi$ to 14 decimal places. format long pie = pi_pascal(10000000) err = pi - pie pie = 3.141592653589817 err = -2.398081733190338e-14 Matrix Exponential Finally, I love this one. The solution to the (potentially infinite) set of ordinary differential equations $\dot{x_1} = x_1$ $\dot{x_j} = x_j + (j-1) x_{j-1}$ is $x_j = e^t (t + 1)^{j-1}$ This means that the matrix exponential of the simple diagonal matrix D = diag(1:7,-1) D = 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 4 0 0 0 0 0 0 0 0 5 0 0 0 0 0 0 0 0 6 0 0 0 0 0 0 0 0 7 0 is expm_D = round(expm(D)) expm_D = 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 2 1 0 0 0 0 0 1 3 3 1 0 0 0 0 1 4 6 4 1 0 0 0 1 5 10 10 5 1 0 0 1 6 15 20 15 6 1 0 1 7 21 35 35 21 7 1 Thanks Thanks to Nick Higham for pascal.m, gallery.m and section 28.4 of N. J. Higham, Accuracy and Stability of Numerical Algorithms, Second edition, SIAM, 2002. http://epubs.siam.org/doi/book/10.1137/1.9780898718027

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Published with MATLAB® R2018a has hundreds of properties of the % triangle and there are dozens of other Web pages devoted to it. % Here are a few facts that I find most interesting. %% Blaise Pascal % % <> % %% % (1623-1662) % was a 17th century French mathematician, physicist, inventor and % theologian. His _Traité du triangle arithmétique_ (_Treatise on % Arithmetical Triangle_) was published posthumously in 1665. % But this was not the first publication about the triangle. % Various versions appear in Indian, Chinese, Persian, Italian % and other manuscripts centuries before Pascal. %% Binomial Coefficients % The _binomial coefficient_ usually denoted by ${n} \choose {k}$ % is the number of ways of picking $k$ unordered outcomes from $n$ % possibilities. These coefficients appear in the expansion of % the binomial $(x+1)^n$. For example, when $n = 7$ syms x n = 7; x7 = expand((x+1)^n) %% % Formally, the binomial coefficients are given by % % $${{n} \choose {k}} = \frac {n!} {k! (n-k)!}$$ %% % But premature floating point overflow of the factorials makes this an % unsatisfactory basis for computation. A better way employs the recursion % % $$ {{n} \choose {k}} = {{n-1} \choose {k}} + {{n-1} \choose {k-1}}$$ % % This is used by the MATLAB function |nchoosek(n,k)|. %% Pascal Matrices % MATLAB offers two Pascal matrices. One is symmetric, positive % definite and has the binomial coefficients on the antidiagonals. P = pascal(7) %% % The other is lower triangular, with the binomial coefficients in the % rows. (We will see why the even numbered columns have minus signs % in a moment.) L = pascal(7,1) %% % The individual elements are % % P(i,j) = P(j,i) = nchoosek(i+j-2,j-1) % % And (temporarily ignoring the minus signs) for |i| $\ge$ |j| % % L(i,j) = nchoosek(i-1,j-1) %% % The first fun fact is that |L| is the (lower) Cholesky factor of |P|. L = chol(P)' %% % So we can reconstruct |P| from |L|. P = L*L' %% Pascal Triangle % The traditional Pascal triangle is obtained by rotating P clockwise % 45 degrees, or by sliding the rows of L to the right in half increments. % Each element of the resulting triangle is the sum of the two above it. triprint(L) %% Square Root of Identity % When the even numbered columns of |L| are given minus signs % the matrix becomes a square root of the identity. L = pascal(n,1) L_squared = L^2 %% % Here is an exercise for you. % What is |sqrt(eye(n))|? Why isn't it |L|? %% Cube Root of Identity % When I first saw this, I was amazed. Rotate L counterclockwise. % The result is a cube root of the identity. X = rot90(L,-1) X_cubed = X^3 %% Sierpinski % Which binomial coefficients are odd? It's a fledgling fractal. odd = @(x) mod(x,2)==1; n = 56; L = abs(pascal(n,1)); spy(odd(L)) title('odd(L)') %% Fibonacci % The sums of the antidiagonals of |L| are the Fibonacci numbers. n = 12; A = fliplr(abs(pascal(n,1))) for k = 1:n F(k) = sum(diag(A,n-k)); end F %% pi % The elements in the third column of lower triangular Pascal matrix % are the _triangle numbers_. % The n-th triangle number is the number of bowling pins in the n-th row % of an array of bowling pins. % % $$t_n = {{n+1} \choose {2}}$$ L = pascal(12,1); t = L(3:end,3)' %% % Here's an unusual series relating the triangle numbers to $\pi$. % The signs go + + - - + + - - . % % pi - 2 = 1 + 1/3 - 1/6 - 1/10 + 1/15 + 1/21 - 1/28 - 1/36 + 1/45 + 1/55 - ... %% type pi_pascal %% % Ten million terms gives $\pi$ to 14 decimal places. format long pie = pi_pascal(10000000) err = pi - pie %% Matrix Exponential % Finally, I love this one. % The solution to the (potentially infinite) set of ordinary differential % equations % % $\dot{x_1} = x_1$ % % $\dot{x_j} = x_j + (j-1) x_{j-1}$ % % is % % $x_j = e^t (t + 1)^{j-1}$ %% % This means that the matrix exponential of the simple diagonal matrix D = diag(1:7,-1) %% % is expm_D = round(expm(D)) %% Thanks % Thanks to Nick Higham for |pascal.m|, |gallery.m| and section 28.4 of % % N. J. Higham, _Accuracy and Stability of Numerical Algorithms_, % Second edition, SIAM, 2002. % % % ##### SOURCE END ##### 1edd50d2d04740ebb72fefb92312caa5 -->

Happy Valentine’s Day

Thu, 02/15/2018 - 01:04
Contents ezplot('(x^2+y^2-1)^3 = x^2*y^3', ... [-1.5,1.5,-1.2,1.5]) colormap([.5 .2 .2]) Thanks

Edward Scheinerman, The Mathematics Lover's Companion, Yale University Press, 2017 https://yalebooks.yale.edu/book/9780300223002/mathematics-lovers-companion

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Published with MATLAB® R2018a

##### SOURCE END ##### eaf18f6e8bbb4a9eb58892ad89366290 -->

The Historic MATLAB Users’ Guide

Mon, 02/05/2018 - 22:30

In the 1970s and early 1980s, while I was working on the LINPACK and EISPACK projects that I discussed in two previous posts, I was a Professor of Mathematics and then of Computer Science at the University of New Mexico in Albuquerque. I was teaching courses in Linear Algebra and Numerical Analysis. I wanted my students to have easy access to LINPACK and EISPACK without writing Fortran programs. By "easy access" I meant not going through the remote batch processing and the repeated edit-compile-link-load-execute process that was ordinarily required on the campus central mainframe computer.

So, I studied Niklaus Wirth's book Algorithms + Data Structures = Programs and learned how to parse programming languages. Wirth calls his example language PL/0. (A zinger aimed at PL/I, the monolithic, committee-designed language being promoted by IBM and the U.S. Defense Department at the time.) PL/0 was a pedagogical subset of Wirth's Pascal, which, in turn, was his response to Algol. Quoting Wirth, "In the realm of data types, however, PL/0 adheres to the demand of simplicity without compromise: integers are its only data type."

Following Wirth's approach, I wrote the first MATLAB -- an acronym for Matrix Laboratory -- in Fortran, as a dialect of PL/0 with matrix as the only data type. The project was a kind of hobby, a new aspect of programming for me to learn and something for my students to use. There was never any formal outside support and certainly no business plan. MathWorks was a few years in the future.

This first MATLAB was not a programming language; it was just a simple interactive matrix calculator. There were no m-files, no toolboxes, no graphics. And no ODEs or FFTs. This snapshot of the start-up screen shows all the reserved words and functions. There are only 71 of them. If you wanted to add another function, you would get the source code from me, write a Fortran subroutine, add your new name to the parse table, and recompile MATLAB.

Here is the first Users' Guide, in its entirety, from 1981.

ContentsUsers' Guide MATLAB Users' Guide May, 1981 Cleve Moler Department of Computer Science University of New Mexico ABSTRACT. MATLAB is an interactive computer program that serves as a convenient "laboratory" for computations involving matrices. It provides easy access to matrix software developed by the LINPACK and EISPACK projects. The program is written in Fortran and is designed to be readily installed under any operating system which permits interactive execution of Fortran programs. MATLAB is an interactive computer program that serves as a convenient "laboratory" for computations involving matrices. It provides easy access to matrix software developed by the LINPACK and EISPACK projects [1-3]. The capabilities range from standard tasks such as solving simultaneous linear equations and inverting matrices, through symmetric and nonsymmetric eigenvalue problems, to fairly sophisticated matrix tools such as the singular value decomposition. It is expected that one of MATLAB's primary uses will be in the classroom. It should be useful in introductory courses in applied linear algebra, as well as more advanced courses in numerical analysis, matrix theory, statistics and applications of matrices to other disciplines. In nonacademic settings, MATLAB can serve as a "desk calculator" for the quick solution of small problems involving matrices. The program is written in Fortran and is designed to be readily installed under any operating system which permits interactive execution of Fortran programs. The resources required are fairly modest. There are less than 7000 lines of Fortran source code, including the LINPACK and EISPACK subroutines used. With proper use of overlays, it is possible run the system on a minicomputer with only 32K bytes of memory. The size of the matrices that can be handled in MATLAB depends upon the amount of storage that is set aside when the system is compiled on a particular machine. We have found that an allocation of 5000 words for matrix elements is usually quite satisfactory. This provides room for several 20 by 20 matrices, for example. One implementation on a virtual memory system provides 100,000 elements. Since most of the algorithms used access memory in a sequential fashion, the large amount of allocated storage causes no difficulties. In some ways, MATLAB resembles SPEAKEASY [4] and, to a lesser extent, APL. All are interactive terminal languages that ordinarily accept single-line commands or statements, process them immediately, and print the results. All have arrays or matrices as principal data types. But for MATLAB, the matrix is the only data type (although scalars, vectors and text are special cases), the underlying system is portable and requires fewer resources, and the supporting subroutines are more powerful and, in some cases, have better numerical properties. Together, LINPACK and EISPACK represent the state of the art in software for matrix computation. EISPACK is a package of over 70 Fortran subroutines for various matrix eigenvalue computations that are based for the most part on Algol procedures published by Wilkinson, Reinsch and their colleagues [5]. LINPACK is a package of 40 Fortran subroutines (in each of four data types) for solving and analyzing simultaneous linear equations and related matrix problems. Since MATLAB is not primarily concerned with either execution time efficiency or storage savings, it ignores most of the special matrix properties that LINPACK and EISPACK subroutines use to advantage. Consequently, only 8 subroutines from LINPACK and 5 from EISPACK are actually involved. In more advanced applications, MATLAB can be used in conjunction with other programs in several ways. It is possible to define new MATLAB functions and add them to the system. With most operating systems, it is possible to use the local file system to pass matrices between MATLAB and other programs. MATLAB command and statement input can be obtained from a local file instead of from the terminal. The most power and flexibility is obtained by using MATLAB as a subroutine which is called by other programs. This document first gives an overview of MATLAB from the user's point of view. Several extended examples involving data fitting, partial differential equations, eigenvalue sensitivity and other topics are included. A formal definition of the MATLAB language and an brief description of the parser and interpreter are given. The system was designed and programmed using techniques described by Wirth [6], implemented in nonrecursive, portable Fortran. There is a brief discussion of some of the matrix algorithms and of their numerical properties. The final section describes how MATLAB can be used with other programs. The appendix includes the HELP documentation available on-line.1. Elementary operations MATLAB works with essentially only one kind of object, a rectangular matrix with complex elements. If the imaginary parts of the elements are all zero, they are not printed, but they still occupy storage. In some situations, special meaning is attached to 1 by 1 matrices, that is scalars, and to 1 by n and m by 1 matrices, that is row and column vectors. Matrices can be introduced into MATLAB in four different ways: -- Explicit list of elements, -- Use of FOR and WHILE statements, -- Read from an external file, -- Execute an external Fortran program. The explicit list is surrounded by angle brackets, '<' and '>', and uses the semicolon ';' to indicate the ends of the rows. For example, the input line A = <1 2 3; 4 5 6; 7 8 9> will result in the output A = 1. 2. 3. 4. 5. 6. 7. 8. 9. The matrix A will be saved for later use. The individual elements are separated by commas or blanks and can be any MATLAB expressions, for example x = < -1.3, 4/5, 4*atan(1) > results in X = -1.3000 0.8000 3.1416 The elementary functions available include sqrt, log, exp, sin, cos, atan, abs, round, real, imag, and conjg. Large matrices can be spread across several input lines, with the carriage returns replacing the semicolons. The above matrix could also have been produced by A = < 1 2 3 4 5 6 7 8 9 > Matrices can be input from the local file system. Say a file named 'xyz' contains five lines of text, A = < 1 2 3 4 5 6 7 8 9 >; then the MATLAB statement EXEC('xyz') reads the matrix and assigns it to A . The FOR statement allows the generation of matrices whose elements are given by simple formulas. Our example matrix A could also have been produced by for i = 1:3, for j = 1:3, a(i,j) = 3*(i-1)+j; The semicolon at the end of the line suppresses the printing, which in this case would have been nine versions of A with changing elements. Several statements may be given on a line, separated by semicolons or commas. Two consecutive periods anywhere on a line indicate continuation. The periods and any following characters are deleted, then another line is input and concatenated onto the previous line. Two consecutive slashes anywhere on a line cause the remainder of the line to be ignored. This is useful for inserting comments. Names of variables are formed by a letter, followed by any number of letters and digits, but only the first 4 characters are remembered. The special character prime (') is used to denote the transpose of a matrix, so x = x' changes the row vector above into the column vector X = -1.3000 0.8000 3.1416 Individual matrix elements may be referenced by enclosing their subscripts in parentheses. When any element is changed, the entire matrix is reprinted. For example, using the above matrix, a(3,3) = a(1,3) + a(3,1) results in A = 1. 2. 3. 4. 5. 6. 7. 8. 10. Addition, subtraction and multiplication of matrices are denoted by +, -, and * . The operations are performed whenever the matrices have the proper dimensions. For example, with the above A and x, the expressions A + x and x*A are incorrect because A is 3 by 3 and x is now 3 by 1. However, b = A*x is correct and results in the output B = 9.7248 17.6496 28.7159 Note that both upper and lower case letters are allowed for input (on those systems which have both), but that lower case is converted to upper case. There are two "matrix division" symbols in MATLAB, \ and / . (If your terminal does not have a backslash, use $ instead, or see CHAR.) If A and B are matrices, then A\B and B/A correspond formally to left and right multiplication of B by the inverse of A , that is inv(A)*B and B*inv(A), but the result is obtained directly without the computation of the inverse. In the scalar case, 3\1 and 1/3 have the same value, namely one-third. In general, A\B denotes the solution X to the equation A*X = B and B/A denotes the solution to X*A = B. Left division, A\B, is defined whenever B has as many rows as A . If A is square, it is factored using Gaussian elimination. The factors are used to solve the equations A*X(:,j) = B(:,j) where B(:,j) denotes the j-th column of B. The result is a matrix X with the same dimensions as B. If A is nearly singular (according to the LINPACK condition estimator, RCOND), a warning message is printed. If A is not square, it is factored using Householder orthogonalization with column pivoting. The factors are used to solve the under- or overdetermined equations in a least squares sense. The result is an m by n matrix X where m is the number of columns of A and n is the number of columns of B . Each column of X has at most k nonzero components, where k is the effective rank of A . Right division, B/A, can be defined in terms of left division by B/A = (A'\B')'. For example, since our vector b was computed as A*x, the statement y = A\b results in Y = -1.3000 0.8000 3.1416 Of course, y is not exactly equal to x because of the roundoff errors involved in both A*x and A\b , but we are not printing enough digits to see the difference. The result of the statement e = x - y depends upon the particular computer being used. In one case it produces E = 1.0e-15 * .3053 -.2498 .0000 The quantity 1.0e-15 is a scale factor which multiplies all the components which follow. Thus our vectors x and y actually agree to about 15 decimal places on this computer. It is also possible to obtain element-by-element multiplicative operations. If A and B have the same dimensions, then A .* B denotes the matrix whose elements are simply the products of the individual elements of A and B . The expressions A ./ B and A .\ B give the quotients of the individual elements. There are several possible output formats. The statement long, x results in X = -1.300000000000000 .800000000000000 3.141592653589793 The statement short restores the original format. The expression A**p means A to the p-th power. It is defined if A is a square matrix and p is a scalar. If p is an integer greater than one, the power is computed by repeated multiplication. For other values of p the calculation involves the eigenvalues and eigenvectors of A. Previously defined matrices and matrix expressions can be used inside brackets to generate larger matrices, for example C = <A, b; <4 2 0>*x, x'> results in C = 1.0000 2.0000 3.0000 9.7248 4.0000 5.0000 6.0000 17.6496 7.0000 8.0000 10.0000 28.7159 -3.6000 -1.3000 0.8000 3.1416 There are four predefined variables, EPS, FLOP, RAND and EYE. The variable EPS is used as a tolerance is determining such things as near singularity and rank. Its initial value is the distance from 1.0 to the next largest floating point number on the particular computer being used. The user may reset this to any other value, including zero. EPS is changed by CHOP, which is described in section 12. The value of RAND is a random variable, with a choice of a uniform or a normal distribution. The name EYE is used in place of I to denote identity matrices because I is often used as a subscript or as sqrt(-1). The dimensions of EYE are determined by context. For example, B = A + 3*EYE adds 3 to the diagonal elements of A and X = EYE/A is one of several ways in MATLAB to invert a matrix. FLOP provides a count of the number of floating point operations, or "flops", required for each calculation. A statement may consist of an expression alone, in which case a variable named ANS is created and the result stored in ANS for possible future use. Thus A\A - EYE is the same as ANS = A\A - EYE (Roundoff error usually causes this result to be a matrix of "small" numbers, rather than all zeros.) All computations are done using either single or double precision real arithmetic, whichever is appropriate for the particular computer. There is no mixed-precision arithmetic. The Fortran COMPLEX data type is not used because many systems create unnecessary underflows and overflows with complex operations and because some systems do not allow double precision complex arithmetic.2. MATLAB functions Much of MATLAB's computational power comes from the various matrix functions available. The current list includes: INV(A) - Inverse. DET(A) - Determinant. COND(A) - Condition number. RCOND(A) - A measure of nearness to singularity. EIG(A) - Eigenvalues and eigenvectors. SCHUR(A) - Schur triangular form. HESS(A) - Hessenberg or tridiagonal form. POLY(A) - Characteristic polynomial. SVD(A) - Singular value decomposition. PINV(A,eps) - Pseudoinverse with optional tolerance. RANK(A,eps) - Matrix rank with optional tolerance. LU(A) - Factors from Gaussian elimination. CHOL(A) - Factor from Cholesky factorization. QR(A) - Factors from Householder orthogonalization. RREF(A) - Reduced row echelon form. ORTH(A) - Orthogonal vectors spanning range of A. EXP(A) - e to the A. LOG(A) - Natural logarithm. SQRT(A) - Square root. SIN(A) - Trigonometric sine. COS(A) - Cosine. ATAN(A) - Arctangent. ROUND(A) - Round the elements to nearest integers. ABS(A) - Absolute value of the elements. REAL(A) - Real parts of the elements. IMAG(A) - Imaginary parts of the elements. CONJG(A) - Complex conjugate. SUM(A) - Sum of the elements. PROD(A) - Product of the elements. DIAG(A) - Extract or create diagonal matrices. TRIL(A) - Lower triangular part of A. TRIU(A) - Upper triangular part of A. NORM(A,p) - Norm with p = 1, 2 or 'Infinity'. EYE(m,n) - Portion of identity matrix. RAND(m,n) - Matrix with random elements. ONES(m,n) - Matrix of all ones. MAGIC(n) - Interesting test matrices. HILBERT(n) - Inverse Hilbert matrices. ROOTS(C) - Roots of polynomial with coefficients C. DISPLAY(A,p) - Print base p representation of A. KRON(A,B) - Kronecker tensor product of A and B. PLOT(X,Y) - Plot Y as a function of X . RAT(A) - Find "simple" rational approximation to A. USER(A) - Function defined by external program. Some of these functions have different interpretations when the argument is a matrix or a vector and some of them have additional optional arguments. Details are given in the HELP document in the appendix. Several of these functions can be used in a generalized assignment statement with two or three variables on the left hand side. For example <X,D> = EIG(A) stores the eigenvectors of A in the matrix X and a diagonal matrix containing the eigenvalues in the matrix D. The statement EIG(A) simply computes the eigenvalues and stores them in ANS. Future versions of MATLAB will probably include additional functions, since they can easily be added to the system.3. Rows, columns and submatrices Individual elements of a matrix can be accessed by giving their subscripts in parentheses, eg. A(1,2), x(i), TAB(ind(k)+1). An expression used as a subscript is rounded to the nearest integer. Individual rows and columns can be accessed using a colon ':' (or a '|') for the free subscript. For example, A(1,:) is the first row of A and A(:,j) is the j-th column. Thus A(i,:) = A(i,:) + c*A(k,:) adds c times the k-th row of A to the i-th row. The colon is used in several other ways in MATLAB, but all of the uses are based on the following definition. j:k is the same as <j, j+1, ..., k> j:k is empty if j > k . j:i:k is the same as <j, j+i, j+2i, ..., k> j:i:k is empty if i > 0 and j > k or if i < 0 and j < k . The colon is usually used with integers, but it is possible to use arbitrary real scalars as well. Thus 1:4 is the same as <1, 2, 3, 4> 0: 0.1: 0.5 is the same as <0.0, 0.1, 0.2, 0.3, 0.4, 0.5> In general, a subscript can be a vector. If X and V are vectors, then X(V) is <X(V(1)), X(V(2)), ..., X(V(n))> . This can also be used with matrices. If V has m components and W has n components, then A(V,W) is the m by n matrix formed from the elements of A whose subscripts are the elements of V and W. Combinations of the colon notation and the indirect subscripting allow manipulation of various submatrices. For example, A(<1,5>,:) = A(<5,1>,:) interchanges rows 1 and 5 of A. A(2:k,1:n) is the submatrix formed from rows 2 through k and columns 1 through n of A . A(:,<3 1 2>) is a permutation of the first three columns. The notation A(:) has a special meaning. On the right hand side of an assignment statement, it denotes all the elements of A, regarded as a single column. When an expression is assigned to A(:), the current dimensions of A, rather than of the expression, are used.4. FOR, WHILE and IF The FOR clause allows statements to be repeated a specific number of times. The general form is FOR variable = expr, statement, ..., statement, END The END and the comma before it may be omitted. In general, the expression may be a matrix, in which case the columns are stored one at a time in the variable and the following statements, up to the END or the end of the line, are executed. The expression is often of the form j:k, and its "columns" are simply the scalars from j to k. Some examples (assume n has already been assigned a value): for i = 1:n, for j = 1:n, A(i,j) = 1/(i+j-1); generates the Hilbert matrix. for j = 2:n-1, for i = j:n-1, ... A(i,j) = 0; end; A(j,j) = j; end; A changes all but the "outer edge" of the lower triangle and then prints the final matrix. for h = 1.0: -0.1: -1.0, (<h, cos(pi*h)>) prints a table of cosines. <X,D> = EIG(A); for v = X, v, A*v displays eigenvectors, one at a time. The WHILE clause allows statements to be repeated an indefinite number of times. The general form is WHILE expr relop expr, statement,..., statement, END where relop is =, <, >, <=, >=, or <> (not equal) . The statements are repeatedly executed as long as the indicated comparison between the real parts of the first components of the two expressions is true. Here are two examples. (Exercise for the reader: What do these segments do?) eps = 1; while 1 + eps > 1, eps = eps/2; eps = 2*eps E = 0*A; F = E + EYE; n = 1; while NORM(E+F-E,1) > 0, E = E + F; F = A*F/n; n = n + 1; E The IF clause allows conditional execution of statements. The general form is IF expr relop expr, statement, ..., statement, ELSE statement, ..., statement The first group of statements are executed if the relation is true and the second group are executed if the relation is false. The ELSE and the statements following it may be omitted. For example, if abs(i-j) = 2, A(i,j) = 0;5. Commands, text, files and macros. MATLAB has several commands which control the output format and the overall execution of the system. The HELP command allows on-line access to short portions of text describing various operations, functions and special characters. The entire HELP document is reproduced in an appendix. Results are usually printed in a scaled fixed point format that shows 4 or 5 significant figures. The commands SHORT, LONG, SHORT E, LONG E and LONG Z alter the output format, but do not alter the precision of the computations or the internal storage. The WHO, WHAT and WHY commands provide information about the functions and variables that are currently defined. The CLEAR command erases all variables, except EPS, FLOP, RAND and EYE. The statement A = <> indicates that a "0 by 0" matrix is to be stored in A. This causes A to be erased so that its storage can be used for other variables. The RETURN and EXIT commands cause return to the underlying operating system through the Fortran RETURN statement. MATLAB has a limited facility for handling text. Any string of characters delineated by quotes (with two quotes used to allow one quote within the string) is saved as a vector of integer values with '1' = 1, 'A' = 10, ' ' = 36, etc. (The complete list is in the appendix under CHAR.) For example '2*A + 3' is the same as <2 43 10 36 41 36 3> It is possible, though seldom very meaningful, to use such strings in matrix operations. More frequently, the text is used as a special argument to various functions. NORM(A,'inf') computes the infinity norm of A . DISPLAY(T) prints the text stored in T . EXEC('file') obtains MATLAB input from an external file. SAVE('file') stores all the current variables in a file. LOAD('file') retrieves all the variables from a file. PRINT('file',X) prints X on a file. DIARY('file') makes a copy of the complete MATLAB session. The text can also be used in a limited string substitution macro facility. If a variable, say T, contains the source text for a MATLAB statement or expression, then the construction > T < causes T to be executed or evaluated. For example T = '2*A + 3'; S = 'B = >T< + 5' A = 4; > S < produces B = 16. Some other examples are given under MACRO in the appendix. This facility is useful for fairly short statements and expressions. More complicated MATLAB "programs" should use the EXEC facility. The operations which access external files cannot be handled in a completely machine-independent manner by portable Fortran code. It is necessary for each particular installation to provide a subroutine which associates external text files with Fortran logical unit numbers.6. Census example Our first extended example involves predicting the population of the United States in 1980 using extrapolation of various fits to the census data from 1900 through 1970. There are eight observations, so we begin with the MATLAB statement n = 8 The values of the dependent variable, the population in millions, can be entered with y = < 75.995 91.972 105.711 123.203 ... 131.669 150.697 179.323 203.212>' In order to produce a reasonably scaled matrix, the independent variable, time, is transformed from the interval [1900,1970] to [-1.00,0.75]. This can be accomplished directly with t = -1.0:0.25:0.75 or in a fancier, but perhaps clearer, way with t = 1900:10:1970; t = (t - 1940*ones(t))/40 Either of these is equivalent to t = <-1 -.75 -.50 -.25 0 .25 .50 .75> The interpolating polynomial of degree n-1 involves an Vandermonde matrix of order n with elements that might be generated by for i = 1:n, for j = 1:n, a(i,j) = t(i)**(j-1); However, this results in an error caused by 0**0 when i = 5 and j = 1 . The preferable approach is A = ones(n,n); for i = 1:n, for j = 2:n, a(i,j) = t(i)*a(i,j-1); Now the statement cond(A) produces the output ANS = 1.1819E+03 which indicates that transformation of the time variable has resulted in a reasonably well conditioned matrix. The statement c = A\y results in C = 131.6690 41.0406 103.5396 262.4535 -326.0658 -662.0814 341.9022 533.6373 These are the coefficients in the interpolating polynomial n-1 c + c t + ... + c t 1 2 n Our transformation of the time variable has resulted in t = 1 corresponding to the year 1980. Consequently, the extrapolated population is simply the sum of the coefficients. This can be computed by p = sum(c) The result is P = 426.0950 which indicates a 1980 population of over 426 million. Clearly, using the seventh degree interpolating polynomial to extrapolate even a fairly short distance beyond the end of the data interval is not a good idea. The coefficients in least squares fits by polynomials of lower degree can be computed using fewer than n columns of the matrix. for k = 1:n, c = A(:,1:k)\y, p = sum(c) would produce the coefficients of these fits, as well as the resulting extrapolated population. If we do not want to print all the coefficients, we can simply generate a small table of populations predicted by polynomials of degrees zero through seven. We also compute the maximum deviation between the fitted and observed values. for k = 1:n, X = A(:,1:k); c = X\y; ... d(k) = k-1; p(k) = sum(c); e(k) = norm(X*c-y,'inf'); <d, p, e> The resulting output is 0 132.7227 70.4892 1 211.5101 9.8079 2 227.7744 5.0354 3 241.9574 3.8941 4 234.2814 4.0643 5 189.7310 2.5066 6 118.3025 1.6741 7 426.0950 0.0000 The zeroth degree fit, 132.7 million, is the result of fitting a constant to the data and is simply the average. The results obtained with polynomials of degree one through four all appear reasonable. The maximum deviation of the degree four fit is slightly greater than the degree three, even though the sum of the squares of the deviations is less. The coefficients of the highest powers in the fits of degree five and six turn out to be negative and the predicted populations of less than 200 million are probably unrealistic. The hopefully absurd prediction of the interpolating polynomial concludes the table. We wish to emphasize that roundoff errors are not significant here. Nearly identical results would be obtained on other computers, or with other algorithms. The results simply indicate the difficulties associated with extrapolation of polynomial fits of even modest degree. A stabilized fit by a seventh degree polynomial can be obtained using the pseudoinverse, but it requires a fairly delicate choice of a tolerance. The statement s = svd(A) produces the singular values S = 3.4594 2.2121 1.0915 0.4879 0.1759 0.0617 0.0134 0.0029 We see that the last three singular values are less than 0.1 , consequently, A can be approximately by a matrix of rank five with an error less than 0.1 . The Moore-Penrose pseudoinverse of this rank five matrix is obtained from the singular value decomposition with the following statements c = pinv(A,0.1)*y, p = sum(c), e = norm(a*c-y,'inf') The output is C = 134.7972 67.5055 23.5523 9.2834 3.0174 2.6503 -2.8808 3.2467 P = 241.1720 E = 3.9469 The resulting seventh degree polynomial has coefficients which are much smaller than those of the interpolating polynomial given earlier. The predicted population and the maximum deviation are reasonable. Any choice of the tolerance between the fifth and sixth singular values would produce the same results, but choices outside this range result in pseudoinverses of different rank and do not work as well. The one term exponential approximation y(t) = k exp(pt) can be transformed into a linear approximation by taking logarithms. log(y(t)) = log k + pt = c + c t 1 2 The following segment makes use of the fact that a function of a vector is the function applied to the individual components. X = A(:,1:2); c = X\log(y) p = exp(sum(c)) e = norm(exp(X*c)-y,'inf') The resulting output is C = 4.9083 0.5407 P = 232.5134 E = 4.9141 The predicted population and maximum deviation appear satisfactory and indicate that the exponential model is a reasonable one to consider. As a curiousity, we return to the degree six polynomial. Since the coefficient of the high order term is negative and the value of the polynomial at t = 1 is positive, it must have a root at some value of t greater than one. The statements X = A(:,1:7); c = X\y; c = c(7:-1:1); //reverse the order of the coefficients z = roots(c) produce Z = 1.1023- 0.0000*i 0.3021+ 0.7293*i -0.8790+ 0.6536*i -1.2939- 0.0000*i -0.8790- 0.6536*i 0.3021- 0.7293*i There is only one real, positive root. The corresponding time on the original scale is 1940 + 40*real(z(1)) = 1984.091 We conclude that the United States population should become zero early in February of 1984.7. Partial differential equation example Our second extended example is a boundary value problem for Laplace's equation. The underlying physical problem involves the conductivity of a medium with cylindrical inclusions and is considered by Keller and Sachs [7]. Find a function u(x,y) satisfying Laplace's equation u + u = 0 xx yy The domain is a unit square with a quarter circle of radius rho removed from one corner. There are Neumann conditions on the top and bottom edges and Dirichlet conditions on the remainder of the boundary. u = 0 n ------------- | . | . | . | . u = 1 | . | . | . u = 0 | | | | | | | | u = 1 | | | | | | ------------------------ u = 0 n The effective conductivity of an medium is then given by the integral along the left edge, 1 sigma = integral u (0,y) dy 0 n It is of interest to study the relation between the radius rho and the conductivity sigma. In particular, as rho approaches one, sigma becomes infinite. Keller and Sachs use a finite difference approximation. The following technique makes use of the fact that the equation is actually Laplace's equation and leads to a much smaller matrix problem to solve. Consider an approximate solution of the form n 2j-1 u = sum c r cos(2j-1)t j=1 j where r,t are polar coordinates (t is theta). The coefficients are to be determined. For any set of coefficients, this function already satisfies the differential equation because the basis functions are harmonic; it satisfies the normal derivative boundary condition on the bottom edge of the domain because we used cos t in preference to sin t ; and it satisfies the boundary condition on the left edge of the domain because we use only odd multiples of t . The computational task is to find coefficients so that the boundary conditions on the remaining edges are satisfied as well as possible. To accomplish this, pick m points (r,t) on the remaining edges. It is desirable to have m > n and in practice we usually choose m to be two or three times as large as n . Typical values of n are 10 or 20 and of m are 20 to 60. An m by n matrix A is generated. The i,j element is the j-th basis function, or its normal derivative, evaluated at the i-th boundary point. A right hand side with m components is also generated. In this example, the elements of the right hand side are either zero or one. The coefficients are then found by solving the overdetermined set of equations Ac = b in a least squares sense. Once the coefficients have been determined, the approximate solution is defined everywhere on the domain. It is then possible to compute the effective conductivity sigma . In fact, a very simple formula results, n j-1 sigma = sum (-1) c j=1 j To use MATLAB for this problem, the following "program" is first stored in the local computer file system, say under the name "PDE". //Conductivity example. //Parameters --- rho //radius of cylindrical inclusion n //number of terms in solution m //number of boundary points //initialize operation counter flop = <0 0>; //initialize variables m1 = round(m/3); //number of points on each straight edge m2 = m - m1; //number of points with Dirichlet conditions pi = 4*atan(1); //generate points in Cartesian coordinates //right hand edge for i = 1:m1, x(i) = 1; y(i) = (1-rho)*(i-1)/(m1-1); //top edge for i = m2+1:m, x(i) = (1-rho)*(m-i)/(m-m2-1); y(i) = 1; //circular edge for i = m1+1:m2, t = pi/2*(i-m1)/(m2-m1+1); ... x(i) = 1-rho*sin(t); y(i) = 1-rho*cos(t); //convert to polar coordinates for i = 1:m-1, th(i) = atan(y(i)/x(i)); ... r(i) = sqrt(x(i)**2+y(i)**2); th(m) = pi/2; r(m) = 1; //generate matrix //Dirichlet conditions for i = 1:m2, for j = 1:n, k = 2*j-1; ... a(i,j) = r(i)**k*cos(k*th(i)); //Neumann conditions for i = m2+1:m, for j = 1:n, k = 2*j-1; ... a(i,j) = k*r(i)**(k-1)*sin((k-1)*th(i)); //generate right hand side for i = 1:m2, b(i) = 1; for i = m2+1:m, b(i) = 0; //solve for coefficients c = A\b //compute effective conductivity c(2:2:n) = -c(2:2:n); sigma = sum(c) //output total operation count ops = flop(2) The program can be used within MATLAB by setting the three parameters and then accessing the file. For example, rho = .9; n = 15; m = 30; exec('PDE') The resulting output is RHO = .9000 N = 15. M = 30. C = 2.2275 -2.2724 1.1448 0.1455 -0.1678 -0.0005 -0.3785 0.2299 0.3228 -0.2242 -0.1311 0.0924 0.0310 -0.0154 -0.0038 SIGM = 5.0895 OPS = 16204. A total of 16204 floating point operations were necessary to set up the matrix, solve for the coefficients and compute the conductivity. The operation count is roughly proportional to m*n**2. The results obtained for sigma as a function of rho by this approach are essentially the same as those obtained by the finite difference technique of Keller and Sachs, but the computational effort involved is much less.8. Eigenvalue sensitivity example In this example, we construct a matrix whose eigenvalues are moderately sensitive to perturbations and then analyze that sensitivity. We begin with the statement B = <3 0 7; 0 2 0; 0 0 1> which produces B = 3. 0. 7. 0. 2. 0. 0. 0. 1. Obviously, the eigenvalues of B are 1, 2 and 3 . Moreover, since B is not symmetric, these eigenvalues are slightly sensitive to perturbation. (The value b(1,3) = 7 was chosen so that the elements of the matrix A below are less than 1000.) We now generate a similarity transformation to disguise the eigenvalues and make them more sensitive. L = <1 0 0; 2 1 0; -3 4 1>, M = L\L' L = 1. 0. 0. 2. 1. 0. -3. 4. 1. M = 1.0000 2.0000 -3.0000 -2.0000 -3.0000 10.0000 11.0000 18.0000 -48.0000 The matrix M has determinant equal to 1 and is moderately badly conditioned. The similarity transformation is A = M*B/M A = -64.0000 82.0000 21.0000 144.0000 -178.0000 -46.0000 -771.0000 962.0000 248.0000 Because det(M) = 1 , the elements of A would be exact integers if there were no roundoff. So, A = round(A) A = -64. 82. 21. 144. -178. -46. -771. 962. 248. This, then, is our test matrix. We can now forget how it was generated and analyze its eigenvalues. <X,D> = eig(A) D = 3.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 2.0000 X = -.0891 3.4903 41.8091 .1782 -9.1284 -62.7136 -.9800 46.4473 376.2818 Since A is similar to B, its eigenvalues are also 1, 2 and 3. They happen to be computed in another order by the EISPACK subroutines. The fact that the columns of X, which are the eigenvectors, are so far from being orthonormal is our first indication that the eigenvalues are sensitive. To see this sensitivity, we display more figures of the computed eigenvalues. long, diag(D) ANS = 2.999999999973599 1.000000000015625 2.000000000011505 We see that, on this computer, the last five significant figures are contaminated by roundoff error. A somewhat superficial explanation of this is provided by short, cond(X) ANS = 3.2216e+05 The condition number of X gives an upper bound for the relative error in the computed eigenvalues. However, this condition number is affected by scaling. X = X/diag(X(3,:)), cond(X) X = .0909 .0751 .1111 -.1818 -.1965 -.1667 1.0000 1.0000 1.0000 ANS = 1.7692e+03 Rescaling the eigenvectors so that their last components are all equal to one has two consequences. The condition of X is decreased by over two orders of magnitude. (This is about the minimum condition that can be obtained by such diagonal scaling.) Moreover, it is now apparent that the three eigenvectors are nearly parallel. More detailed information on the sensitivity of the individual eigenvalues involves the left eigenvectors. Y = inv(X'), Y'*A*X Y = -511.5000 259.5000 252.0000 616.0000 -346.0000 -270.0000 159.5000 -86.5000 -72.0000 ANS = 3.0000 .0000 .0000 .0000 1.0000 .0000 .0000 .0000 2.0000 We are now in a position to compute the sensitivities of the individual eigenvalues. for j = 1:3, c(j) = norm(Y(:,j))*norm(X(:,j)); end, C C = 833.1092 450.7228 383.7564 These three numbers are the reciprocals of the cosines of the angles between the left and right eigenvectors. It can be shown that perturbation of the elements of A can result in a perturbation of the j-th eigenvalue which is c(j) times as large. In this example, the first eigenvalue has the largest sensitivity. We now proceed to show that A is close to a matrix with a double eigenvalue. The direction of the required perturbation is given by E = -1.e-6*Y(:,1)*X(:,1)' E = 1.0e-03 * .0465 -.0930 .5115 -.0560 .1120 -.6160 -.0145 .0290 -.1595 With some trial and error which we do not show, we bracket the point where two eigenvalues of a perturbed A coalesce and then become complex. eig(A + .4*E), eig(A + .5*E) ANS = 1.1500 2.5996 2.2504 ANS = 2.4067 + .1753*i 2.4067 - .1753*i 1.1866 + 0.0000*i Now, a bisecting search, driven by the imaginary part of one of the eigenvalues, finds the point where two eigenvalues are nearly equal. r = .4; s = .5; while s-r > 1.e-14, t = (r+s)/2; d = eig(A+t*E); ... if imag(d(1))=0, r = t; else, s = t; long, T T = .450380734134507 Finally, we display the perturbed matrix, which is obviously close to the original, and its pair of nearly equal eigenvalues. (We have dropped a few digits from the long output.) A+t*E, eig(A+t*E) A -63.999979057 81.999958114 21.000230369 143.999974778 -177.999949557 -46.000277434 -771.000006530 962.000013061 247.999928164 ANS = 2.415741150 2.415740621 1.168517777 The first two eigenvectors of A + t*E are almost indistinguishable indicating that the perturbed matrix is almost defective. <X,D> = eig(A+t*E); X = X/diag(X(3,:)) X = .096019578 .096019586 .071608466 -.178329614 -.178329608 -.199190520 1.000000000 1.000000000 1.000000000 short, cond(X) ANS = 3.3997e+099. Syntax diagrams A formal description of the language acceptable to MATLAB, as well as a flow chart of the MATLAB program, is provided by the syntax diagrams or syntax graphs of Wirth [6]. There are eleven non-terminal symbols in the language: line, statement, clause, expression, term, factor, number, integer, name, command, text . The diagrams define each of the non-terminal symbols using the others and the terminal symbols: letter -- A through Z, digit -- 0 through 9, char -- ( ) ; : + - * / \ = . , < > quote -- ' line |-----> statement >----| | | |-----> clause >-------| | | -------|-----> expr >---------|------> | | | | | |-----> command >------| | | | | | | |-> > >-> expr >-> < >-| | | | | | | |----------------------| | | | | |-< ; <-| | |--------| |---------| |-< , <-| statement |-> name >--------------------------------| | | | | | |--> : >---| | | | | | | | |-> ( >---|-> expr >-|---> ) >-| | | | | -----| |-----< , <----| |--> = >--> expr >---> | | | |--< , <---| | | | | | |-> < >---> name >---> > >----------------| clause |---> FOR >---> name >---> = >---> expr >--------------| | | | |-> WHILE >-| | |-| |-> expr >---------------------- | | |-> IF >-| | | | | | | | -----| < <= = <> >= > |----> | | | | | | | | | ----------------------> expr >--| | | |---> ELSE >--------------------------------------------| | | |---> END >--------------------------------------------| expr |-> + >-| | | -------|-------|-------> term >----------> | | | | |-> - >-| | |-< + <-| | | | | | |--|-< - <-|--| | | |-< : <-| term ---------------------> factor >----------------------> | | | |-< * <-| | | |-------| | | |-------| | |--| |--|-< / <-|--| |--| |-< . <-| | | |-< . <-| |-< \ <-| factor |----------------> number >---------------| | | |-> name >--------------------------------| | | | | | |--> : >---| | | | | | | | |-> ( >---|-> expr >-|---> ) >-| | | | | | |-----< , <----| | | | -----|------------> ( >-----> expr >-----> ) >-|-|-------|-----> | | | | | | |--------------| | |-> ' >-| | | | | | | |------------> < >-|---> expr >---|-> > >-| | | | | | | | |--< <---| | | | | | | | | |--< ; <---| | | | | | | | | |--< , <---| | | | | | |------------> > >-----> expr >-----> < >-| | | | | |-----> factor >---> ** >---> factor >----| | | | |------------> ' >-----> text >-----> ' >-------------| number |----------| |-> + >-| | | | | -----> int >-----> . >---> int >-----> E >---------> int >----> | | | | | | | | | |-> - >-| | | | | | |---------------------------------------------| int ------------> digit >-----------> | | |-----------| name |--< letter <--| | | ------> letter >--|--------------|-----> | | |--< digit <--| command |--> name >--| | | --------> name >--------|------------|----> | | |--> char >--| | | |---> ' >----| text |-> letter >--| | | |-> digit >---| ----------------| |--------------> | |-> char >----| | | | | | | |-> ' >-> ' >-| | | | |---------------------|10. The parser-interpreter The structure of the parser-interpreter is similar to that of Wirth's compiler [6] for his simple language, PL/0 , except that MATLAB is programmed in Fortran, which does not have explicit recursion. The interrelation of the primary subroutines is shown in the following diagram. MAIN | MATLAB |--CLAUSE | | | PARSE-----|--EXPR----TERM----FACTOR | | | | | |-------|-------| | | | | | STACK1 STACK2 STACKG | |--STACKP--PRINT | |--COMAND | | | |--CGECO | | | |--CGEFA | | |--MATFN1--|--CGESL | | | |--CGEDI | | | |--CPOFA | | | |--IMTQL2 | | | |--HTRIDI | | |--MATFN2--|--HTRIBK | | | |--CORTH | | | |--COMQR3 | | |--MATFN3-----CSVDC | | | |--CQRDC |--MATFN4--| | |--CQRSL | | | |--FILES |--MATFN5--| |--SAVLOD Subroutine PARSE controls the interpretation of each statement. It calls subroutines that process the various syntactic quantities such as command, expression, term and factor. A fairly simple program stack mechanism allows these subroutines to recursively "call" each other along the lines allowed by the syntax diagrams. The four STACK subroutines manage the variable memory and perform elementary operations, such as matrix addition and transposition. The four subroutines MATFN1 though MATFN4 are called whenever "serious" matrix computations are required. They are interface routines which call the various LINPACK and EISPACK subroutines. MATFN5 primarily handles the file access tasks. Two large real arrays, STKR and STKI, are used to store all the matrices. Four integer arrays are used to store the names, the row and column dimensions, and the pointers into the real stacks. The following diagram illustrates this storage scheme. TOP IDSTK MSTK NSTK LSTK STKR STKI -- -- -- -- -- -- -- -- -------- -------- | |--->| | | | | | | | | | |----------->| | | | -- -- -- -- -- -- -- -- -------- -------- | | | | | | | | | | | | | | | -- -- -- -- -- -- -- -------- -------- . . . . . . . . . . . . . . . . . . -- -- -- -- -- -- -- -------- -------- BOT | | | | | | | | | | | | | | | -- -- -- -- -- -- -- -- -------- -------- | |--->| X| | | | | 2| | 1| | |----------->| 3.14 | | 0.00 | -- -- -- -- -- -- -- -- -------- -------- | A| | | | | 2| | 2| | |--------- | 0.00 | | 1.00 | -- -- -- -- -- -- -- \ -------- -------- | E| P| S| | | 1| | 1| | |------- ->| 11.00 | | 0.00 | -- -- -- -- -- -- -- \ -------- -------- | F| L| O| P| | 1| | 2| | |------ \ | 21.00 | | 0.00 | -- -- -- -- -- -- -- \ \ -------- -------- | E| Y| E| | |-1| |-1| | |--- \ | | 12.00 | | 0.00 | -- -- -- -- -- -- -- \ | | -------- -------- | R| A| N| D| | 1| | 1| | |- \ | | | 22.00 | | 0.00 | -- -- -- -- -- -- -- \ | \ \ -------- -------- | \ \ ->| 1.E-15 | | 0.00 | \ \ \ -------- -------- \ \ ->| 0.00 | | 0.00 | \ \ -------- -------- \ \ | 0.00 | | 0.00 | \ \ -------- -------- \ ->| 1.00 | | 0.00 | \ -------- -------- --->| URAND | | 0.00 | -------- -------- The top portion of the stack is used for temporary variables and the bottom portion for saved variables. The figure shows the situation after the line A = <11,12; 21,22>, x = <3.14, sqrt(-1)>' has been processed. The four permanent names, EPS, FLOP, RAND and EYE, occupy the last four positions of the variable stacks. RAND has dimensions 1 by 1, but whenever its value is requested, a random number generator is used instead. EYE has dimensions -1 by -1 to indicate that the actual dimensions must be determined later by context. The two saved variables have dimensions 2 by 2 and 2 by 1 and so take up a total of 6 locations. Subsequent statements involving A and x will result in temporary copies being made in the top of the stack for use in the actual calculations. Whenever the top of the stack reaches the bottom, a message indicating memory has been exceeded is printed, but the current variables are not affected. This modular structure makes it possible to implement MATLAB on a system with a limited amount of memory. The object code for the MATFN's and the LINPACK-EISPACK subroutines is rarely needed. Although it is not standard, many Fortran operating systems provide some overlay mechanism so that this code is brought into the main memory only when required. The variables, which occupy a relatively small portion of the memory, remain in place, while the subroutines which process them are loaded a few at a time.11. The numerical algorithms The algorithms underlying the basic MATLAB functions are described in the LINPACK and EISPACK guides [1-3]. The following list gives the subroutines used by these functions. INV(A) - CGECO,CGEDI DET(A) - CGECO,CGEDI LU(A) - CGEFA RCOND(A) - CGECO CHOL(A) - CPOFA SVD(A) - CSVDC COND(A) - CSVDC NORM(A,2) - CSVDC PINV(A,eps) - CSVDC RANK(A,eps) - CSVDC QR(A) - CQRDC,CQRSL ORTH(A) - CQRDC,CSQSL A\B and B/A - CGECO,CGESL if A is square. - CQRDC,CQRSL if A is not square. EIG(A) - HTRIDI,IMTQL2,HTRIBK if A is Hermitian. - CORTH,COMQR2 if A is not Hermitian. SCHUR(A) - same as EIG. HESS(A) - same as EIG. Minor modifications were made to all these subroutines. The LINPACK routines were changed to replace the Fortran complex arithmetic with explicit references to real and imaginary parts. Since most of the floating point arithmetic is concentrated in a few low-level subroutines which perform vector operations (the Basic Linear Algebra Subprograms), this was not an extensive change. It also facilitated implementation of the FLOP and CHOP features which count and optionally truncate each floating point operation. The EISPACK subroutine COMQR2 was modified to allow access to the Schur triangular form, ordinarily just an intermediate result. IMTQL2 was modified to make computation of the eigenvectors optional. Both subroutines were modified to eliminate the machine-dependent accuracy parameter and all the EISPACK subroutines were changed to include FLOP and CHOP. The algorithms employed for the POLY and ROOTS functions illustrate an interesting aspect of the modern approach to eigenvalue computation. POLY(A) generates the characteristic polynomial of A and ROOTS(POLY(A)) finds the roots of that polynomial, which are, of course, the eigenvalues of A . But both POLY and ROOTS use EISPACK eigenvalues subroutines, which are based on similarity transformations. So the classical approach which characterizes eigenvalues as roots of the characteristic polynomial is actually reversed. If A is an n by n matrix, POLY(A) produces the coefficients C(1) through C(n+1), with C(1) = 1, in DET(z*EYE-A) = C(1)*z**n + ... + C(n)*z + C(n+1) . The algorithm can be expressed compactly using MATLAB: Z = EIG(A); C = 0*ONES(n+1,1); C(1) = 1; for j = 1:n, C(2:j+1) = C(2:j+1) - Z(j)*C(1:j); C This recursion is easily derived by expanding the product (z - z(1))*(z - z(2))* ... * (z-z(n)) . It is possible to prove that POLY(A) produces the coefficients in the characteristic polynomial of a matrix within roundoff error of A . This is true even if the eigenvalues of A are badly conditioned. The traditional algorithms for obtaining the characteristic polynomial which do not use the eigenvalues do not have such satisfactory numerical properties. If C is a vector with n+1 components, ROOTS(C) finds the roots of the polynomial of degree n , p(z) = C(1)*z**n + ... + C(n)*z + C(n+1) . The algorithm simply involves computing the eigenvalues of the companion matrix: A = 0*ONES(n,n) for j = 1:n, A(1,j) = -C(j+1)/C(1); for i = 2:n, A(i,i-1) = 1; EIG(A) It is possible to prove that the results produced are the exact eigenvalues of a matrix within roundoff error of the companion matrix A, but this does not mean that they are the exact roots of a polynomial with coefficients within roundoff error of those in C . There are more accurate, more efficient methods for finding polynomial roots, but this approach has the crucial advantage that it does not require very much additional code. The elementary functions EXP, LOG, SQRT, SIN, COS and ATAN are applied to square matrices by diagonalizing the matrix, applying the functions to the individual eigenvalues and then transforming back. For example, EXP(A) is computed by <X,D> = EIG(A); for j = 1:n, D(j,j) = EXP(D(j,j)); X*D/X This is essentially method number 14 out of the 19 'dubious' possibilities described in [8]. It is dubious because it doesn't always work. The matrix of eigenvectors X can be arbitrarily badly conditioned and all accuracy lost in the computation of X*D/X. A warning message is printed if RCOND(X) is very small, but this only catches the extreme cases. An example of a case not detected is when A has a double eigenvalue, but theoretically only one linearly independent eigenvector associated with it. The computed eigenvalues will be separated by something on the order of the square root of the roundoff level. This separation will be reflected in RCOND(X) which will probably not be small enough to trigger the error message. The computed EXP(A) will be accurate to only half precision. Better methods are known for computing EXP(A), but they do not easily extend to the other five functions and would require a considerable amount of additional code. The expression A**p is evaluated by repeated multiplication if p is an integer greater than 1. Otherwise it is evaluated by <X,D> = EIG(A); for j = 1:n, D(j,j) = EXP(p*LOG(D(j,j))) X*D/X This suffers from the same potential loss of accuracy if X is badly conditioned. It was partly for this reason that the case p = 1 is included in the general case. Comparison of A**1 with A gives some idea of the loss of accuracy for other values of p and for the elementary functions. RREF, the reduced row echelon form, is of some interest in theoretical linear algebra, although it has little computational value. It is included in MATLAB for pedagogical reasons. The algorithm is essentially Gauss-Jordan elimination with detection of negligible columns applied to rectangular matrices. There are three separate places in MATLAB where the rank of a matrix is implicitly computed: in RREF(A), in A\B for non- square A, and in the pseudoinverse PINV(A). Three different algorithms with three different criteria for negligibility are used and so it is possible that three different values could be produced for the same matrix. With RREF(A), the rank of A is the number of nonzero rows. The elimination algorithm used for RREF is the fastest of the three rank-determining algorithms, but it is the least sophisticated numerically and the least reliable. With A\B, the algorithm is essentially that used by example subroutine SQRST in chapter 9 of the LINPACK guide. With PINV(A), the algorithm is based on the singular value decomposition and is described in chapter 11 of the LINPACK guide. The SVD algorithm is the most time-consuming, but the most reliable and is therefore also used for RANK(A). The uniformly distributed random numbers in RAND are obtained from the machine-independent random number generator URAND described in [9]. It is possible to switch to normally distributed random numbers, which are obtained using a transformation also described in [9]. The computation of 2 2 sqrt(a + b ) is required in many matrix algorithms, particularly those involving complex arithmetic. A new approach to carrying out this operation is described by Moler and Morrison [10]. It is a cubically convergent algorithm which starts with a and b , rather than with their squares, and thereby avoids destructive arithmetic underflows and overflows. In MATLAB, the algorithm is used for complex modulus, Euclidean vector norm, plane rotations, and the shift calculation in the eigenvalue and singular value iterations.12. FLOP and CHOP Detailed information about the amount of work involved in matrix calculations and the resulting accuracy is provided by FLOP and CHOP. The basic unit of work is the "flop", or floating point operation. Roughly, one flop is one execution of a Fortran statement like S = S + X(I)*Y(I) or Y(I) = Y(I) + T*X(I) In other words, it consists of one floating point multiplication, together with one floating point addition and the associated indexing and storage reference operations. MATLAB will print the number of flops required for a particular statement when the statement is terminated by an extra comma. For example, the line n = 20; RAND(n)*RAND(n);, ends with an extra comma. Two 20 by 20 random matrices are generated and multiplied together. The result is assigned to ANS, but the semicolon suppresses its printing. The only output is 8800 flops This is n**3 + 2*n**2 flops, n**2 for each random matrix and n**3 for the product. FLOP is a predefined vector with two components. FLOP(1) is the number of flops used by the most recently executed statement, except that statements with zero flops are ignored. For example, after executing the previous statement, flop(1)/n**3 results in ANS = 1.1000 FLOP(2) is the cumulative total of all the flops used since the beginning of the MATLAB session. The statement FLOP = <0 0> resets the total. There are several difficulties associated with keeping a precise count of floating point operations. An addition or subtraction that is not paired with a multiplication is usually counted as a flop. The same is true of an isolated multiplication that is not paired with an addition. Each floating point division counts as a flop. But the number of operations required by system dependent library functions such as square root cannot be counted, so most elementary functions are arbitrarily counted as using only one flop. The biggest difficulty occurs with complex arithmetic. Almost all operations on the real parts of matrices are counted. However, the operations on the complex parts of matrices are counted only when they involve nonzero elements. This means that simple operations on nonreal matrices require only about twice as many flops as the same operations on real matrices. This factor of two is not necessarily an accurate measure of the relative costs of real and complex arithmetic. The result of each floating point operation may also be "chopped" to simulate a computer with a shorter word length. The details of this chopping operation depend upon the format of the floating point word. Usually, the fraction in the floating point word can be regarded as consisting of several octal or hexadecimal digits. The least significant of these digits can be set to zero by a logical masking operation. Thus the statement CHOP(p) causes the p least significant octal or hexadecimal digits in the result of each floating point operation to be set to zero. For example, if the computer being used has an IBM 360 long floating point word with 14 hexadecimal digits in the fraction, then CHOP(8) results in simulation of a computer with only 6 hexadecimal digits in the fraction, i.e. a short floating point word. On a computer such as the CDC 6600 with 16 octal digits, CHOP(8) results in about the same accuracy because the remaining 8 octal digits represent the same number of bits as 6 hexadecimal digits. Some idea of the effect of CHOP on any particular system can be obtained by executing the following statements. long, t = 1/10 long z, t = 1/10 chop(8) long, t = 1/10 long z, t = 1/10 The following Fortran subprograms illustrate more details of FLOP and CHOP. The first subprogram is a simplified example of a system-dependent function used within MATLAB itself. The common variable FLP is essentially the first component of the variable FLOP. The common variable CHP is initially zero, but it is set to p by the statement CHOP(p). To shorten the DATA statement, we assume there are only 6 hexadecimal digits. We also assume an extension of Fortran that allows .AND. to be used as a binary operation between two real variables. REAL FUNCTION FLOP(X) REAL X INTEGER FLP,CHP COMMON FLP,CHP REAL MASK(5) DATA MASK/ZFFFFFFF0,ZFFFFFF00,ZFFFFF000,ZFFFF0000,ZFFF00000/ FLP = FLP + 1 IF (CHP .EQ. 0) FLOP = X IF (CHP .GE. 1 .AND. CHP .LE. 5) FLOP = X .AND. MASK(CHP) IF (CHP .GE. 6) FLOP = 0.0 RETURN END The following subroutine illustrates a typical use of the previous function within MATLAB. It is a simplified version of the Basic Linear Algebra Subprogram that adds a scalar multiple of one vector to another. We assume here that the vectors are stored with a memory increment of one. SUBROUTINE SAXPY(N,TR,TI,XR,XI,YR,YI) REAL TR,TI,XR(N),XI(N),YR(N),YI(N),FLOP IF (N .LE. 0) RETURN IF (TR .EQ. 0.0 .AND. TI .EQ. 0.0) RETURN DO 10 I = 1, N YR(I) = FLOP(YR(I) + TR*XR(I) - TI*XI(I)) YI(I) = YI(I) + TR*XI(I) + TI*XR(I) IF (YI(I) .NE. 0.0D0) YI(I) = FLOP(YI(I)) 10 CONTINUE RETURN END The saxpy operation is perhaps the most fundamental operation within LINPACK. It is used in the computation of the LU, the QR and the SVD factorizations, and in several other places. We see that adding a multiple of one vector with n components to another uses n flops if the vectors are real and between n and 2*n flops if the vectors have nonzero imaginary components. The permanent MATLAB variable EPS is reset by the statement CHOP(p). Its new value is usually the smallest inverse power of two that satisfies the Fortran logical test FLOP(1.0+EPS) .GT. 1.0 However, if EPS had been directly reset to a larger value, the old value is retained.13. Communicating with other programs There are four different ways MATLAB can be used in conjunction with other programs: -- USER, -- EXEC, -- SAVE and LOAD, -- MATZ, CALL and RETURN . Let us illustrate each of these by the following simple example. n = 6 for i = 1:n, for j = 1:n, a(i,j) = abs(i-j); A X = inv(A) The example A could be introduced into MATLAB by writing the following Fortran subroutine. SUBROUTINE USER(A,M,N,S,T) DOUBLE PRECISION A(1),S,T N = IDINT(A(1)) M = N DO 10 J = 1, N DO 10 I = 1, N K = I + (J-1)*M A(K) = IABS(I-J) 10 CONTINUE RETURN END This subroutine should be compiled and linked into MATLAB in place of the original version of USER. Then the MATLAB statements n = 6 A = user(n) X = inv(A) do the job. The example A could be generated by storing the following text in a file named, say, EXAMPLE . for i = 1:n, for j = 1:n, a(i,j) = abs(i-j); Then the MATLAB statements n = 6 exec('EXAMPLE',0) X = inv(A) have the desired effect. The 0 as the optional second parameter of exec indicates that the text in the file should not be printed on the terminal. The matrices A and X could also be stored in files. Two separate main programs would be involved. The first is: PROGRAM MAINA DOUBLE PRECISION A(10,10) N = 6 DO 10 J = 1, N DO 10 I = 1, N A(I,J) = IABS(I-J) 10 CONTINUE OPEN(UNIT=1,FILE='A') WRITE(1,101) N,N 101 FORMAT('A ',2I4) DO 20 J = 1, N WRITE(1,102) (A(I,J),I=1,N) 20 CONTINUE 102 FORMAT(4Z18) END The OPEN statement may take different forms on different systems. It attaches Fortran logical unit number 1 to the file named A. The FORMAT number 102 may also be system dependent. This particular one is appropriate for hexadecimal computers with an 8 byte double precision floating point word. Check, or modify, MATLAB subroutine SAVLOD. After this program is executed, enter MATLAB and give the following statements: load('A') X = inv(A) save('X',X) If all goes according to plan, this will read the matrix A from the file A, invert it, store the inverse in X and then write the matrix X on the file X . The following program can then access X . PROGRAM MAINX DOUBLE PRECISION X(10,10) OPEN(UNIT=1,FILE='X') REWIND 1 READ (1,101) ID,M,N 101 FORMAT(A4,2I4) DO 10 J = 1, N READ(1,102) (X(I,J),I=1,M) 10 CONTINUE 102 FORMAT(4Z18) ... ... The most elaborate mechanism involves using MATLAB as a subroutine within another program. Communication with the MATLAB stack is accomplished using subroutine MATZ which is distributed with MATLAB, but which is not used by MATLAB itself. The preample of MATZ is: SUBROUTINE MATZ(A,LDA,M,N,IDA,JOB,IERR) INTEGER LDA,M,N,IDA(1),JOB,IERR DOUBLE PRECISION A(LDA,N) C C ACCESS MATLAB VARIABLE STACK C A IS AN M BY N MATRIX, STORED IN AN ARRAY WITH C LEADING DIMENSION LDA. C IDA IS THE NAME OF A. C IF IDA IS AN INTEGER K LESS THAN 10, THEN THE NAME IS 'A'K C OTHERWISE, IDA(1:4) IS FOUR CHARACTERS, FORMAT 4A1. C JOB = 0 GET REAL A FROM MATLAB, C = 1 PUT REAL A INTO MATLAB, C = 10 GET IMAG PART OF A FROM MATLAB, C = 11 PUT IMAG PART OF A INTO MATLAB. C RETURN WITH NONZERO IERR AFTER MATLAB ERROR MESSAGE. C C USES MATLAB ROUTINES STACKG, STACKP AND ERROR The preample of subroutine MATLAB is: SUBROUTINE MATLAB(INIT) C INIT = 0 FOR FIRST ENTRY, NONZERO FOR SUBSEQUENT ENTRIES To do our example, write the following program: DOUBLE PRECISION A(10,10),X(10,10) INTEGER IDA(4),IDX(4) DATA LDA/10/ DATA IDA/'A',' ',' ',' '/, IDX/'X',' ',' ',' '/ CALL MATLAB(0) N = 6 DO 10 J = 1, N DO 10 I = 1, N A(I,J) = IABS(I-J) 10 CONTINUE CALL MATZ(A,LDA,N,N,IDA,1,IERR) IF (IERR .NE. 0) GO TO ... CALL MATLAB(1) CALL MATZ(X,LDA,N,N,IDX,0,IERR) IF (IERR .NE. 0) GO TO ... ... ... When this program is executed, the call to MATLAB(0) produces the MATLAB greeting, then waits for input. The command return sends control back to our example program. The matrix A is generated by the program and sent to the stack by the first call to MATZ. The call to MATLAB(1) produces the MATLAB prompt. Then the statements X = inv(A) return will invert our matrix, put the result on the stack and go back to our program. The second call to MATZ will retrieve X . By the way, this matrix X is interesting. Take a look at round(2*(n-1)*X). References [1] J. J. Dongarra, J. R. Bunch, C. B. Moler and G. W. Stewart, LINPACK Users' Guide, Society for Industrial and Applied Mathematics, Philadelphia, 1979. [2] B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines -- EISPACK Guide, Lecture Notes in Computer Science, volume 6, second edition, Springer-Verlag, 1976. [3] B. S. Garbow, J. M. Boyle, J. J. Dongarra, C. B. Moler, Matrix Eigensystem Routines -- EISPACK Guide Extension, Lecture Notes in Computer Science, volume 51, Springer- Verlag, 1977. [4] S. Cohen and S. Piper, SPEAKEASY III Reference Manual, Speakeasy Computing Corp., Chicago, Ill., 1979. [5] J. H. Wilkinson and C. Reinsch, Handbook for Automatic Computation, volume II, Linear Algebra, Springer-Verlag, 1971. [6] Niklaus Wirth, Algorithms + Data Structures = Programs, Prentice-Hall, 1976. [7] H. B. Keller and D. Sachs, "Calculations of the Conductivity of a Medium Containing Cylindrical Inclusions", J. Applied Physics 35, 537-538, 1964. [8] C. B. Moler and C. F. Van Loan, Nineteen Dubious Ways to Compute the Exponential of a Matrix, SIAM Review 20, 801- 836, 1979. [9] G. E. Forsythe, M. A. Malcolm and C. B. Moler, Computer Methods for Mathematical Computations, Prentice-Hall, 1977. [10] C. B. Moler and D. R. Morrison, "Replacing square roots by Pythagorean sums", University of New Mexico, Computer Science Department, technical report, submitted for publication, 1980.Appendix. The HELP document NEWS MATLAB NEWS dated May, 1981. This describes recent or local changes. The new features added since the November, 1980, printing of the Users' Guide include DIARY, EDIT, KRON, MACRO, PLOT, RAT, TRIL, TRIU and six element-by-element operations: .* ./ .\ .*. ./. .\. Some additional capabilities have been added to EXIT, RANDOM, RCOND, SIZE and SVD. INTRO Welcome to MATLAB. Here are a few sample statements: A = <1 2; 3 4> b = <5 6>' x = A\b <V,D> = eig(A), norm(A-V*D/V) help \ , help eig exec('demo',7) For more information, see the MATLAB Users' Guide which is contained in file ... or may be obtained from ... . HELP HELP gives assistance. HELP HELP obviously prints this message. To see all the HELP messages, list the file ... . < < > Brackets used in forming vectors and matrices. <6.9 9.64 SQRT(-1)> is a vector with three elements separated by blanks. <6.9, 9.64, sqrt(-1)> is the same thing. <1+I 2-I 3> and <1 +I 2 -I 3> are not the same. The first has three elements, the second has five. <11 12 13; 21 22 23> is a 2 by 3 matrix . The semicolon ends the first row. Vectors and matrices can be used inside < > brackets. <A B; C> is allowed if the number of rows of A equals the number of rows of B and the number of columns of A plus the number of columns of B equals the number of columns of C . This rule generalizes in a hopefully obvious way to allow fairly complicated constructions. A = < > stores an empty matrix in A , thereby removing it from the list of current variables. For the use of < and > on the left of the = in multiple assignment statements, see LU, EIG, SVD and so on. In WHILE and IF clauses, <> means less than or greater than, i.e. not equal, < means less than, > means greater than, <= means less than or equal, >= means greater than or equal. For the use of > and < to delineate macros, see MACRO. > See < . Also see MACRO. ( ( ) Used to indicate precedence in arithmetic expressions in the usual way. Used to enclose arguments of functions in the usual way. Used to enclose subscripts of vectors and matrices in a manner somewhat more general than the usual way. If X and V are vectors, then X(V) is <X(V(1)), X(V(2)), ..., X(V(N))> . The components of V are rounded to nearest integers and used as subscripts. An error occurs if any such subscript is less than 1 or greater than the dimension of X . Some examples: X(3) is the third element of X . X(<1 2 3>) is the first three elements of X . So is X(<SQRT(2), SQRT(3), 4*ATAN(1)>) . If X has N components, X(N:-1:1) reverses them. The same indirect subscripting is used in matrices. If V has M components and W has N components, then A(V,W) is the M by N matrix formed from the elements of A whose subscripts are the elements of V and W . For example... A(<1,5>,:) = A(<5,1>,:) interchanges rows 1 and 5 of A . ) See ( . = Used in assignment statements and to mean equality in WHILE and IF clauses. . Decimal point. 314/100, 3.14 and .314E1 are all the same. Element-by-element multiplicative operations are obtained using .* , ./ , or .\ . For example, C = A ./ B is the matrix with elements c(i,j) = a(i,j)/b(i,j) . Kronecker tensor products and quotients are obtained with .*. , ./. and .\. . See KRON. Two or more points at the end of the line indicate continuation. The total line length limit is 1024 characters. , Used to separate matrix subscripts and function arguments. Used at the end of FOR, WHILE and IF clauses. Used to separate statements in multi-statement lines. In this situation, it may be replaced by semicolon to suppress printing. ; Used inside brackets to end rows. Used after an expression or statement to suppress printing. See SEMI. \ Backslash or matrix left division. A\B is roughly the same as INV(A)*B , except it is computed in a different way. If A is an N by N matrix and B is a column vector with N components, or a matrix with several such columns, then X = A\B is the solution to the equation A*X = B computed by Gaussian elimination. A warning message is printed if A is badly scaled or nearly singular. A\EYE produces the inverse of A . If A is an M by N matrix with M < or > N and B is a column vector with M components, or a matrix with several such columns, then X = A\B is the solution in the least squares sense to the under- or overdetermined system of equations A*X = B . The effective rank, K, of A is determined from the QR decomposition with pivoting. A solution X is computed which has at most K nonzero components per column. If K < N this will usually not be the same solution as PINV(A)*B . A\EYE produces a generalized inverse of A . If A and B have the same dimensions, then A .\ B has elements a(i,j)\b(i,j) . Also, see EDIT. / Slash or matrix right division. B/A is roughly the same as B*INV(A) . More precisely, B/A = (A'\B')' . See \ . IF A and B have the same dimensions, then A ./ B has elements a(i,j)/b(i,j) . Two or more slashes together on a line indicate a logical end of line. Any following text is ignored. ' Transpose. X' is the complex conjugate transpose of X . Quote. 'ANY TEXT' is a vector whose components are the MATLAB internal codes for the characters. A quote within the text is indicated by two quotes. See DISP and FILE . + Addition. X + Y . X and Y must have the same dimensions. - Subtraction. X - Y . X and Y must have the same dimensions. * Matrix multiplication, X*Y . Any scalar (1 by 1 matrix) may multiply anything. Otherwise, the number of columns of X must equal the number of rows of Y . Element-by-element multiplication is obtained with X .* Y . The Kronecker tensor product is denoted by X .*. Y . Powers. X**p is X to the p power. p must be a scalar. If X is a matrix, see FUN . : Colon. Used in subscripts, FOR iterations and possibly elsewhere. J:K is the same as <J, J+1, ..., K> J:K is empty if J > K . J:I:K is the same as <J, J+I, J+2I, ..., K> J:I:K is empty if I > 0 and J > K or if I < 0 and J < K . The colon notation can be used to pick out selected rows, columns and elements of vectors and matrices. A(:) is all the elements of A, regarded as a single column. A(:,J) is the J-th column of A A(J:K) is A(J),A(J+1),...,A(K) A(:,J:K) is A(:,J),A(:,J+1),...,A(:,K) and so on. For the use of the colon in the FOR statement, See FOR . ABS ABS(X) is the absolute value, or complex modulus, of the elements of X . ANS Variable created automatically when expressions are not assigned to anything else. ATAN ATAN(X) is the arctangent of X . See FUN . BASE BASE(X,B) is a vector containing the base B representation of X . This is often used in conjunction with DISPLAY. DISPLAY(X,B) is the same as DISPLAY(BASE(X,B)). For example, DISP(4*ATAN(1),16) prints the hexadecimal representation of pi. CHAR CHAR(K) requests an input line containing a single character to replace MATLAB character number K in the following table. For example, CHAR(45) replaces backslash. CHAR(-K) replaces the alternate character number K. K character alternate name 0 - 9 0 - 9 0 - 9 digits 10 - 35 A - Z a - z letters 36 blank 37 ( ( lparen 38 ) ) rparen 39 ; ; semi 40 : | colon 41 + + plus 42 - - minus 43 * * star 44 / / slash 45 \ $ backslash 46 = = equal 47 . . dot 48 , , comma 49 ' " quote 50 < [ less 51 > ] great CHOL Cholesky factorization. CHOL(X) uses only the diagonal and upper triangle of X . The lower triangular is assumed to be the (complex conjugate) transpose of the upper. If X is positive definite, then R = CHOL(X) produces an upper triangular R so that R'*R = X . If X is not positive definite, an error message is printed. CHOP Truncate arithmetic. CHOP(P) causes P places to be chopped off after each arithmetic operation in subsequent computations. This means P hexadecimal digits on some computers and P octal digits on others. CHOP(0) restores full precision. CLEAR Erases all variables, except EPS, FLOP, EYE and RAND. X = <> erases only variable X . So does CLEAR X . COND Condition number in 2-norm. COND(X) is the ratio of the largest singular value of X to the smallest. CONJG CONJG(X) is the complex conjugate of X . COS COS(X) is the cosine of X . See FUN . DET DET(X) is the determinant of the square matrix X . DIAG If V is a row or column vector with N components, DIAG(V,K) is a square matrix of order N+ABS(K) with the elements of V on the K-th diagonal. K = 0 is the main diagonal, K > 0 is above the main diagonal and K < 0 is below the main diagonal. DIAG(V) simply puts V on the main diagonal. eg. DIAG(-M:M) + DIAG(ONES(2*M,1),1) + DIAG(ONES(2*M,1),-1) produces a tridiagonal matrix of order 2*M+1 . IF X is a matrix, DIAG(X,K) is a column vector formed from the elements of the K-th diagonal of X . DIAG(X) is the main diagonal of X . DIAG(DIAG(X)) is a diagonal matrix . DIARY DIARY('file') causes a copy of all subsequent terminal input and most of the resulting output to be written on the file. DIARY(0) turns it off. See FILE. DISP DISPLAY(X) prints X in a compact format. If all the elements of X are integers between 0 and 51, then X is interpreted as MATLAB text and printed accordingly. Otherwise, + , - and blank are printed for positive, negative and zero elements. Imaginary parts are ignored. DISP(X,B) is the same as DISP(BASE(X,B)). EDIT There are no editing features available on most installations and EDIT is not a command. However, on a few systems a command line consisting of a single backslash \ will cause the local file editor to be called with a copy of the previous input line. When the editor returns control to MATLAB, it will execute the line again. EIG Eigenvalues and eigenvectors. EIG(X) is a vector containing the eigenvalues of a square matrix X . <V,D> = EIG(X) produces a diagonal matrix D of eigenvalues and a full matrix V whose columns are the corresponding eigenvectors so that X*V = V*D . ELSE Used with IF . END Terminates the scope of FOR, WHILE and IF statements. Without END's, FOR and WHILE repeat all statements up to the end of the line. Each END is paired with the closest previous unpaired FOR or WHILE and serves to terminate its scope. The line FOR I=1:N, FOR J=1:N, A(I,J)=1/(I+J-1); A would cause A to be printed N**2 times, once for each new element. On the other hand, the line FOR I=1:N, FOR J=1:N, A(I,J)=1/(I+J-1); END, END, A will lead to only the final printing of A . Similar considerations apply to WHILE. EXIT terminates execution of loops or of MATLAB itself. EPS Floating point relative accuracy. A permanent variable whose value is initially the distance from 1.0 to the next largest floating point number. The value is changed by CHOP, and other values may be assigned. EPS is used as a default tolerance by PINV and RANK. EXEC EXEC('file',k) obtains subsequent MATLAB input from an external file. The printing of input is controlled by the optional parameter k . If k = 1 , the input is echoed. If k = 2 , the MATLAB prompt <> is printed. If k = 4 , MATLAB pauses before each prompt and waits for a null line to continue. If k = 0 , there is no echo, prompt or pause. This is the default if the exec command is followed by a semicolon. If k = 7 , there will be echos, prompts and pauses. This is useful for demonstrations on video terminals. If k = 3 , there will be echos and prompts, but no pauses. This is the the default if the exec command is not followed by a semicolon. EXEC(0) causes subsequent input to be obtained from the terminal. An end-of-file has the same effect. EXEC's may be nested, i.e. the text in the file may contain EXEC of another file. EXEC's may also be driven by FOR and WHILE loops. EXIT Causes termination of a FOR or WHILE loop. If not in a loop, terminates execution of MATLAB. EXP EXP(X) is the exponential of X , e to the X . See FUN . EYE Identity matrix. EYE(N) is the N by N identity matrix. EYE(M,N) is an M by N matrix with 1's on the diagonal and zeros elsewhere. EYE(A) is the same size as A . EYE with no arguments is an identity matrix of whatever order is appropriate in the context. For example, A + 3*EYE adds 3 to each diagonal element of A . FILE The EXEC, SAVE, LOAD, PRINT and DIARY functions access files. The 'file' parameter takes different forms for different operating systems. On most systems, 'file' may be a string of up to 32 characters in quotes. For example, SAVE('A') or EXEC('matlab/demo.exec') . The string will be used as the name of a file in the local operating system. On all systems, 'file' may be a positive integer k less than 10 which will be used as a FORTRAN logical unit number. Some systems then automatically access a file with a name like FORT.k or FORk.DAT. Other systems require a file with a name like FT0kF001 to be assigned to unit k before MATLAB is executed. Check your local installation for details. FLOPS Count of floating point operations. FLOPS is a permanently defined row vector with two elements. FLOPS(1) is the number of floating point operations counted during the previous statement. FLOPS(2) is a cumulative total. FLOPS can be used in the same way as any other vector. FLOPS(2) = 0 resets the cumulative total. In addition, FLOPS(1) will be printed whenever a statement is terminated by an extra comma. For example, X = INV(A);, or COND(A), (as the last statement on the line). HELP FLPS gives more details. FLPS More detail on FLOPS. It is not feasible to count absolutely all floating point operations, but most of the important ones are counted. Each multiply and add in a real vector operation such as a dot product or a 'saxpy' counts one flop. Each multiply and add in a complex vector operation counts two flops. Other additions, subtractions and multiplications count one flop each if the result is real and two flops if it is not. Real divisions count one and complex divisions count two. Elementary functions count one if real and two if complex. Some examples. If A and B are real N by N matrices, then A + B counts N**2 flops, A*B counts N**3 flops, A**100 counts 99*N**3 flops, LU(A) counts roughly (1/3)*N**3 flops. FOR Repeat statements a specific number of times. FOR variable = expr, statement, ..., statement, END The END at the end of a line may be omitted. The comma before the END may also be omitted. The columns of the expression are stored one at a time in the variable and then the following statements, up to the END, are executed. The expression is often of the form X:Y, in which case its columns are simply scalars. Some examples (assume N has already been assigned a value). FOR I = 1:N, FOR J = 1:N, A(I,J) = 1/(I+J-1); FOR J = 2:N-1, A(J,J) = J; END; A FOR S = 1.0: -0.1: 0.0, ... steps S with increments of -0.1 . FOR E = EYE(N), ... sets E to the unit N-vectors. FOR V = A, ... has the same effect as FOR J = 1:N, V = A(:,J); ... except J is also set here. FUN For matrix arguments X , the functions SIN, COS, ATAN, SQRT, LOG, EXP and X**p are computed using eigenvalues D and eigenvectors V . If <V,D> = EIG(X) then f(X) = V*f(D)/V . This method may give inaccurate results if V is badly conditioned. Some idea of the accuracy can be obtained by comparing X**1 with X . For vector arguments, the function is applied to each component. HESS Hessenberg form. The Hessenberg form of a matrix is zero below the first subdiagonal. If the matrix is symmetric or Hermitian, the form is tridiagonal. <P,H> = HESS(A) produces a unitary matrix P and a Hessenberg matrix H so that A = P*H*P'. By itself, HESS(A) returns H. HILB Inverse Hilbert matrix. HILB(N) is the inverse of the N by N matrix with elements 1/(i+j-1), which is a famous example of a badly conditioned matrix. The result is exact for N less than about 15, depending upon the computer. IF Conditionally execute statements. Simple form... IF expression rop expression, statements where rop is =, <, >, <=, >=, or <> (not equal) . The statements are executed once if the indicated comparison between the real parts of the first components of the two expressions is true, otherwise the statements are skipped. Example. IF ABS(I-J) = 1, A(I,J) = -1; More complicated forms use END in the same way it is used with FOR and WHILE and use ELSE as an abbreviation for END, IF expression not rop expression . Example FOR I = 1:N, FOR J = 1:N, ... IF I = J, A(I,J) = 2; ELSE IF ABS(I-J) = 1, A(I,J) = -1; ... ELSE A(I,J) = 0; An easier way to accomplish the same thing is A = 2*EYE(N); FOR I = 1:N-1, A(I,I+1) = -1; A(I+1,I) = -1; IMAG IMAG(X) is the imaginary part of X . INV INV(X) is the inverse of the square matrix X . A warning message is printed if X is badly scaled or nearly singular. KRON KRON(X,Y) is the Kronecker tensor product of X and Y . It is also denoted by X .*. Y . The result is a large matrix formed by taking all possible products between the elements of X and those of Y . For example, if X is 2 by 3, then X .*. Y is < x(1,1)*Y x(1,2)*Y x(1,3)*Y x(2,1)*Y x(2,2)*Y x(2,3)*Y > The five-point discrete Laplacian for an n-by-n grid can be generated by T = diag(ones(n-1,1),1); T = T + T'; I = EYE(T); A = T.*.I + I.*.T - 4*EYE; Just in case they might be useful, MATLAB includes constructions called Kronecker tensor quotients, denoted by X ./. Y and X .\. Y . They are obtained by replacing the elementwise multiplications in X .*. Y with divisions. LINES An internal count is kept of the number of lines of output since the last input. Whenever this count approaches a limit, the user is asked whether or not to suppress printing until the next input. Initially the limit is 25. LINES(N) resets the limit to N . LOAD LOAD('file') retrieves all the variables from the file . See FILE and SAVE for more details. To prepare your own file for LOADing, change the READs to WRITEs in the code given under SAVE. LOG LOG(X) is the natural logarithm of X . See FUN . Complex results are produced if X is not positive, or has nonpositive eigenvalues. LONG Determine output format. All computations are done in complex arithmetic and double precision if it is available. SHORT and LONG merely switch between different output formats. SHORT Scaled fixed point format with about 5 digits. LONG Scaled fixed point format with about 15 digits. SHORT E Floating point format with about 5 digits. LONG E Floating point format with about 15 digits. LONG Z System dependent format, often hexadecimal. LU Factors from Gaussian elimination. <L,U> = LU(X) stores a upper triangular matrix in U and a 'psychologically lower triangular matrix', i.e. a product of lower triangular and permutation matrices, in L , so that X = L*U . By itself, LU(X) returns the output from CGEFA . MACRO The macro facility involves text and inward pointing angle brackets. If STRING is the source text for any MATLAB expression or statement, then t = 'STRING'; encodes the text as a vector of integers and stores that vector in t . DISP(t) will print the text and >t< causes the text to be interpreted, either as a statement or as a factor in an expression. For example t = '1/(i+j-1)'; disp(t) for i = 1:n, for j = 1:n, a(i,j) = >t<; generates the Hilbert matrix of order n. Another example showing indexed text, S = <'x = 3 ' 'y = 4 ' 'z = sqrt(x*x+y*y)'> for k = 1:3, >S(k,:)< It is necessary that the strings making up the "rows" of the "matrix" S have the same lengths. MAGIC Magic square. MAGIC(N) is an N by N matrix constructed from the integers 1 through N**2 with equal row and column sums. NORM For matrices.. NORM(X) is the largest singular value of X . NORM(X,1) is the 1-norm of X . NORM(X,2) is the same as NORM(X) . NORM(X,'INF') is the infinity norm of X . NORM(X,'FRO') is the F-norm, i.e. SQRT(SUM(DIAG(X'*X))) . For vectors.. NORM(V,P) = (SUM(V(I)**P))**(1/P) . NORM(V) = NORM(V,2) . NORM(V,'INF') = MAX(ABS(V(I))) . ONES All ones. ONES(N) is an N by N matrix of ones. ONES(M,N) is an M by N matrix of ones . ONES(A) is the same size as A and all ones . ORTH Orthogonalization. Q = ORTH(X) is a matrix with orthonormal columns, i.e. Q'*Q = EYE, which span the same space as the columns of X . PINV Pseudoinverse. X = PINV(A) produces a matrix X of the same dimensions as A' so that A*X*A = A , X*A*X = X and AX and XA are Hermitian . The computation is based on SVD(A) and any singular values less than a tolerance are treated as zero. The default tolerance is NORM(SIZE(A),'inf')*NORM(A)*EPS. This tolerance may be overridden with X = PINV(A,tol). See RANK. PLOT PLOT(X,Y) produces a plot of the elements of Y against those of X . PLOT(Y) is the same as PLOT(1:n,Y) where n is the number of elements in Y . PLOT(X,Y,P) or PLOT(X,Y,p1,...,pk) passes the optional parameter vector P or scalars p1 through pk to the plot routine. The default plot routine is a crude printer-plot. It is hoped that an interface to local graphics equipment can be provided. An interesting example is t = 0:50; PLOT( t.*cos(t), t.*sin(t) ) POLY Characteristic polynomial. If A is an N by N matrix, POLY(A) is a column vector with N+1 elements which are the coefficients of the characteristic polynomial, DET(lambda*EYE - A) . If V is a vector, POLY(V) is a vector whose elements are the coefficients of the polynomial whose roots are the elements of V . For vectors, ROOTS and POLY are inverse functions of each other, up to ordering, scaling, and roundoff error. ROOTS(POLY(1:20)) generates Wilkinson's famous example. PRINT PRINT('file',X) prints X on the file using the current format determined by SHORT, LONG Z, etc. See FILE. PROD PROD(X) is the product of all the elements of X . QR Orthogonal-triangular decomposition. <Q,R> = QR(X) produces an upper triangular matrix R of the same dimension as X and a unitary matrix Q so that X = Q*R . <Q,R,E> = QR(X) produces a permutation matrix E , an upper triangular R with decreasing diagonal elements and a unitary Q so that X*E = Q*R . By itself, QR(X) returns the output of CQRDC . TRIU(QR(X)) is R . RAND Random numbers and matrices. RAND(N) is an N by N matrix with random entries. RAND(M,N) is an M by N matrix with random entries. RAND(A) is the same size as A . RAND with no arguments is a scalar whose value changes each time it is referenced. Ordinarily, random numbers are uniformly distributed in the interval (0.0,1.0) . RAND('NORMAL') switches to a normal distribution with mean 0.0 and variance 1.0 . RAND('UNIFORM') switches back to the uniform distribution. RAND('SEED') returns the current value of the seed for the generator. RAND('SEED',n) sets the seed to n . RAND('SEED',0) resets the seed to 0, its value when MATLAB is first entered. RANK Rank. K = RANK(X) is the number of singular values of X that are larger than NORM(SIZE(X),'inf')*NORM(X)*EPS. K = RANK(X,tol) is the number of singular values of X that are larger than tol . RCOND RCOND(X) is an estimate for the reciprocal of the condition of X in the 1-norm obtained by the LINPACK condition estimator. If X is well conditioned, RCOND(X) is near 1.0 . If X is badly conditioned, RCOND(X) is near 0.0 . <R, Z> = RCOND(A) sets R to RCOND(A) and also produces a vector Z so that NORM(A*Z,1) = R*NORM(A,1)*NORM(Z,1) So, if RCOND(A) is small, then Z is an approximate null vector. RAT An experimental function which attempts to remove the roundoff error from results that should be "simple" rational numbers. RAT(X) approximates each element of X by a continued fraction of the form a/b = d1 + 1/(d2 + 1/(d3 + ... + 1/dk)) with k <= len, integer di and abs(di) <= max . The default values of the parameters are len = 5 and max = 100. RAT(len,max) changes the default values. Increasing either len or max increases the number of possible fractions. <A,B> = RAT(X) produces integer matrices A and B so that A ./ B = RAT(X) Some examples: long T = hilb(6), X = inv(T) <A,B> = rat(X) H = A ./ B, S = inv(H) short e d = 1:8, e = ones(d), A = abs(d'*e - e'*d) X = inv(A) rat(X) display(ans) REAL REAL(X) is the real part of X . RETURN From the terminal, causes return to the operating system or other program which invoked MATLAB. From inside an EXEC, causes return to the invoking EXEC, or to the terminal. RREF RREF(A) is the reduced row echelon form of the rectangular matrix. RREF(A,B) is the same as RREF(<A,B>) . ROOTS Find polynomial roots. ROOTS(C) computes the roots of the polynomial whose coefficients are the elements of the vector C . If C has N+1 components, the polynomial is C(1)*X**N + ... + C(N)*X + C(N+1) . See POLY. ROUND ROUND(X) rounds the elements of X to the nearest integers. SAVE SAVE('file') stores all the current variables in a file. SAVE('file',X) saves only X . See FILE . The variables may be retrieved later by LOAD('file') or by your own program using the following code for each matrix. The lines involving XIMAG may be eliminated if everything is known to be real. attach lunit to 'file' REAL or DOUBLE PRECISION XREAL(MMAX,NMAX) REAL or DOUBLE PRECISION XIMAG(MMAX,NMAX) READ(lunit,101) ID,M,N,IMG DO 10 J = 1, N READ(lunit,102) (XREAL(I,J), I=1,M) IF (IMG .NE. 0) READ(lunit,102) (XIMAG(I,J),I=1,M) 10 CONTINUE The formats used are system dependent. The following are typical. See SUBROUTINE SAVLOD in your local implementation of MATLAB. 101 FORMAT(4A1,3I4) 102 FORMAT(4Z18) 102 FORMAT(4O20) 102 FORMAT(4D25.18) SCHUR Schur decomposition. <U,T> = SCHUR(X) produces an upper triangular matrix T , with the eigenvalues of X on the diagonal, and a unitary matrix U so that X = U*T*U' and U'*U = EYE . By itself, SCHUR(X) returns T . SHORT See LONG . SEMI Semicolons at the end of lines will cause, rather than suppress, printing. A second SEMI restores the initial interpretation. SIN SIN(X) is the sine of X . See FUN . SIZE If X is an M by N matrix, then SIZE(X) is <M, N> . Can also be used with a multiple assignment, <M, N> = SIZE(X) . SQRT SQRT(X) is the square root of X . See FUN . Complex results are produced if X is not positive, or has nonpositive eigenvalues. STOP Use EXIT instead. SUM SUM(X) is the sum of all the elements of X . SUM(DIAG(X)) is the trace of X . SVD Singular value decomposition. <U,S,V> = SVD(X) produces a diagonal matrix S , of the same dimension as X and with nonnegative diagonal elements in decreasing order, and unitary matrices U and V so that X = U*S*V' . By itself, SVD(X) returns a vector containing the singular values. <U,S,V> = SVD(X,0) produces the "economy size" decomposition. If X is m by n with m > n, then only the first n columns of U are computed and S is n by n . TRIL Lower triangle. TRIL(X) is the lower triangular part of X. TRIL(X,K) is the elements on and below the K-th diagonal of X. K = 0 is the main diagonal, K > 0 is above the main diagonal and K < 0 is below the main diagonal. TRIU Upper triangle. TRIU(X) is the upper triangular part of X. TRIU(X,K) is the elements on and above the K-th diagonal of X. K = 0 is the main diagonal, K > 0 is above the main diagonal and K < 0 is below the main diagonal. USER Allows personal Fortran subroutines to be linked into MATLAB . The subroutine should have the heading SUBROUTINE USER(A,M,N,S,T) REAL or DOUBLE PRECISION A(M,N),S,T The MATLAB statement Y = USER(X,s,t) results in a call to the subroutine with a copy of the matrix X stored in the argument A , its column and row dimensions in M and N , and the scalar parameters s and t stored in S and T . If s and t are omitted, they are set to 0.0 . After the return, A is stored in Y . The dimensions M and N may be reset within the subroutine. The statement Y = USER(K) results in a call with M = 1, N = 1 and A(1,1) = FLOAT(K) . After the subroutine has been written, it must be compiled and linked to the MATLAB object code within the local operating system. WHAT Lists commands and functions currently available. WHILE Repeat statements an indefinite number of times. WHILE expr rop expr, statement, ..., statement, END where rop is =, <, >, <=, >=, or <> (not equal) . The END at the end of a line may be omitted. The comma before the END may also be omitted. The commas may be replaced by semicolons to avoid printing. The statements are repeatedly executed as long as the indicated comparison between the real parts of the first components of the two expressions is true. Example (assume a matrix A is already defined). E = 0*A; F = E + EYE; N = 1; WHILE NORM(E+F-E,1) > 0, E = E + F; F = A*F/N; N = N + 1; E WHO Lists current variables. WHY Provides succinct answers to any questions. // \n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


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and % projects that I discussed in two previous posts, I was a % Professor of Mathematics and then of Computer Science at the % University of New Mexico in Albuquerque. % I was teaching courses in Linear Algebra and Numerical Analysis. % I wanted my students to have easy access to LINPACK and EISPACK % without writing Fortran programs. By "easy access" I meant not % going through the remote batch processing and the repeated % edit-compile-link-load-execute process that was ordinarily required % on the campus central mainframe computer. % % So, I studied Niklaus Wirth's book _Algorithms + Data Structures = % Programs_ and learned how to parse programming languages. Wirth % calls his example language PL/0. (A _zinger_ aimed at PL/I, the % monolithic, committee-designed language being promoted by IBM and % the U.S. Defense Department at the time.) PL/0 was a pedagogical % subset of Wirth's Pascal, which, in turn, was his response to Algol. % Quoting Wirth, "In the realm of data types, however, PL/0 adheres to % the demand of simplicity without compromise: integers are its only % data type." % % Following Wirth's approach, I wrote the first MATLAB REPLACE_WITH_DASH_DASH % an acronym for Matrix Laboratory REPLACE_WITH_DASH_DASH in Fortran, as a dialect of PL/0 % with matrix as the only data type. The project was a kind of hobby, % a new aspect of programming for me to learn and something for my % students to use. There was never any formal outside support and % certainly no business plan. MathWorks was a few years in the future. % % This first MATLAB was not a programming language; it was just a simple % interactive matrix calculator. There were no % m-files, no toolboxes, no graphics. And no ODEs or FFTs. % This snapshot of the start-up screen shows all the reserved words % and functions. There are only 71 of them. If you wanted to add another % function, you would get the source code from me, write a Fortran % subroutine, add your new name to the parse table, and recompile MATLAB. % % <> % % Here is the first Users' Guide, in its entirety, from 1981. %% Users' Guide % MATLAB Users' Guide % May, 1981 % % % Cleve Moler % Department of Computer Science % University of New Mexico % % % ABSTRACT. MATLAB is an interactive computer program % that serves as a convenient "laboratory" for % computations involving matrices. It provides easy % access to matrix software developed by the LINPACK and % EISPACK projects. The program is written in Fortran % and is designed to be readily installed under any % operating system which permits interactive execution of % Fortran programs. % % % MATLAB is an interactive computer program that serves as a % convenient "laboratory" for computations involving matrices. It % provides easy access to matrix software developed by the LINPACK % and EISPACK projects [1-3]. The capabilities range from standard % tasks such as solving simultaneous linear equations and inverting % matrices, through symmetric and nonsymmetric eigenvalue problems, % to fairly sophisticated matrix tools such as the singular value % decomposition. % % It is expected that one of MATLAB's primary uses will be in % the classroom. It should be useful in introductory courses in % applied linear algebra, as well as more advanced courses in % numerical analysis, matrix theory, statistics and applications of % matrices to other disciplines. In nonacademic settings, MATLAB % can serve as a "desk calculator" for the quick solution of small % problems involving matrices. % % The program is written in Fortran and is designed to be % readily installed under any operating system which permits % interactive execution of Fortran programs. The resources % required are fairly modest. There are less than 7000 lines of % Fortran source code, including the LINPACK and EISPACK % subroutines used. With proper use of overlays, it is possible % run the system on a minicomputer with only 32K bytes of memory. % % The size of the matrices that can be handled in MATLAB % depends upon the amount of storage that is set aside when the % system is compiled on a particular machine. We have found that % an allocation of 5000 words for matrix elements is usually quite % satisfactory. This provides room for several 20 by 20 matrices, % for example. One implementation on a virtual memory system % provides 100,000 elements. Since most of the algorithms used % access memory in a sequential fashion, the large amount of % allocated storage causes no difficulties. % % In some ways, MATLAB resembles SPEAKEASY [4] and, to a % lesser extent, APL. All are interactive terminal languages that % ordinarily accept single-line commands or statements, process % them immediately, and print the results. All have arrays or % matrices as principal data types. But for MATLAB, the matrix is % the only data type (although scalars, vectors and text are % special cases), the underlying system is portable and requires % fewer resources, and the supporting subroutines are more powerful % and, in some cases, have better numerical properties. % % Together, LINPACK and EISPACK represent the state of the art % in software for matrix computation. EISPACK is a package of over % 70 Fortran subroutines for various matrix eigenvalue computations % that are based for the most part on Algol procedures published by % Wilkinson, Reinsch and their colleagues [5]. LINPACK is a % package of 40 Fortran subroutines (in each of four data types) % for solving and analyzing simultaneous linear equations and % related matrix problems. Since MATLAB is not primarily concerned % with either execution time efficiency or storage savings, it % ignores most of the special matrix properties that LINPACK and % EISPACK subroutines use to advantage. Consequently, only 8 % subroutines from LINPACK and 5 from EISPACK are actually % involved. % % In more advanced applications, MATLAB can be used in % conjunction with other programs in several ways. It is possible % to define new MATLAB functions and add them to the system. With % most operating systems, it is possible to use the local file % system to pass matrices between MATLAB and other programs. % MATLAB command and statement input can be obtained from a local % file instead of from the terminal. The most power and % flexibility is obtained by using MATLAB as a subroutine which is % called by other programs. % % This document first gives an overview of MATLAB from the % user's point of view. Several extended examples involving data % fitting, partial differential equations, eigenvalue sensitivity % and other topics are included. A formal definition of the MATLAB % language and an brief description of the parser and interpreter % are given. The system was designed and programmed using % techniques described by Wirth [6], implemented in nonrecursive, % portable Fortran. There is a brief discussion of some of the % matrix algorithms and of their numerical properties. The final % section describes how MATLAB can be used with other programs. % The appendix includes the HELP documentation available on-line. %% 1. Elementary operations % % % MATLAB works with essentially only one kind of object, a % rectangular matrix with complex elements. If the imaginary parts % of the elements are all zero, they are not printed, but they % still occupy storage. In some situations, special meaning is % attached to 1 by 1 matrices, that is scalars, and to 1 by n and m % by 1 matrices, that is row and column vectors. % % Matrices can be introduced into MATLAB in four different % ways: % REPLACE_WITH_DASH_DASH Explicit list of elements, % REPLACE_WITH_DASH_DASH Use of FOR and WHILE statements, % REPLACE_WITH_DASH_DASH Read from an external file, % REPLACE_WITH_DASH_DASH Execute an external Fortran program. % % The explicit list is surrounded by angle brackets, '<' and % '>', and uses the semicolon ';' to indicate the ends of the rows. % For example, the input line % % A = <1 2 3; 4 5 6; 7 8 9> % % will result in the output % % A = % % 1. 2. 3. % 4. 5. 6. % 7. 8. 9. % % The matrix A will be saved for later use. The individual % elements are separated by commas or blanks and can be any MATLAB % expressions, for example % % x = < -1.3, 4/5, 4*atan(1) > % % results in % % X = % % -1.3000 0.8000 3.1416 % % The elementary functions available include sqrt, log, exp, sin, % cos, atan, abs, round, real, imag, and conjg. % % Large matrices can be spread across several input lines, % with the carriage returns replacing the semicolons. The above % matrix could also have been produced by % % A = < 1 2 3 % 4 5 6 % 7 8 9 > % % % Matrices can be input from the local file system. Say a % file named 'xyz' contains five lines of text, % % % A = < % 1 2 3 % 4 5 6 % 7 8 9 % >; % % then the MATLAB statement EXEC('xyz') reads the matrix and % assigns it to A . % % The FOR statement allows the generation of matrices whose % elements are given by simple formulas. Our example matrix A % could also have been produced by % % for i = 1:3, for j = 1:3, a(i,j) = 3*(i-1)+j; % % The semicolon at the end of the line suppresses the printing, % which in this case would have been nine versions of A with % changing elements. % % Several statements may be given on a line, separated by % semicolons or commas. % % Two consecutive periods anywhere on a line indicate % continuation. The periods and any following characters are % deleted, then another line is input and concatenated onto the % previous line. % % Two consecutive slashes anywhere on a line cause the % remainder of the line to be ignored. This is useful for % inserting comments. % % Names of variables are formed by a letter, followed by any % number of letters and digits, but only the first 4 characters are % remembered. % % The special character prime (') is used to denote the % transpose of a matrix, so % % x = x' % % changes the row vector above into the column vector % % X = % % -1.3000 % 0.8000 % 3.1416 % % % Individual matrix elements may be referenced by enclosing % their subscripts in parentheses. When any element is changed, % the entire matrix is reprinted. For example, using the above % matrix, % a(3,3) = a(1,3) + a(3,1) % % results in % % A = % % 1. 2. 3. % 4. 5. 6. % 7. 8. 10. % % % Addition, subtraction and multiplication of matrices are % denoted by +, -, and * . The operations are performed whenever % the matrices have the proper dimensions. For example, with the % above A and x, the expressions A + x and x*A are incorrect % because A is 3 by 3 and x is now 3 by 1. However, % % b = A*x % % is correct and results in the output % % B = % % 9.7248 % 17.6496 % 28.7159 % % Note that both upper and lower case letters are allowed for input % (on those systems which have both), but that lower case is % converted to upper case. % % There are two "matrix division" symbols in MATLAB, \ and / . % (If your terminal does not have a backslash, use $ instead, or % see CHAR.) If A and B are matrices, then A\B and B/A correspond % formally to left and right multiplication of B by the inverse of % A , that is inv(A)*B and B*inv(A), but the result is obtained % directly without the computation of the inverse. In the scalar % case, 3\1 and 1/3 have the same value, namely one-third. In % general, A\B denotes the solution X to the equation A*X = B and % B/A denotes the solution to X*A = B. % % Left division, A\B, is defined whenever B has as many rows % as A . If A is square, it is factored using Gaussian % elimination. The factors are used to solve the equations % A*X(:,j) = B(:,j) where B(:,j) denotes the j-th column of B. The % result is a matrix X with the same dimensions as B. If A is % nearly singular (according to the LINPACK condition estimator, % RCOND), a warning message is printed. If A is not square, it is % factored using Householder orthogonalization with column % pivoting. The factors are used to solve the under- or % overdetermined equations in a least squares sense. The result is % an m by n matrix X where m is the number of columns of A and n is % the number of columns of B . Each column of X has at most k % nonzero components, where k is the effective rank of A . % % Right division, B/A, can be defined in terms of left % division by B/A = (A'\B')'. % % For example, since our vector b was computed as A*x, the % statement % % y = A\b % % results in % % Y = % % -1.3000 % 0.8000 % 3.1416 % % Of course, y is not exactly equal to x because of the % roundoff errors involved in both A*x and A\b , but we are not % printing enough digits to see the difference. The result of the % statement % % e = x - y % % depends upon the particular computer being used. In one case it % produces % % E = % % 1.0e-15 * % % .3053 % -.2498 % .0000 % % The quantity 1.0e-15 is a scale factor which multiplies all the % components which follow. Thus our vectors x and y actually % agree to about 15 decimal places on this computer. % % It is also possible to obtain element-by-element % multiplicative operations. If A and B have the same dimensions, % then A .* B denotes the matrix whose elements are simply the % products of the individual elements of A and B . The expressions % A ./ B and A .\ B give the quotients of the individual elements. % % There are several possible output formats. The statement % % long, x % % results in % % X = % -1.300000000000000 % .800000000000000 % 3.141592653589793 % % The statement % % short % % restores the original format. % % The expression A**p means A to the p-th power. It is % defined if A is a square matrix and p is a scalar. If p is an % integer greater than one, the power is computed by repeated % multiplication. For other values of p the calculation involves % the eigenvalues and eigenvectors of A. % % Previously defined matrices and matrix expressions can be % used inside brackets to generate larger matrices, for example % % C = *x, x'> % % results in % % % C = % % 1.0000 2.0000 3.0000 9.7248 % 4.0000 5.0000 6.0000 17.6496 % 7.0000 8.0000 10.0000 28.7159 % -3.6000 -1.3000 0.8000 3.1416 % % % There are four predefined variables, EPS, FLOP, RAND and % EYE. The variable EPS is used as a tolerance is determining such % things as near singularity and rank. Its initial value is the % distance from 1.0 to the next largest floating point number on % the particular computer being used. The user may reset this to % any other value, including zero. EPS is changed by CHOP, which is % described in section 12. % % The value of RAND is a random variable, with a choice of a % uniform or a normal distribution. % % The name EYE is used in place of I to denote identity % matrices because I is often used as a subscript or as sqrt(-1). % The dimensions of EYE are determined by context. For example, % % B = A + 3*EYE % % adds 3 to the diagonal elements of A and % % X = EYE/A % % is one of several ways in MATLAB to invert a matrix. % % FLOP provides a count of the number of floating point % operations, or "flops", required for each calculation. % % A statement may consist of an expression alone, in which % case a variable named ANS is created and the result stored in ANS % for possible future use. Thus % % A\A - EYE % % is the same as % % ANS = A\A - EYE % % (Roundoff error usually causes this result to be a matrix of % "small" numbers, rather than all zeros.) % % All computations are done using either single or double % precision real arithmetic, whichever is appropriate for the % particular computer. There is no mixed-precision arithmetic. % The Fortran COMPLEX data type is not used because many systems % create unnecessary underflows and overflows with complex % operations and because some systems do not allow double precision % complex arithmetic. %% 2. MATLAB functions % % Much of MATLAB's computational power comes from the various % matrix functions available. The current list includes: % % INV(A) - Inverse. % DET(A) - Determinant. % COND(A) - Condition number. % RCOND(A) - A measure of nearness to singularity. % EIG(A) - Eigenvalues and eigenvectors. % SCHUR(A) - Schur triangular form. % HESS(A) - Hessenberg or tridiagonal form. % POLY(A) - Characteristic polynomial. % SVD(A) - Singular value decomposition. % PINV(A,eps) - Pseudoinverse with optional tolerance. % RANK(A,eps) - Matrix rank with optional tolerance. % LU(A) - Factors from Gaussian elimination. % CHOL(A) - Factor from Cholesky factorization. % QR(A) - Factors from Householder orthogonalization. % RREF(A) - Reduced row echelon form. % ORTH(A) - Orthogonal vectors spanning range of A. % EXP(A) - e to the A. % LOG(A) - Natural logarithm. % SQRT(A) - Square root. % SIN(A) - Trigonometric sine. % COS(A) - Cosine. % ATAN(A) - Arctangent. % ROUND(A) - Round the elements to nearest integers. % ABS(A) - Absolute value of the elements. % REAL(A) - Real parts of the elements. % IMAG(A) - Imaginary parts of the elements. % CONJG(A) - Complex conjugate. % SUM(A) - Sum of the elements. % PROD(A) - Product of the elements. % DIAG(A) - Extract or create diagonal matrices. % TRIL(A) - Lower triangular part of A. % TRIU(A) - Upper triangular part of A. % NORM(A,p) - Norm with p = 1, 2 or 'Infinity'. % EYE(m,n) - Portion of identity matrix. % RAND(m,n) - Matrix with random elements. % ONES(m,n) - Matrix of all ones. % MAGIC(n) - Interesting test matrices. % HILBERT(n) - Inverse Hilbert matrices. % ROOTS(C) - Roots of polynomial with coefficients C. % DISPLAY(A,p) - Print base p representation of A. % KRON(A,B) - Kronecker tensor product of A and B. % PLOT(X,Y) - Plot Y as a function of X . % RAT(A) - Find "simple" rational approximation to A. % USER(A) - Function defined by external program. % % Some of these functions have different interpretations when % the argument is a matrix or a vector and some of them have % additional optional arguments. Details are given in the HELP % document in the appendix. % % Several of these functions can be used in a generalized % assignment statement with two or three variables on the left hand % side. For example % % = EIG(A) % % stores the eigenvectors of A in the matrix X and a diagonal % matrix containing the eigenvalues in the matrix D. The statement % % EIG(A) % % simply computes the eigenvalues and stores them in ANS. % % Future versions of MATLAB will probably include additional % functions, since they can easily be added to the system. % %% 3. Rows, columns and submatrices % % % Individual elements of a matrix can be accessed by giving % their subscripts in parentheses, eg. A(1,2), x(i), TAB(ind(k)+1). % An expression used as a subscript is rounded to the nearest % integer. % % Individual rows and columns can be accessed using a colon % ':' (or a '|') for the free subscript. For example, A(1,:) is the % first row of A and A(:,j) is the j-th column. Thus % % A(i,:) = A(i,:) + c*A(k,:) % % adds c times the k-th row of A to the i-th row. % % The colon is used in several other ways in MATLAB, but all % of the uses are based on the following definition. % % j:k is the same as % j:k is empty if j > k . % j:i:k is the same as % j:i:k is empty if i > 0 and j > k or if i < 0 and j < k . % % The colon is usually used with integers, but it is possible to % use arbitrary real scalars as well. Thus % % 1:4 is the same as <1, 2, 3, 4> % 0: 0.1: 0.5 is the same as <0.0, 0.1, 0.2, 0.3, 0.4, 0.5> % % % In general, a subscript can be a vector. If X and V are % vectors, then X(V) is . This can % also be used with matrices. If V has m components and W has n % components, then A(V,W) is the m by n matrix formed from the % elements of A whose subscripts are the elements of V and W. % Combinations of the colon notation and the indirect subscripting % allow manipulation of various submatrices. For example, % % A(<1,5>,:) = A(<5,1>,:) interchanges rows 1 and 5 of A. % A(2:k,1:n) is the submatrix formed from rows 2 through k % and columns 1 through n of A . % A(:,<3 1 2>) is a permutation of the first three columns. % % % The notation A(:) has a special meaning. On the right hand % side of an assignment statement, it denotes all the elements of % A, regarded as a single column. When an expression is assigned % to A(:), the current dimensions of A, rather than of the % expression, are used. %% 4. FOR, WHILE and IF % % % The FOR clause allows statements to be repeated a specific % number of times. The general form is % % FOR variable = expr, statement, ..., statement, END % % The END and the comma before it may be omitted. In general, the % expression may be a matrix, in which case the columns are stored % one at a time in the variable and the following statements, up to % the END or the end of the line, are executed. The expression is % often of the form j:k, and its "columns" are simply the scalars % from j to k. Some examples (assume n has already been assigned a % value): % % for i = 1:n, for j = 1:n, A(i,j) = 1/(i+j-1); % % generates the Hilbert matrix. % % for j = 2:n-1, for i = j:n-1, ... % A(i,j) = 0; end; A(j,j) = j; end; A % % changes all but the "outer edge" of the lower triangle and then % prints the final matrix. % % for h = 1.0: -0.1: -1.0, () % % prints a table of cosines. % % = EIG(A); for v = X, v, A*v % % displays eigenvectors, one at a time. % % The WHILE clause allows statements to be repeated an % indefinite number of times. The general form is % % WHILE expr relop expr, statement,..., statement, END % % where relop is =, <, >, <=, >=, or <> (not equal) . The % statements are repeatedly executed as long as the indicated % comparison between the real parts of the first components of the % two expressions is true. Here are two examples. (Exercise for % the reader: What do these segments do?) % % eps = 1; % while 1 + eps > 1, eps = eps/2; % eps = 2*eps % % E = 0*A; F = E + EYE; n = 1; % while NORM(E+F-E,1) > 0, E = E + F; F = A*F/n; n = n + 1; % E % % % The IF clause allows conditional execution of statements. % The general form is % % IF expr relop expr, statement, ..., statement, % ELSE statement, ..., statement % % The first group of statements are executed if the relation is % true and the second group are executed if the relation is false. % The ELSE and the statements following it may be omitted. For % example, % % if abs(i-j) = 2, A(i,j) = 0; %% 5. Commands, text, files and macros. % % % MATLAB has several commands which control the output format % and the overall execution of the system. % % The HELP command allows on-line access to short portions of % text describing various operations, functions and special % characters. The entire HELP document is reproduced in an % appendix. % % Results are usually printed in a scaled fixed point format % that shows 4 or 5 significant figures. The commands SHORT, LONG, % SHORT E, LONG E and LONG Z alter the output format, but do not % alter the precision of the computations or the internal storage. % % The WHO, WHAT and WHY commands provide information about the % functions and variables that are currently defined. % % The CLEAR command erases all variables, except EPS, FLOP, % RAND and EYE. The statement A = <> indicates that a "0 by 0" % matrix is to be stored in A. This causes A to be erased so that % its storage can be used for other variables. % % The RETURN and EXIT commands cause return to the underlying % operating system through the Fortran RETURN statement. % % MATLAB has a limited facility for handling text. Any string % of characters delineated by quotes (with two quotes used to allow % one quote within the string) is saved as a vector of integer % values with '1' = 1, 'A' = 10, ' ' = 36, etc. (The complete list % is in the appendix under CHAR.) For example % % '2*A + 3' is the same as <2 43 10 36 41 36 3> % % It is possible, though seldom very meaningful, to use such % strings in matrix operations. More frequently, the text is used % as a special argument to various functions. % % NORM(A,'inf') computes the infinity norm of A . % DISPLAY(T) prints the text stored in T . % EXEC('file') obtains MATLAB input from an external file. % SAVE('file') stores all the current variables in a file. % LOAD('file') retrieves all the variables from a file. % PRINT('file',X) prints X on a file. % DIARY('file') makes a copy of the complete MATLAB session. % % The text can also be used in a limited string substitution % macro facility. If a variable, say T, contains the source text % for a MATLAB statement or expression, then the construction % % > T < % % causes T to be executed or evaluated. For example % % T = '2*A + 3'; % S = 'B = >T< + 5' % A = 4; % > S < % % produces % % B = % % 16. % % Some other examples are given under MACRO in the appendix. This % facility is useful for fairly short statements and expressions. % More complicated MATLAB "programs" should use the EXEC facility. % % The operations which access external files cannot be handled % in a completely machine-independent manner by portable Fortran % code. It is necessary for each particular installation to % provide a subroutine which associates external text files with % Fortran logical unit numbers. %% 6. Census example % % % Our first extended example involves predicting the % population of the United States in 1980 using extrapolation of % various fits to the census data from 1900 through 1970. There % are eight observations, so we begin with the MATLAB statement % % n = 8 % % The values of the dependent variable, the population in millions, % can be entered with % % y = < 75.995 91.972 105.711 123.203 ... % 131.669 150.697 179.323 203.212>' % % In order to produce a reasonably scaled matrix, the independent % variable, time, is transformed from the interval [1900,1970] to % [-1.00,0.75]. This can be accomplished directly with % % t = -1.0:0.25:0.75 % % or in a fancier, but perhaps clearer, way with % % t = 1900:10:1970; t = (t - 1940*ones(t))/40 % % Either of these is equivalent to % % t = <-1 -.75 -.50 -.25 0 .25 .50 .75> % % The interpolating polynomial of degree n-1 involves an % Vandermonde matrix of order n with elements that might be % generated by % % for i = 1:n, for j = 1:n, a(i,j) = t(i)**(j-1); % % However, this results in an error caused by 0**0 when i = 5 and % j = 1 . The preferable approach is % % A = ones(n,n); % for i = 1:n, for j = 2:n, a(i,j) = t(i)*a(i,j-1); % % Now the statement % % cond(A) % % produces the output % % ANS = % % 1.1819E+03 % % which indicates that transformation of the time variable has % resulted in a reasonably well conditioned matrix. % % The statement % % c = A\y % % results in % % C = % % 131.6690 % 41.0406 % 103.5396 % 262.4535 % -326.0658 % -662.0814 % 341.9022 % 533.6373 % % These are the coefficients in the interpolating polynomial % % n-1 % c + c t + ... + c t % 1 2 n % % Our transformation of the time variable has resulted in t = 1 % corresponding to the year 1980. Consequently, the extrapolated % population is simply the sum of the coefficients. This can be % computed by % % p = sum(c) % % The result is % % P = % % 426.0950 % % which indicates a 1980 population of over 426 million. Clearly, % using the seventh degree interpolating polynomial to extrapolate % even a fairly short distance beyond the end of the data interval % is not a good idea. % % The coefficients in least squares fits by polynomials of % lower degree can be computed using fewer than n columns of the % matrix. % % for k = 1:n, c = A(:,1:k)\y, p = sum(c) % % would produce the coefficients of these fits, as well as the % resulting extrapolated population. If we do not want to print % all the coefficients, we can simply generate a small table of % populations predicted by polynomials of degrees zero through % seven. We also compute the maximum deviation between the fitted % and observed values. % % for k = 1:n, X = A(:,1:k); c = X\y; ... % d(k) = k-1; p(k) = sum(c); e(k) = norm(X*c-y,'inf'); % % % The resulting output is % % 0 132.7227 70.4892 % 1 211.5101 9.8079 % 2 227.7744 5.0354 % 3 241.9574 3.8941 % 4 234.2814 4.0643 % 5 189.7310 2.5066 % 6 118.3025 1.6741 % 7 426.0950 0.0000 % % The zeroth degree fit, 132.7 million, is the result of fitting a % constant to the data and is simply the average. The results % obtained with polynomials of degree one through four all appear % reasonable. The maximum deviation of the degree four fit is % slightly greater than the degree three, even though the sum of % the squares of the deviations is less. The coefficients of the % highest powers in the fits of degree five and six turn out to be % negative and the predicted populations of less than 200 million % are probably unrealistic. The hopefully absurd prediction of the % interpolating polynomial concludes the table. % % We wish to emphasize that roundoff errors are not % significant here. Nearly identical results would be obtained on % other computers, or with other algorithms. The results simply % indicate the difficulties associated with extrapolation of % polynomial fits of even modest degree. % % A stabilized fit by a seventh degree polynomial can be % obtained using the pseudoinverse, but it requires a fairly % delicate choice of a tolerance. The statement % % s = svd(A) % % produces the singular values % % S = % % 3.4594 % 2.2121 % 1.0915 % 0.4879 % 0.1759 % 0.0617 % 0.0134 % 0.0029 % % We see that the last three singular values are less than 0.1 , % consequently, A can be approximately by a matrix of rank five % with an error less than 0.1 . The Moore-Penrose pseudoinverse of % this rank five matrix is obtained from the singular value % decomposition with the following statements % % c = pinv(A,0.1)*y, p = sum(c), e = norm(a*c-y,'inf') % % The output is % % % % % % % % % % % % % C = % % 134.7972 % 67.5055 % 23.5523 % 9.2834 % 3.0174 % 2.6503 % -2.8808 % 3.2467 % % P = % % 241.1720 % % E = % % 3.9469 % % The resulting seventh degree polynomial has coefficients which % are much smaller than those of the interpolating polynomial given % earlier. The predicted population and the maximum deviation are % reasonable. Any choice of the tolerance between the fifth and % sixth singular values would produce the same results, but choices % outside this range result in pseudoinverses of different rank and % do not work as well. % % The one term exponential approximation % % y(t) = k exp(pt) % % can be transformed into a linear approximation by taking % logarithms. % % log(y(t)) = log k + pt % % = c + c t % 1 2 % % The following segment makes use of the fact that a function of a % vector is the function applied to the individual components. % % X = A(:,1:2); % c = X\log(y) % p = exp(sum(c)) % e = norm(exp(X*c)-y,'inf') % % The resulting output is % % % % % % C = % % 4.9083 % 0.5407 % % P = % % 232.5134 % % E = % % 4.9141 % % The predicted population and maximum deviation appear % satisfactory and indicate that the exponential model is a % reasonable one to consider. % % As a curiousity, we return to the degree six polynomial. % Since the coefficient of the high order term is negative and the % value of the polynomial at t = 1 is positive, it must have a root % at some value of t greater than one. The statements % % X = A(:,1:7); % c = X\y; % c = c(7:-1:1); //reverse the order of the coefficients % z = roots(c) % % produce % % Z = % % 1.1023- 0.0000*i % 0.3021+ 0.7293*i % -0.8790+ 0.6536*i % -1.2939- 0.0000*i % -0.8790- 0.6536*i % 0.3021- 0.7293*i % % There is only one real, positive root. The corresponding time on % the original scale is % % 1940 + 40*real(z(1)) % % = 1984.091 % % We conclude that the United States population should become zero % early in February of 1984. % % % % %% 7. Partial differential equation example % % % Our second extended example is a boundary value problem for % Laplace's equation. The underlying physical problem involves the % conductivity of a medium with cylindrical inclusions and is % considered by Keller and Sachs [7]. % % Find a function u(x,y) satisfying Laplace's equation % % u + u = 0 % xx yy % % The domain is a unit square with a quarter circle of radius rho % removed from one corner. There are Neumann conditions on the top % and bottom edges and Dirichlet conditions on the remainder of the % boundary. % % % u = 0 % n % % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH- % | . % | . % | . % | . u = 1 % | . % | . % | . % u = 0 | | % | | % | | % | | u = 1 % | | % | | % | | % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % % u = 0 % n % % % The effective conductivity of an medium is then given by the % integral along the left edge, % % 1 % sigma = integral u (0,y) dy % 0 n % % It is of interest to study the relation between the radius rho % and the conductivity sigma. In particular, as rho approaches % one, sigma becomes infinite. % Keller and Sachs use a finite difference approximation. The % following technique makes use of the fact that the equation is % actually Laplace's equation and leads to a much smaller matrix % problem to solve. % % Consider an approximate solution of the form % % n 2j-1 % u = sum c r cos(2j-1)t % j=1 j % % where r,t are polar coordinates (t is theta). The coefficients % are to be determined. For any set of coefficients, this function % already satisfies the differential equation because the basis % functions are harmonic; it satisfies the normal derivative % boundary condition on the bottom edge of the domain because we % used cos t in preference to sin t ; and it satisfies the % boundary condition on the left edge of the domain because we use % only odd multiples of t . % % The computational task is to find coefficients so that the % boundary conditions on the remaining edges are satisfied as well % as possible. To accomplish this, pick m points (r,t) on the % remaining edges. It is desirable to have m > n and in practice % we usually choose m to be two or three times as large as n . % Typical values of n are 10 or 20 and of m are 20 to 60. An % m by n matrix A is generated. The i,j element is the j-th % basis function, or its normal derivative, evaluated at the i-th % boundary point. A right hand side with m components is also % generated. In this example, the elements of the right hand side % are either zero or one. The coefficients are then found by % solving the overdetermined set of equations % % Ac = b % % in a least squares sense. % % Once the coefficients have been determined, the approximate % solution is defined everywhere on the domain. It is then % possible to compute the effective conductivity sigma . In fact, % a very simple formula results, % % n j-1 % sigma = sum (-1) c % j=1 j % % To use MATLAB for this problem, the following "program" is % first stored in the local computer file system, say under the % name "PDE". % % % % % //Conductivity example. % //Parameters REPLACE_WITH_DASH_DASH- % rho //radius of cylindrical inclusion % n //number of terms in solution % m //number of boundary points % //initialize operation counter % flop = <0 0>; % //initialize variables % m1 = round(m/3); //number of points on each straight edge % m2 = m - m1; //number of points with Dirichlet conditions % pi = 4*atan(1); % //generate points in Cartesian coordinates % //right hand edge % for i = 1:m1, x(i) = 1; y(i) = (1-rho)*(i-1)/(m1-1); % //top edge % for i = m2+1:m, x(i) = (1-rho)*(m-i)/(m-m2-1); y(i) = 1; % //circular edge % for i = m1+1:m2, t = pi/2*(i-m1)/(m2-m1+1); ... % x(i) = 1-rho*sin(t); y(i) = 1-rho*cos(t); % //convert to polar coordinates % for i = 1:m-1, th(i) = atan(y(i)/x(i)); ... % r(i) = sqrt(x(i)**2+y(i)**2); % th(m) = pi/2; r(m) = 1; % //generate matrix % //Dirichlet conditions % for i = 1:m2, for j = 1:n, k = 2*j-1; ... % a(i,j) = r(i)**k*cos(k*th(i)); % //Neumann conditions % for i = m2+1:m, for j = 1:n, k = 2*j-1; ... % a(i,j) = k*r(i)**(k-1)*sin((k-1)*th(i)); % //generate right hand side % for i = 1:m2, b(i) = 1; % for i = m2+1:m, b(i) = 0; % //solve for coefficients % c = A\b % //compute effective conductivity % c(2:2:n) = -c(2:2:n); % sigma = sum(c) % //output total operation count % ops = flop(2) % % % % % The program can be used within MATLAB by setting the three % parameters and then accessing the file. For example, % % rho = .9; % n = 15; % m = 30; % exec('PDE') % % The resulting output is % RHO = % % .9000 % % N = % % 15. % % M = % % 30. % % C = % % 2.2275 % -2.2724 % 1.1448 % 0.1455 % -0.1678 % -0.0005 % -0.3785 % 0.2299 % 0.3228 % -0.2242 % -0.1311 % 0.0924 % 0.0310 % -0.0154 % -0.0038 % % SIGM = % % 5.0895 % % OPS = % % 16204. % % % A total of 16204 floating point operations were necessary to % set up the matrix, solve for the coefficients and compute the % conductivity. The operation count is roughly proportional to % m*n**2. The results obtained for sigma as a function of rho by % this approach are essentially the same as those obtained by the % finite difference technique of Keller and Sachs, but the % computational effort involved is much less. % % % % % %% 8. Eigenvalue sensitivity example % % % In this example, we construct a matrix whose eigenvalues are % moderately sensitive to perturbations and then analyze that % sensitivity. We begin with the statement % % B = <3 0 7; 0 2 0; 0 0 1> % % which produces % % B = % % 3. 0. 7. % 0. 2. 0. % 0. 0. 1. % % % Obviously, the eigenvalues of B are 1, 2 and 3 . Moreover, % since B is not symmetric, these eigenvalues are slightly % sensitive to perturbation. (The value b(1,3) = 7 was chosen so % that the elements of the matrix A below are less than 1000.) % % We now generate a similarity transformation to disguise the % eigenvalues and make them more sensitive. % % L = <1 0 0; 2 1 0; -3 4 1>, M = L\L' % % L = % % 1. 0. 0. % 2. 1. 0. % -3. 4. 1. % % M = % % 1.0000 2.0000 -3.0000 % -2.0000 -3.0000 10.0000 % 11.0000 18.0000 -48.0000 % % The matrix M has determinant equal to 1 and is moderately badly % conditioned. The similarity transformation is % % A = M*B/M % % A = % % -64.0000 82.0000 21.0000 % 144.0000 -178.0000 -46.0000 % -771.0000 962.0000 248.0000 % % Because det(M) = 1 , the elements of A would be exact integers % if there were no roundoff. So, % A = round(A) % % A = % % -64. 82. 21. % 144. -178. -46. % -771. 962. 248. % % % This, then, is our test matrix. We can now forget how it % was generated and analyze its eigenvalues. % % = eig(A) % % D = % % 3.0000 0.0000 0.0000 % 0.0000 1.0000 0.0000 % 0.0000 0.0000 2.0000 % % X = % % -.0891 3.4903 41.8091 % .1782 -9.1284 -62.7136 % -.9800 46.4473 376.2818 % % Since A is similar to B, its eigenvalues are also 1, 2 and 3. % They happen to be computed in another order by the EISPACK % subroutines. The fact that the columns of X, which are the % eigenvectors, are so far from being orthonormal is our first % indication that the eigenvalues are sensitive. To see this % sensitivity, we display more figures of the computed eigenvalues. % % long, diag(D) % % ANS = % % 2.999999999973599 % 1.000000000015625 % 2.000000000011505 % % We see that, on this computer, the last five significant figures % are contaminated by roundoff error. A somewhat superficial % explanation of this is provided by % % short, cond(X) % % ANS = % % 3.2216e+05 % % The condition number of X gives an upper bound for the relative % error in the computed eigenvalues. However, this condition % number is affected by scaling. % % X = X/diag(X(3,:)), cond(X) % % X = % % .0909 .0751 .1111 % -.1818 -.1965 -.1667 % 1.0000 1.0000 1.0000 % % ANS = % % 1.7692e+03 % % % Rescaling the eigenvectors so that their last components are % all equal to one has two consequences. The condition of X is % decreased by over two orders of magnitude. (This is about the % minimum condition that can be obtained by such diagonal scaling.) % Moreover, it is now apparent that the three eigenvectors are % nearly parallel. % % More detailed information on the sensitivity of the % individual eigenvalues involves the left eigenvectors. % % Y = inv(X'), Y'*A*X % % Y = % % -511.5000 259.5000 252.0000 % 616.0000 -346.0000 -270.0000 % 159.5000 -86.5000 -72.0000 % % ANS = % % 3.0000 .0000 .0000 % .0000 1.0000 .0000 % .0000 .0000 2.0000 % % We are now in a position to compute the sensitivities of the % individual eigenvalues. % % for j = 1:3, c(j) = norm(Y(:,j))*norm(X(:,j)); end, C % % C = % % 833.1092 % 450.7228 % 383.7564 % % These three numbers are the reciprocals of the cosines of the % angles between the left and right eigenvectors. It can be shown % that perturbation of the elements of A can result in a % perturbation of the j-th eigenvalue which is c(j) times as large. % In this example, the first eigenvalue has the largest % sensitivity. % % We now proceed to show that A is close to a matrix with a % double eigenvalue. The direction of the required perturbation is % given by % % E = -1.e-6*Y(:,1)*X(:,1)' % % E = % % 1.0e-03 * % % .0465 -.0930 .5115 % -.0560 .1120 -.6160 % -.0145 .0290 -.1595 % % With some trial and error which we do not show, we bracket the % point where two eigenvalues of a perturbed A coalesce and then % become complex. % % eig(A + .4*E), eig(A + .5*E) % % ANS = % % 1.1500 % 2.5996 % 2.2504 % % ANS = % % 2.4067 + .1753*i % 2.4067 - .1753*i % 1.1866 + 0.0000*i % % Now, a bisecting search, driven by the imaginary part of one of % the eigenvalues, finds the point where two eigenvalues are nearly % equal. % % r = .4; s = .5; % % while s-r > 1.e-14, t = (r+s)/2; d = eig(A+t*E); ... % if imag(d(1))=0, r = t; else, s = t; % % long, T % % T = % % .450380734134507 % % % Finally, we display the perturbed matrix, which is obviously % close to the original, and its pair of nearly equal eigenvalues. % (We have dropped a few digits from the long output.) % % A+t*E, eig(A+t*E) % % A % % -63.999979057 81.999958114 21.000230369 % 143.999974778 -177.999949557 -46.000277434 % -771.000006530 962.000013061 247.999928164 % % ANS = % % 2.415741150 % 2.415740621 % 1.168517777 % % % The first two eigenvectors of A + t*E are almost % indistinguishable indicating that the perturbed matrix is almost % defective. % % = eig(A+t*E); X = X/diag(X(3,:)) % % X = % % .096019578 .096019586 .071608466 % -.178329614 -.178329608 -.199190520 % 1.000000000 1.000000000 1.000000000 % % short, cond(X) % % ANS = % % 3.3997e+09 %% 9. Syntax diagrams % % % A formal description of the language acceptable to MATLAB, % as well as a flow chart of the MATLAB program, is provided by the % syntax diagrams or syntax graphs of Wirth [6]. There are eleven % non-terminal symbols in the language: % % line, statement, clause, expression, term, % factor, number, integer, name, command, text . % % The diagrams define each of the non-terminal symbols using the % others and the terminal symbols: % % letter REPLACE_WITH_DASH_DASH A through Z, % digit REPLACE_WITH_DASH_DASH 0 through 9, % char REPLACE_WITH_DASH_DASH ( ) ; : + - * / \ = . , < > % quote REPLACE_WITH_DASH_DASH ' % % % line % % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> statement >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| % | | % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> clause >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| % | | % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> expr >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> % | | | | % | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> command >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| | % | | | | % | |-> > >-> expr >-> < >-| | % | | | | % | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| | % | | % | |-< ; <-| | % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| % |-< , <-| % % % % % statement % % |-> name >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| % | | | % | | |REPLACE_WITH_DASH_DASH> : >REPLACE_WITH_DASH_DASH-| | % | | | | | % | |-> ( >REPLACE_WITH_DASH_DASH-|-> expr >-|REPLACE_WITH_DASH_DASH-> ) >-| % | | | | % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-< , = >REPLACE_WITH_DASH_DASH> expr >REPLACE_WITH_DASH_DASH-> % | | % | |REPLACE_WITH_DASH_DASH< , < >REPLACE_WITH_DASH_DASH-> name >REPLACE_WITH_DASH_DASH-> > >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| % % % % % % % % % % % % % % % % clause % % |REPLACE_WITH_DASH_DASH-> FOR >REPLACE_WITH_DASH_DASH-> name >REPLACE_WITH_DASH_DASH-> = >REPLACE_WITH_DASH_DASH-> expr >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| % | | % | |-> WHILE >-| | % |-| |-> expr >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH | % | |-> IF >-| | | | | | | | % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| < <= = <> >= > |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> % | | | | | | | | % | REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> expr >REPLACE_WITH_DASH_DASH| % | | % |REPLACE_WITH_DASH_DASH-> ELSE >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| % | | % |REPLACE_WITH_DASH_DASH-> END >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| % % % % % expr % % |-> + >-| % | | % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> term >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> % | | | | % |-> - >-| | |-< + <-| | % | | | | % |REPLACE_WITH_DASH_DASH|-< - <-|REPLACE_WITH_DASH_DASH| % | | % |-< : <-| % % % % % term % % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> factor >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> % | | % | |-< * <-| | % | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| | | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| | % |REPLACE_WITH_DASH_DASH| |REPLACE_WITH_DASH_DASH|-< / <-|REPLACE_WITH_DASH_DASH| |REPLACE_WITH_DASH_DASH| % |-< . <-| | | |-< . <-| % |-< \ <-| % % % % % % % % % % % % factor % % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> number >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| % | | % |-> name >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| % | | | % | | |REPLACE_WITH_DASH_DASH> : >REPLACE_WITH_DASH_DASH-| | % | | | | | % | |-> ( >REPLACE_WITH_DASH_DASH-|-> expr >-|REPLACE_WITH_DASH_DASH-> ) >-| % | | | | % | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-< , ( >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> expr >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> ) >-|-|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> % | | | | | % | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| | |-> ' >-| | % | | | | | % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> < >-|REPLACE_WITH_DASH_DASH-> expr >REPLACE_WITH_DASH_DASH-|-> > >-| | % | | | | | % | |REPLACE_WITH_DASH_DASH< > >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> expr >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> < >-| | % | | | % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> factor >REPLACE_WITH_DASH_DASH-> ** >REPLACE_WITH_DASH_DASH-> factor >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| | % | | % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> ' >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> text >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> ' >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| % % % % % number % % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| |-> + >-| % | | | | % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> int >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> . >REPLACE_WITH_DASH_DASH-> int >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> E >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> int >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> % | | | | | | % | | | |-> - >-| | % | | | | % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| % % % % % int % % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> digit >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> % | | % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| % % % % % name % % |REPLACE_WITH_DASH_DASH< letter letter >REPLACE_WITH_DASH_DASH|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-> % | | % |REPLACE_WITH_DASH_DASH< digit name >REPLACE_WITH_DASH_DASH| % | | % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> name >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> % | | % |REPLACE_WITH_DASH_DASH> char >REPLACE_WITH_DASH_DASH| % | | % |REPLACE_WITH_DASH_DASH-> ' >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| % % text % % |-> letter >REPLACE_WITH_DASH_DASH| % | | % |-> digit >REPLACE_WITH_DASH_DASH-| % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH> % | |-> char >REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH| | % | | | | % | |-> ' >-> ' >-| | % | | % |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| %% 10. The parser-interpreter % % % The structure of the parser-interpreter is similar to that % of Wirth's compiler [6] for his simple language, PL/0 , except % that MATLAB is programmed in Fortran, which does not have % explicit recursion. The interrelation of the primary subroutines % is shown in the following diagram. % % % % % % % % % % MAIN % | % MATLAB |REPLACE_WITH_DASH_DASHCLAUSE % | | | % PARSEREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-|REPLACE_WITH_DASH_DASHEXPRREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHTERMREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHFACTOR % | | | | % | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-|REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-| % | | | | % | STACK1 STACK2 STACKG % | % |REPLACE_WITH_DASH_DASHSTACKPREPLACE_WITH_DASH_DASHPRINT % | % |REPLACE_WITH_DASH_DASHCOMAND % | % | % | |REPLACE_WITH_DASH_DASHCGECO % | | % | |REPLACE_WITH_DASH_DASHCGEFA % | | % |REPLACE_WITH_DASH_DASHMATFN1REPLACE_WITH_DASH_DASH|REPLACE_WITH_DASH_DASHCGESL % | | % | |REPLACE_WITH_DASH_DASHCGEDI % | | % | |REPLACE_WITH_DASH_DASHCPOFA % | % | % | |REPLACE_WITH_DASH_DASHIMTQL2 % | | % | |REPLACE_WITH_DASH_DASHHTRIDI % | | % |REPLACE_WITH_DASH_DASHMATFN2REPLACE_WITH_DASH_DASH|REPLACE_WITH_DASH_DASHHTRIBK % | | % | |REPLACE_WITH_DASH_DASHCORTH % | | % | |REPLACE_WITH_DASH_DASHCOMQR3 % | % | % |REPLACE_WITH_DASH_DASHMATFN3REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH-CSVDC % | % | % | |REPLACE_WITH_DASH_DASHCQRDC % |REPLACE_WITH_DASH_DASHMATFN4REPLACE_WITH_DASH_DASH| % | |REPLACE_WITH_DASH_DASHCQRSL % | % | % | |REPLACE_WITH_DASH_DASHFILES % |REPLACE_WITH_DASH_DASHMATFN5REPLACE_WITH_DASH_DASH| % |REPLACE_WITH_DASH_DASHSAVLOD % % Subroutine PARSE controls the interpretation of each % statement. It calls subroutines that process the various % syntactic quantities such as command, expression, term and % factor. A fairly simple program stack mechanism allows these % subroutines to recursively "call" each other along the lines % allowed by the syntax diagrams. The four STACK subroutines % manage the variable memory and perform elementary operations, % such as matrix addition and transposition. % % The four subroutines MATFN1 though MATFN4 are called % whenever "serious" matrix computations are required. They are % interface routines which call the various LINPACK and EISPACK % subroutines. MATFN5 primarily handles the file access tasks. % % Two large real arrays, STKR and STKI, are used to store all % the matrices. Four integer arrays are used to store the names, % the row and column dimensions, and the pointers into the real % stacks. The following diagram illustrates this storage scheme. % % TOP IDSTK MSTK NSTK LSTK STKR STKI % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | |REPLACE_WITH_DASH_DASH->| | | | | | | | | | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH->| | | | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | | | | | | | | | | | | | | | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % . . . . . . % . . . . . . % . . . . . . % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % BOT | | | | | | | | | | | | | | | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | |REPLACE_WITH_DASH_DASH->| X| | | | | 2| | 1| | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH->| 3.14 | | 0.00 | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | A| | | | | 2| | 2| | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH- | 0.00 | | 1.00 | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH \ REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | E| P| S| | | 1| | 1| | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH- ->| 11.00 | | 0.00 | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH \ REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | F| L| O| P| | 1| | 2| | |REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH \ | 21.00 | | 0.00 | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH \ \ REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | E| Y| E| | |-1| |-1| | |REPLACE_WITH_DASH_DASH- \ | | 12.00 | | 0.00 | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH \ | | REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | R| A| N| D| | 1| | 1| | |- \ | | | 22.00 | | 0.00 | % REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASH \ | \ \ REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % | \ \ ->| 1.E-15 | | 0.00 | % \ \ \ REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % \ \ ->| 0.00 | | 0.00 | % \ \ REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % \ \ | 0.00 | | 0.00 | % \ \ REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % \ ->| 1.00 | | 0.00 | % \ REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % REPLACE_WITH_DASH_DASH->| URAND | | 0.00 | % REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH REPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASHREPLACE_WITH_DASH_DASH % % The top portion of the stack is used for temporary variables % and the bottom portion for saved variables. The figure shows the % situation after the line % A = <11,12; 21,22>, x = <3.14, sqrt(-1)>' % % has been processed. The four permanent names, EPS, FLOP, RAND % and EYE, occupy the last four positions of the variable stacks. % RAND has dimensions 1 by 1, but whenever its value is requested, % a random number generator is used instead. EYE has dimensions -1 % by -1 to indicate that the actual dimensions must be determined % later by context. The two saved variables have dimensions 2 by 2 % and 2 by 1 and so take up a total of 6 locations. % % Subsequent statements involving A and x will result in % temporary copies being made in the top of the stack for use in % the actual calculations. Whenever the top of the stack reaches % the bottom, a message indicating memory has been exceeded is % printed, but the current variables are not affected. % % This modular structure makes it possible to implement MATLAB % on a system with a limited amount of memory. The object code for % the MATFN's and the LINPACK-EISPACK subroutines is rarely needed. % Although it is not standard, many Fortran operating systems % provide some overlay mechanism so that this code is brought into % the main memory only when required. The variables, which occupy % a relatively small portion of the memory, remain in place, while % the subroutines which process them are loaded a few at a time. %% 11. The numerical algorithms % % % The algorithms underlying the basic MATLAB functions are % described in the LINPACK and EISPACK guides [1-3]. The following % list gives the subroutines used by these functions. % % INV(A) - CGECO,CGEDI % DET(A) - CGECO,CGEDI % LU(A) - CGEFA % RCOND(A) - CGECO % CHOL(A) - CPOFA % SVD(A) - CSVDC % COND(A) - CSVDC % NORM(A,2) - CSVDC % PINV(A,eps) - CSVDC % RANK(A,eps) - CSVDC % QR(A) - CQRDC,CQRSL % ORTH(A) - CQRDC,CSQSL % A\B and B/A - CGECO,CGESL if A is square. % - CQRDC,CQRSL if A is not square. % EIG(A) - HTRIDI,IMTQL2,HTRIBK if A is Hermitian. % - CORTH,COMQR2 if A is not Hermitian. % SCHUR(A) - same as EIG. % HESS(A) - same as EIG. % % % Minor modifications were made to all these subroutines. The % LINPACK routines were changed to replace the Fortran complex % arithmetic with explicit references to real and imaginary parts. % Since most of the floating point arithmetic is concentrated in a % few low-level subroutines which perform vector operations (the % Basic Linear Algebra Subprograms), this was not an extensive % change. It also facilitated implementation of the FLOP and CHOP % features which count and optionally truncate each floating point % operation. % % The EISPACK subroutine COMQR2 was modified to allow access % to the Schur triangular form, ordinarily just an intermediate % result. IMTQL2 was modified to make computation of the % eigenvectors optional. Both subroutines were modified to % eliminate the machine-dependent accuracy parameter and all the % EISPACK subroutines were changed to include FLOP and CHOP. % % The algorithms employed for the POLY and ROOTS functions % illustrate an interesting aspect of the modern approach to % eigenvalue computation. POLY(A) generates the characteristic % polynomial of A and ROOTS(POLY(A)) finds the roots of that % polynomial, which are, of course, the eigenvalues of A . But both % POLY and ROOTS use EISPACK eigenvalues subroutines, which are % based on similarity transformations. So the classical approach % which characterizes eigenvalues as roots of the characteristic % polynomial is actually reversed. % % If A is an n by n matrix, POLY(A) produces the coefficients % C(1) through C(n+1), with C(1) = 1, in % % DET(z*EYE-A) = C(1)*z**n + ... + C(n)*z + C(n+1) . % % The algorithm can be expressed compactly using MATLAB: % % Z = EIG(A); % C = 0*ONES(n+1,1); C(1) = 1; % for j = 1:n, C(2:j+1) = C(2:j+1) - Z(j)*C(1:j); % C % % This recursion is easily derived by expanding the product % % (z - z(1))*(z - z(2))* ... * (z-z(n)) . % % It is possible to prove that POLY(A) produces the coefficients in % the characteristic polynomial of a matrix within roundoff error % of A . This is true even if the eigenvalues of A are badly % conditioned. The traditional algorithms for obtaining the % characteristic polynomial which do not use the eigenvalues do not % have such satisfactory numerical properties. % % If C is a vector with n+1 components, ROOTS(C) finds the % roots of the polynomial of degree n , % % p(z) = C(1)*z**n + ... + C(n)*z + C(n+1) . % % The algorithm simply involves computing the eigenvalues of the % companion matrix: % % A = 0*ONES(n,n) % for j = 1:n, A(1,j) = -C(j+1)/C(1); % for i = 2:n, A(i,i-1) = 1; % EIG(A) % % It is possible to prove that the results produced are the exact % eigenvalues of a matrix within roundoff error of the companion % matrix A, but this does not mean that they are the exact roots of % a polynomial with coefficients within roundoff error of those in % C . There are more accurate, more efficient methods for finding % polynomial roots, but this approach has the crucial advantage % that it does not require very much additional code. % % The elementary functions EXP, LOG, SQRT, SIN, COS and ATAN % are applied to square matrices by diagonalizing the matrix, % applying the functions to the individual eigenvalues and then % transforming back. For example, EXP(A) is computed by % % = EIG(A); % for j = 1:n, D(j,j) = EXP(D(j,j)); % X*D/X % % This is essentially method number 14 out of the 19 'dubious' % possibilities described in [8]. It is dubious because it doesn't % always work. The matrix of eigenvectors X can be arbitrarily % badly conditioned and all accuracy lost in the computation of % X*D/X. A warning message is printed if RCOND(X) is very small, % but this only catches the extreme cases. An example of a case % not detected is when A has a double eigenvalue, but theoretically % only one linearly independent eigenvector associated with it. % The computed eigenvalues will be separated by something on the % order of the square root of the roundoff level. This separation % will be reflected in RCOND(X) which will probably not be small % enough to trigger the error message. The computed EXP(A) will be % accurate to only half precision. Better methods are known for % computing EXP(A), but they do not easily extend to the other five % functions and would require a considerable amount of additional % code. % % The expression A**p is evaluated by repeated multiplication % if p is an integer greater than 1. Otherwise it is evaluated by % % = EIG(A); % for j = 1:n, D(j,j) = EXP(p*LOG(D(j,j))) % X*D/X % % This suffers from the same potential loss of accuracy if X is % badly conditioned. It was partly for this reason that the case p % = 1 is included in the general case. Comparison of A**1 with A % gives some idea of the loss of accuracy for other values of p and % for the elementary functions. % % RREF, the reduced row echelon form, is of some interest in % theoretical linear algebra, although it has little computational % value. It is included in MATLAB for pedagogical reasons. The % algorithm is essentially Gauss-Jordan elimination with detection % of negligible columns applied to rectangular matrices. % % There are three separate places in MATLAB where the rank of % a matrix is implicitly computed: in RREF(A), in A\B for non- % square A, and in the pseudoinverse PINV(A). Three different % algorithms with three different criteria for negligibility are % used and so it is possible that three different values could be % produced for the same matrix. With RREF(A), the rank of A is the % number of nonzero rows. The elimination algorithm used for RREF % is the fastest of the three rank-determining algorithms, but it % is the least sophisticated numerically and the least reliable. % With A\B, the algorithm is essentially that used by example % subroutine SQRST in chapter 9 of the LINPACK guide. With % PINV(A), the algorithm is based on the singular value % decomposition and is described in chapter 11 of the LINPACK % guide. The SVD algorithm is the most time-consuming, but the % most reliable and is therefore also used for RANK(A). % % The uniformly distributed random numbers in RAND are % obtained from the machine-independent random number generator % URAND described in [9]. It is possible to switch to normally % distributed random numbers, which are obtained using a % transformation also described in [9]. % % The computation of % % 2 2 % sqrt(a + b ) % % is required in many matrix algorithms, particularly those % involving complex arithmetic. A new approach to carrying out % this operation is described by Moler and Morrison [10]. It is a % cubically convergent algorithm which starts with a and b , % rather than with their squares, and thereby avoids destructive % arithmetic underflows and overflows. In MATLAB, the algorithm is % used for complex modulus, Euclidean vector norm, plane rotations, % and the shift calculation in the eigenvalue and singular value % iterations. %% 12. FLOP and CHOP % % Detailed information about the amount of work involved in % matrix calculations and the resulting accuracy is provided by % FLOP and CHOP. The basic unit of work is the "flop", or floating % point operation. Roughly, one flop is one execution of a Fortran % statement like % % S = S + X(I)*Y(I) % % or % % Y(I) = Y(I) + T*X(I) % % In other words, it consists of one floating point multiplication, % together with one floating point addition and the associated % indexing and storage reference operations. % % MATLAB will print the number of flops required for a % particular statement when the statement is terminated by an extra % comma. For example, the line % % n = 20; RAND(n)*RAND(n);, % % ends with an extra comma. Two 20 by 20 random matrices are % generated and multiplied together. The result is assigned to % ANS, but the semicolon suppresses its printing. The only output % is % % 8800 flops % % This is n**3 + 2*n**2 flops, n**2 for each random matrix and % n**3 for the product. % % FLOP is a predefined vector with two components. FLOP(1) is % the number of flops used by the most recently executed statement, % except that statements with zero flops are ignored. For example, % after executing the previous statement, % % flop(1)/n**3 % % results in % % ANS = % % 1.1000 % % % FLOP(2) is the cumulative total of all the flops used since % the beginning of the MATLAB session. The statement % % FLOP = <0 0> % % resets the total. % % There are several difficulties associated with keeping a % precise count of floating point operations. An addition or % subtraction that is not paired with a multiplication is usually % counted as a flop. The same is true of an isolated multiplication % that is not paired with an addition. Each floating point % division counts as a flop. But the number of operations required % by system dependent library functions such as square root cannot % be counted, so most elementary functions are arbitrarily counted % as using only one flop. % % The biggest difficulty occurs with complex arithmetic. % Almost all operations on the real parts of matrices are counted. % However, the operations on the complex parts of matrices are % counted only when they involve nonzero elements. This means that % simple operations on nonreal matrices require only about twice as % many flops as the same operations on real matrices. This factor % of two is not necessarily an accurate measure of the relative % costs of real and complex arithmetic. % % The result of each floating point operation may also be % "chopped" to simulate a computer with a shorter word length. The % details of this chopping operation depend upon the format of the % floating point word. Usually, the fraction in the floating point % word can be regarded as consisting of several octal or % hexadecimal digits. The least significant of these digits can be % set to zero by a logical masking operation. Thus the statement % % CHOP(p) % % causes the p least significant octal or hexadecimal digits in % the result of each floating point operation to be set to zero. % For example, if the computer being used has an IBM 360 long % floating point word with 14 hexadecimal digits in the fraction, % then CHOP(8) results in simulation of a computer with only 6 % hexadecimal digits in the fraction, i.e. a short floating point % word. On a computer such as the CDC 6600 with 16 octal digits, % CHOP(8) results in about the same accuracy because the remaining % 8 octal digits represent the same number of bits as 6 hexadecimal % digits. % % Some idea of the effect of CHOP on any particular system can % be obtained by executing the following statements. % % long, t = 1/10 % long z, t = 1/10 % chop(8) % long, t = 1/10 % long z, t = 1/10 % % % The following Fortran subprograms illustrate more details of % FLOP and CHOP. The first subprogram is a simplified example of a % system-dependent function used within MATLAB itself. The common % variable FLP is essentially the first component of the variable % FLOP. The common variable CHP is initially zero, but it is set % to p by the statement CHOP(p). To shorten the DATA statement, % we assume there are only 6 hexadecimal digits. We also assume an % extension of Fortran that allows .AND. to be used as a binary % operation between two real variables. % % REAL FUNCTION FLOP(X) % REAL X % INTEGER FLP,CHP % COMMON FLP,CHP % REAL MASK(5) % DATA MASK/ZFFFFFFF0,ZFFFFFF00,ZFFFFF000,ZFFFF0000,ZFFF00000/ % FLP = FLP + 1 % IF (CHP .EQ. 0) FLOP = X % IF (CHP .GE. 1 .AND. CHP .LE. 5) FLOP = X .AND. MASK(CHP) % IF (CHP .GE. 6) FLOP = 0.0 % RETURN % END % % % The following subroutine illustrates a typical use of the % previous function within MATLAB. It is a simplified version of % the Basic Linear Algebra Subprogram that adds a scalar multiple % of one vector to another. We assume here that the vectors are % stored with a memory increment of one. % % SUBROUTINE SAXPY(N,TR,TI,XR,XI,YR,YI) % REAL TR,TI,XR(N),XI(N),YR(N),YI(N),FLOP % IF (N .LE. 0) RETURN % IF (TR .EQ. 0.0 .AND. TI .EQ. 0.0) RETURN % DO 10 I = 1, N % YR(I) = FLOP(YR(I) + TR*XR(I) - TI*XI(I)) % YI(I) = YI(I) + TR*XI(I) + TI*XR(I) % IF (YI(I) .NE. 0.0D0) YI(I) = FLOP(YI(I)) % 10 CONTINUE % RETURN % END % % % The saxpy operation is perhaps the most fundamental % operation within LINPACK. It is used in the computation of the % LU, the QR and the SVD factorizations, and in several other % places. We see that adding a multiple of one vector with n % components to another uses n flops if the vectors are real and % between n and 2*n flops if the vectors have nonzero imaginary % components. % % The permanent MATLAB variable EPS is reset by the statement % CHOP(p). Its new value is usually the smallest inverse power of % two that satisfies the Fortran logical test % % FLOP(1.0+EPS) .GT. 1.0 % % However, if EPS had been directly reset to a larger value, the % old value is retained. %% 13. Communicating with other programs % % There are four different ways MATLAB can be used in % conjunction with other programs: % REPLACE_WITH_DASH_DASH USER, % REPLACE_WITH_DASH_DASH EXEC, % REPLACE_WITH_DASH_DASH SAVE and LOAD, % REPLACE_WITH_DASH_DASH MATZ, CALL and RETURN . % % Let us illustrate each of these by the following simple % example. % % n = 6 % for i = 1:n, for j = 1:n, a(i,j) = abs(i-j); % A % X = inv(A) % % % The example A could be introduced into MATLAB by writing % the following Fortran subroutine. % % SUBROUTINE USER(A,M,N,S,T) % DOUBLE PRECISION A(1),S,T % N = IDINT(A(1)) % M = N % DO 10 J = 1, N % DO 10 I = 1, N % K = I + (J-1)*M % A(K) = IABS(I-J) % 10 CONTINUE % RETURN % END % % This subroutine should be compiled and linked into MATLAB in % place of the original version of USER. Then the MATLAB % statements % % n = 6 % A = user(n) % X = inv(A) % % do the job. % % The example A could be generated by storing the following % text in a file named, say, EXAMPLE . % % for i = 1:n, for j = 1:n, a(i,j) = abs(i-j); % % Then the MATLAB statements % % n = 6 % exec('EXAMPLE',0) % X = inv(A) % % have the desired effect. The 0 as the optional second parameter % of exec indicates that the text in the file should not be printed % on the terminal. % % The matrices A and X could also be stored in files. Two % separate main programs would be involved. The first is: % % PROGRAM MAINA % DOUBLE PRECISION A(10,10) % N = 6 % DO 10 J = 1, N % DO 10 I = 1, N % A(I,J) = IABS(I-J) % 10 CONTINUE % OPEN(UNIT=1,FILE='A') % WRITE(1,101) N,N % 101 FORMAT('A ',2I4) % DO 20 J = 1, N % WRITE(1,102) (A(I,J),I=1,N) % 20 CONTINUE % 102 FORMAT(4Z18) % END % % The OPEN statement may take different forms on different systems. % It attaches Fortran logical unit number 1 to the file named A. % The FORMAT number 102 may also be system dependent. This % particular one is appropriate for hexadecimal computers with an 8 % byte double precision floating point word. Check, or modify, % MATLAB subroutine SAVLOD. % % After this program is executed, enter MATLAB and give the % following statements: % % load('A') % X = inv(A) % save('X',X) % % If all goes according to plan, this will read the matrix A from % the file A, invert it, store the inverse in X and then write the % matrix X on the file X . The following program can then access X % . % % PROGRAM MAINX % DOUBLE PRECISION X(10,10) % OPEN(UNIT=1,FILE='X') % REWIND 1 % READ (1,101) ID,M,N % 101 FORMAT(A4,2I4) % DO 10 J = 1, N % READ(1,102) (X(I,J),I=1,M) % 10 CONTINUE % 102 FORMAT(4Z18) % ... % ... % % % The most elaborate mechanism involves using MATLAB as a % subroutine within another program. Communication with the MATLAB % stack is accomplished using subroutine MATZ which is distributed % with MATLAB, but which is not used by MATLAB itself. The % preample of MATZ is: % % SUBROUTINE MATZ(A,LDA,M,N,IDA,JOB,IERR) % INTEGER LDA,M,N,IDA(1),JOB,IERR % DOUBLE PRECISION A(LDA,N) % C % C ACCESS MATLAB VARIABLE STACK % C A IS AN M BY N MATRIX, STORED IN AN ARRAY WITH % C LEADING DIMENSION LDA. % C IDA IS THE NAME OF A. % C IF IDA IS AN INTEGER K LESS THAN 10, THEN THE NAME IS 'A'K % C OTHERWISE, IDA(1:4) IS FOUR CHARACTERS, FORMAT 4A1. % C JOB = 0 GET REAL A FROM MATLAB, % C = 1 PUT REAL A INTO MATLAB, % C = 10 GET IMAG PART OF A FROM MATLAB, % C = 11 PUT IMAG PART OF A INTO MATLAB. % C RETURN WITH NONZERO IERR AFTER MATLAB ERROR MESSAGE. % C % C USES MATLAB ROUTINES STACKG, STACKP AND ERROR % % % The preample of subroutine MATLAB is: % % % SUBROUTINE MATLAB(INIT) % C INIT = 0 FOR FIRST ENTRY, NONZERO FOR SUBSEQUENT ENTRIES % % % To do our example, write the following program: % % DOUBLE PRECISION A(10,10),X(10,10) % INTEGER IDA(4),IDX(4) % DATA LDA/10/ % DATA IDA/'A',' ',' ',' '/, IDX/'X',' ',' ',' '/ % CALL MATLAB(0) % N = 6 % DO 10 J = 1, N % DO 10 I = 1, N % A(I,J) = IABS(I-J) % 10 CONTINUE % CALL MATZ(A,LDA,N,N,IDA,1,IERR) % IF (IERR .NE. 0) GO TO ... % CALL MATLAB(1) % CALL MATZ(X,LDA,N,N,IDX,0,IERR) % IF (IERR .NE. 0) GO TO ... % ... % ... % % When this program is executed, the call to MATLAB(0) produces the % MATLAB greeting, then waits for input. The command % % return % % sends control back to our example program. The matrix A is % generated by the program and sent to the stack by the first call % to MATZ. The call to MATLAB(1) produces the MATLAB prompt. Then % the statements % % X = inv(A) % return % % will invert our matrix, put the result on the stack and go back % to our program. The second call to MATZ will retrieve X . % % By the way, this matrix X is interesting. Take a look at % round(2*(n-1)*X). % % % % References % % [1] J. J. Dongarra, J. R. Bunch, C. B. Moler and G. W. Stewart, % LINPACK Users' Guide, Society for Industrial and Applied % Mathematics, Philadelphia, 1979. % % [2] B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. % Ikebe, V. C. Klema, C. B. Moler, Matrix Eigensystem Routines % REPLACE_WITH_DASH_DASH EISPACK Guide, Lecture Notes in Computer Science, volume % 6, second edition, Springer-Verlag, 1976. % % [3] B. S. Garbow, J. M. Boyle, J. J. Dongarra, C. B. Moler, % Matrix Eigensystem Routines REPLACE_WITH_DASH_DASH EISPACK Guide Extension, % Lecture Notes in Computer Science, volume 51, Springer- % Verlag, 1977. % [4] S. Cohen and S. Piper, SPEAKEASY III Reference Manual, % Speakeasy Computing Corp., Chicago, Ill., 1979. % % [5] J. H. Wilkinson and C. Reinsch, Handbook for Automatic % Computation, volume II, Linear Algebra, Springer-Verlag, % 1971. % % [6] Niklaus Wirth, Algorithms + Data Structures = Programs, % Prentice-Hall, 1976. % % [7] H. B. Keller and D. Sachs, "Calculations of the Conductivity % of a Medium Containing Cylindrical Inclusions", J. Applied % Physics 35, 537-538, 1964. % % [8] C. B. Moler and C. F. Van Loan, Nineteen Dubious Ways to % Compute the Exponential of a Matrix, SIAM Review 20, 801- % 836, 1979. % % [9] G. E. Forsythe, M. A. Malcolm and C. B. Moler, Computer % Methods for Mathematical Computations, Prentice-Hall, 1977. % % [10] C. B. Moler and D. R. Morrison, "Replacing square roots by % Pythagorean sums", University of New Mexico, Computer % Science Department, technical report, submitted for % publication, 1980. %% Appendix. The HELP document % % NEWS MATLAB NEWS dated May, 1981. % This describes recent or local changes. % The new features added since the November, 1980, printing % of the Users' Guide include DIARY, EDIT, KRON, MACRO, PLOT, % RAT, TRIL, TRIU and six element-by-element operations: % .* ./ .\ .*. ./. .\. % Some additional capabilities have been added to EXIT, % RANDOM, RCOND, SIZE and SVD. % % INTRO Welcome to MATLAB. % % Here are a few sample statements: % % A = <1 2; 3 4> % b = <5 6>' % x = A\b % = eig(A), norm(A-V*D/V) % help \ , help eig % exec('demo',7) % % For more information, see the MATLAB Users' Guide which is % contained in file ... or may be obtained from ... . % % HELP HELP gives assistance. % HELP HELP obviously prints this message. % To see all the HELP messages, list the file ... . % % < < > Brackets used in forming vectors and matrices. % <6.9 9.64 SQRT(-1)> is a vector with three elements % separated by blanks. <6.9, 9.64, sqrt(-1)> is the same % thing. <1+I 2-I 3> and <1 +I 2 -I 3> are not the same. % The first has three elements, the second has five. % <11 12 13; 21 22 23> is a 2 by 3 matrix . The semicolon % ends the first row. % % Vectors and matrices can be used inside < > brackets. % is allowed if the number of rows of A equals % the number of rows of B and the number of columns of A % plus the number of columns of B equals the number of % columns of C . This rule generalizes in a hopefully % obvious way to allow fairly complicated constructions. % % A = < > stores an empty matrix in A , thereby removing it % from the list of current variables. % % For the use of < and > on the left of the = in multiple % assignment statements, see LU, EIG, SVD and so on. % % In WHILE and IF clauses, <> means less than or greater % than, i.e. not equal, < means less than, > means greater % than, <= means less than or equal, >= means greater than or % equal. % % For the use of > and < to delineate macros, see MACRO. % % > See < . Also see MACRO. % % ( ( ) Used to indicate precedence in arithmetic expressions % in the usual way. Used to enclose arguments of functions % in the usual way. Used to enclose subscripts of vectors % and matrices in a manner somewhat more general than the % usual way. If X and V are vectors, then X(V) is % . The components of V % are rounded to nearest integers and used as subscripts. An % error occurs if any such subscript is less than 1 or % greater than the dimension of X . Some examples: % X(3) is the third element of X . % X(<1 2 3>) is the first three elements of X . So is % X() . % If X has N components, X(N:-1:1) reverses them. % The same indirect subscripting is used in matrices. If V % has M components and W has N components, then A(V,W) % is the M by N matrix formed from the elements of A whose % subscripts are the elements of V and W . For example... % A(<1,5>,:) = A(<5,1>,:) interchanges rows 1 and 5 of A . % % ) See ( . % % = Used in assignment statements and to mean equality in WHILE % and IF clauses. % % . Decimal point. 314/100, 3.14 and .314E1 are all the % same. % % Element-by-element multiplicative operations are obtained % using .* , ./ , or .\ . For example, C = A ./ B is the % matrix with elements c(i,j) = a(i,j)/b(i,j) . % % Kronecker tensor products and quotients are obtained with % .*. , ./. and .\. . See KRON. % % Two or more points at the end of the line indicate % continuation. The total line length limit is 1024 % characters. % % , Used to separate matrix subscripts and function arguments. % Used at the end of FOR, WHILE and IF clauses. Used to % separate statements in multi-statement lines. In this % situation, it may be replaced by semicolon to suppress % printing. % % ; Used inside brackets to end rows. % Used after an expression or statement to suppress printing. % See SEMI. % \ Backslash or matrix left division. A\B is roughly the % same as INV(A)*B , except it is computed in a different % way. If A is an N by N matrix and B is a column vector % with N components, or a matrix with several such columns, % then X = A\B is the solution to the equation A*X = B % computed by Gaussian elimination. A warning message is % printed if A is badly scaled or nearly singular. % A\EYE produces the inverse of A . % % If A is an M by N matrix with M < or > N and B is a % column vector with M components, or a matrix with several % such columns, then X = A\B is the solution in the least % squares sense to the under- or overdetermined system of % equations A*X = B . The effective rank, K, of A is % determined from the QR decomposition with pivoting. A % solution X is computed which has at most K nonzero % components per column. If K < N this will usually not be % the same solution as PINV(A)*B . % A\EYE produces a generalized inverse of A . % % If A and B have the same dimensions, then A .\ B has % elements a(i,j)\b(i,j) . % % Also, see EDIT. % % / Slash or matrix right division. B/A is roughly the same % as B*INV(A) . More precisely, B/A = (A'\B')' . See \ . % % IF A and B have the same dimensions, then A ./ B has % elements a(i,j)/b(i,j) . % % Two or more slashes together on a line indicate a logical % end of line. Any following text is ignored. % % ' Transpose. X' is the complex conjugate transpose of X . % Quote. 'ANY TEXT' is a vector whose components are the % MATLAB internal codes for the characters. A quote within % the text is indicated by two quotes. See DISP and FILE . % % + Addition. X + Y . X and Y must have the same dimensions. % % - Subtraction. X - Y . X and Y must have the same % dimensions. % % * Matrix multiplication, X*Y . Any scalar (1 by 1 matrix) % may multiply anything. Otherwise, the number of columns of % X must equal the number of rows of Y . % % Element-by-element multiplication is obtained with X .* Y . % % The Kronecker tensor product is denoted by X .*. Y . % % Powers. X**p is X to the p power. p must be a % scalar. If X is a matrix, see FUN . % % : Colon. Used in subscripts, FOR iterations and possibly % elsewhere. % J:K is the same as % J:K is empty if J > K . % J:I:K is the same as % J:I:K is empty if I > 0 and J > K or if I < 0 and J < K . % The colon notation can be used to pick out selected rows, % columns and elements of vectors and matrices. % A(:) is all the elements of A, regarded as a single % column. % A(:,J) is the J-th column of A % A(J:K) is A(J),A(J+1),...,A(K) % A(:,J:K) is A(:,J),A(:,J+1),...,A(:,K) and so on. % For the use of the colon in the FOR statement, See FOR . % % ABS ABS(X) is the absolute value, or complex modulus, of the % elements of X . % % ANS Variable created automatically when expressions are not % assigned to anything else. % % ATAN ATAN(X) is the arctangent of X . See FUN . % % BASE BASE(X,B) is a vector containing the base B representation % of X . This is often used in conjunction with DISPLAY. % DISPLAY(X,B) is the same as DISPLAY(BASE(X,B)). For % example, DISP(4*ATAN(1),16) prints the hexadecimal % representation of pi. % % CHAR CHAR(K) requests an input line containing a single % character to replace MATLAB character number K in the % following table. For example, CHAR(45) replaces backslash. % CHAR(-K) replaces the alternate character number K. % % K character alternate name % 0 - 9 0 - 9 0 - 9 digits % 10 - 35 A - Z a - z letters % 36 blank % 37 ( ( lparen % 38 ) ) rparen % 39 ; ; semi % 40 : | colon % 41 + + plus % 42 - - minus % 43 * * star % 44 / / slash % 45 \ $ backslash % 46 = = equal % 47 . . dot % 48 , , comma % 49 ' " quote % 50 < [ less % 51 > ] great % % CHOL Cholesky factorization. CHOL(X) uses only the diagonal % and upper triangle of X . The lower triangular is assumed % to be the (complex conjugate) transpose of the upper. If % X is positive definite, then R = CHOL(X) produces an % upper triangular R so that R'*R = X . If X is not % positive definite, an error message is printed. % % CHOP Truncate arithmetic. CHOP(P) causes P places to be chopped % off after each arithmetic operation in subsequent % computations. This means P hexadecimal digits on some % computers and P octal digits on others. CHOP(0) restores % full precision. % % CLEAR Erases all variables, except EPS, FLOP, EYE and RAND. % X = <> erases only variable X . So does CLEAR X . % % COND Condition number in 2-norm. COND(X) is the ratio of the % largest singular value of X to the smallest. % % CONJG CONJG(X) is the complex conjugate of X . % % COS COS(X) is the cosine of X . See FUN . % % DET DET(X) is the determinant of the square matrix X . % % DIAG If V is a row or column vector with N components, % DIAG(V,K) is a square matrix of order N+ABS(K) with the % elements of V on the K-th diagonal. K = 0 is the main % diagonal, K > 0 is above the main diagonal and K < 0 is % below the main diagonal. DIAG(V) simply puts V on the % main diagonal. % eg. DIAG(-M:M) + DIAG(ONES(2*M,1),1) + DIAG(ONES(2*M,1),-1) % produces a tridiagonal matrix of order 2*M+1 . % IF X is a matrix, DIAG(X,K) is a column vector formed % from the elements of the K-th diagonal of X . % DIAG(X) is the main diagonal of X . % DIAG(DIAG(X)) is a diagonal matrix . % % DIARY DIARY('file') causes a copy of all subsequent terminal % input and most of the resulting output to be written on the % file. DIARY(0) turns it off. See FILE. % % DISP DISPLAY(X) prints X in a compact format. If all the % elements of X are integers between 0 and 51, then X is % interpreted as MATLAB text and printed accordingly. % Otherwise, + , - and blank are printed for positive, % negative and zero elements. Imaginary parts are ignored. % DISP(X,B) is the same as DISP(BASE(X,B)). % % EDIT There are no editing features available on most % installations and EDIT is not a command. However, on a few % systems a command line consisting of a single backslash \ % will cause the local file editor to be called with a copy % of the previous input line. When the editor returns % control to MATLAB, it will execute the line again. % % EIG Eigenvalues and eigenvectors. % EIG(X) is a vector containing the eigenvalues of a square % matrix X . % = EIG(X) produces a diagonal matrix D of % eigenvalues and a full matrix V whose columns are the % corresponding eigenvectors so that X*V = V*D . % % ELSE Used with IF . % % END Terminates the scope of FOR, WHILE and IF statements. % Without END's, FOR and WHILE repeat all statements up to % the end of the line. Each END is paired with the closest % previous unpaired FOR or WHILE and serves to terminate its % scope. The line % FOR I=1:N, FOR J=1:N, A(I,J)=1/(I+J-1); A % would cause A to be printed N**2 times, once for each new % element. On the other hand, the line % FOR I=1:N, FOR J=1:N, A(I,J)=1/(I+J-1); END, END, A % will lead to only the final printing of A . % Similar considerations apply to WHILE. % EXIT terminates execution of loops or of MATLAB itself. % % EPS Floating point relative accuracy. A permanent variable % whose value is initially the distance from 1.0 to the next % largest floating point number. The value is changed by % CHOP, and other values may be assigned. EPS is used as a % default tolerance by PINV and RANK. % % EXEC EXEC('file',k) obtains subsequent MATLAB input from an % external file. The printing of input is controlled by the % optional parameter k . % If k = 1 , the input is echoed. % If k = 2 , the MATLAB prompt <> is printed. % If k = 4 , MATLAB pauses before each prompt and waits for a % null line to continue. % If k = 0 , there is no echo, prompt or pause. This is the % default if the exec command is followed by a semicolon. % If k = 7 , there will be echos, prompts and pauses. This is % useful for demonstrations on video terminals. % If k = 3 , there will be echos and prompts, but no pauses. % This is the the default if the exec command is not followed % by a semicolon. % EXEC(0) causes subsequent input to be obtained from the % terminal. An end-of-file has the same effect. % EXEC's may be nested, i.e. the text in the file may contain % EXEC of another file. EXEC's may also be driven by FOR and % WHILE loops. % EXIT Causes termination of a FOR or WHILE loop. % If not in a loop, terminates execution of MATLAB. % % EXP EXP(X) is the exponential of X , e to the X . See FUN % . % % EYE Identity matrix. EYE(N) is the N by N identity matrix. % EYE(M,N) is an M by N matrix with 1's on the diagonal and % zeros elsewhere. EYE(A) is the same size as A . EYE % with no arguments is an identity matrix of whatever order % is appropriate in the context. For example, A + 3*EYE % adds 3 to each diagonal element of A . % % FILE The EXEC, SAVE, LOAD, PRINT and DIARY functions access % files. The 'file' parameter takes different forms for % different operating systems. On most systems, 'file' may % be a string of up to 32 characters in quotes. For example, % SAVE('A') or EXEC('matlab/demo.exec') . The string will be % used as the name of a file in the local operating system. % On all systems, 'file' may be a positive integer k less % than 10 which will be used as a FORTRAN logical unit % number. Some systems then automatically access a file with % a name like FORT.k or FORk.DAT. Other systems require a % file with a name like FT0kF001 to be assigned to unit k % before MATLAB is executed. Check your local installation % for details. % % FLOPS Count of floating point operations. % FLOPS is a permanently defined row vector with two % elements. FLOPS(1) is the number of floating point % operations counted during the previous statement. FLOPS(2) % is a cumulative total. FLOPS can be used in the same way % as any other vector. FLOPS(2) = 0 resets the cumulative % total. In addition, FLOPS(1) will be printed whenever a % statement is terminated by an extra comma. For example, % X = INV(A);, % or % COND(A), (as the last statement on the line). % HELP FLPS gives more details. % % FLPS More detail on FLOPS. % It is not feasible to count absolutely all floating point % operations, but most of the important ones are counted. % Each multiply and add in a real vector operation such as a % dot product or a 'saxpy' counts one flop. Each multiply % and add in a complex vector operation counts two flops. % Other additions, subtractions and multiplications count one % flop each if the result is real and two flops if it is not. % Real divisions count one and complex divisions count two. % Elementary functions count one if real and two if complex. % Some examples. If A and B are real N by N matrices, then % A + B counts N**2 flops, % A*B counts N**3 flops, % A**100 counts 99*N**3 flops, % LU(A) counts roughly (1/3)*N**3 flops. % % FOR Repeat statements a specific number of times. % FOR variable = expr, statement, ..., statement, END % The END at the end of a line may be omitted. The comma % before the END may also be omitted. The columns of the % expression are stored one at a time in the variable and % then the following statements, up to the END, are executed. % The expression is often of the form X:Y, in which case its % columns are simply scalars. Some examples (assume N has % already been assigned a value). % FOR I = 1:N, FOR J = 1:N, A(I,J) = 1/(I+J-1); % FOR J = 2:N-1, A(J,J) = J; END; A % FOR S = 1.0: -0.1: 0.0, ... steps S with increments of -0.1 . % FOR E = EYE(N), ... sets E to the unit N-vectors. % FOR V = A, ... has the same effect as % FOR J = 1:N, V = A(:,J); ... except J is also set here. % % FUN For matrix arguments X , the functions SIN, COS, ATAN, % SQRT, LOG, EXP and X**p are computed using eigenvalues D % and eigenvectors V . If = EIG(X) then f(X) = % V*f(D)/V . This method may give inaccurate results if V % is badly conditioned. Some idea of the accuracy can be % obtained by comparing X**1 with X . % For vector arguments, the function is applied to each % component. % % HESS Hessenberg form. The Hessenberg form of a matrix is zero % below the first subdiagonal. If the matrix is symmetric or % Hermitian, the form is tridiagonal.

= HESS(A) % produces a unitary matrix P and a Hessenberg matrix H so % that A = P*H*P'. By itself, HESS(A) returns H. % % HILB Inverse Hilbert matrix. HILB(N) is the inverse of the N % by N matrix with elements 1/(i+j-1), which is a famous % example of a badly conditioned matrix. The result is exact % for N less than about 15, depending upon the computer. % % IF Conditionally execute statements. Simple form... % IF expression rop expression, statements % where rop is =, <, >, <=, >=, or <> (not equal) . The % statements are executed once if the indicated comparison % between the real parts of the first components of the two % expressions is true, otherwise the statements are skipped. % Example. % IF ABS(I-J) = 1, A(I,J) = -1; % More complicated forms use END in the same way it is used % with FOR and WHILE and use ELSE as an abbreviation for END, % IF expression not rop expression . Example % FOR I = 1:N, FOR J = 1:N, ... % IF I = J, A(I,J) = 2; ELSE IF ABS(I-J) = 1, A(I,J) = -1; ... % ELSE A(I,J) = 0; % An easier way to accomplish the same thing is % A = 2*EYE(N); % FOR I = 1:N-1, A(I,I+1) = -1; A(I+1,I) = -1; % % IMAG IMAG(X) is the imaginary part of X . % % INV INV(X) is the inverse of the square matrix X . A warning % message is printed if X is badly scaled or nearly % singular. % % KRON KRON(X,Y) is the Kronecker tensor product of X and Y . It % is also denoted by X .*. Y . The result is a large matrix % formed by taking all possible products between the elements % of X and those of Y . For example, if X is 2 by 3, then % X .*. Y is % % < x(1,1)*Y x(1,2)*Y x(1,3)*Y % x(2,1)*Y x(2,2)*Y x(2,3)*Y > % % The five-point discrete Laplacian for an n-by-n grid can be % generated by % % T = diag(ones(n-1,1),1); T = T + T'; I = EYE(T); % A = T.*.I + I.*.T - 4*EYE; % % Just in case they might be useful, MATLAB includes % constructions called Kronecker tensor quotients, denoted by % X ./. Y and X .\. Y . They are obtained by replacing the % elementwise multiplications in X .*. Y with divisions. % % LINES An internal count is kept of the number of lines of output % since the last input. Whenever this count approaches a % limit, the user is asked whether or not to suppress % printing until the next input. Initially the limit is 25. % LINES(N) resets the limit to N . % % LOAD LOAD('file') retrieves all the variables from the file . % See FILE and SAVE for more details. To prepare your own % file for LOADing, change the READs to WRITEs in the code % given under SAVE. % % LOG LOG(X) is the natural logarithm of X . See FUN . % Complex results are produced if X is not positive, or has % nonpositive eigenvalues. % % LONG Determine output format. All computations are done in % complex arithmetic and double precision if it is available. % SHORT and LONG merely switch between different output % formats. % SHORT Scaled fixed point format with about 5 digits. % LONG Scaled fixed point format with about 15 digits. % SHORT E Floating point format with about 5 digits. % LONG E Floating point format with about 15 digits. % LONG Z System dependent format, often hexadecimal. % % LU Factors from Gaussian elimination. = LU(X) stores a % upper triangular matrix in U and a 'psychologically lower % triangular matrix', i.e. a product of lower triangular and % permutation matrices, in L , so that X = L*U . By itself, % LU(X) returns the output from CGEFA . % % MACRO The macro facility involves text and inward pointing angle % brackets. If STRING is the source text for any MATLAB % expression or statement, then % t = 'STRING'; % encodes the text as a vector of integers and stores that % vector in t . DISP(t) will print the text and % >t< % causes the text to be interpreted, either as a statement or % as a factor in an expression. For example % t = '1/(i+j-1)'; % disp(t) % for i = 1:n, for j = 1:n, a(i,j) = >t<; % generates the Hilbert matrix of order n. % Another example showing indexed text, % S = <'x = 3 ' % 'y = 4 ' % 'z = sqrt(x*x+y*y)'> % for k = 1:3, >S(k,:)< % It is necessary that the strings making up the "rows" of % the "matrix" S have the same lengths. % % MAGIC Magic square. MAGIC(N) is an N by N matrix constructed % from the integers 1 through N**2 with equal row and column % sums. % % NORM For matrices.. % NORM(X) is the largest singular value of X . % NORM(X,1) is the 1-norm of X . % NORM(X,2) is the same as NORM(X) . % NORM(X,'INF') is the infinity norm of X . % NORM(X,'FRO') is the F-norm, i.e. SQRT(SUM(DIAG(X'*X))) . % For vectors.. % NORM(V,P) = (SUM(V(I)**P))**(1/P) . % NORM(V) = NORM(V,2) . % NORM(V,'INF') = MAX(ABS(V(I))) . % % ONES All ones. ONES(N) is an N by N matrix of ones. ONES(M,N) % is an M by N matrix of ones . ONES(A) is the same size as % A and all ones . % % ORTH Orthogonalization. Q = ORTH(X) is a matrix with % orthonormal columns, i.e. Q'*Q = EYE, which span the same % space as the columns of X . % % PINV Pseudoinverse. X = PINV(A) produces a matrix X of the % same dimensions as A' so that A*X*A = A , X*A*X = X and % AX and XA are Hermitian . The computation is based on % SVD(A) and any singular values less than a tolerance are % treated as zero. The default tolerance is % NORM(SIZE(A),'inf')*NORM(A)*EPS. This tolerance may be % overridden with X = PINV(A,tol). See RANK. % % PLOT PLOT(X,Y) produces a plot of the elements of Y against % those of X . PLOT(Y) is the same as PLOT(1:n,Y) where n is % the number of elements in Y . PLOT(X,Y,P) or % PLOT(X,Y,p1,...,pk) passes the optional parameter vector P % or scalars p1 through pk to the plot routine. The default % plot routine is a crude printer-plot. It is hoped that an % interface to local graphics equipment can be provided. % An interesting example is % t = 0:50; % PLOT( t.*cos(t), t.*sin(t) ) % % POLY Characteristic polynomial. % If A is an N by N matrix, POLY(A) is a column vector with % N+1 elements which are the coefficients of the % characteristic polynomial, DET(lambda*EYE - A) . % If V is a vector, POLY(V) is a vector whose elements are % the coefficients of the polynomial whose roots are the % elements of V . For vectors, ROOTS and POLY are inverse % functions of each other, up to ordering, scaling, and % roundoff error. % ROOTS(POLY(1:20)) generates Wilkinson's famous example. % % PRINT PRINT('file',X) prints X on the file using the current % format determined by SHORT, LONG Z, etc. See FILE. % % PROD PROD(X) is the product of all the elements of X . % % QR Orthogonal-triangular decomposition. % = QR(X) produces an upper triangular matrix R of % the same dimension as X and a unitary matrix Q so that % X = Q*R . % = QR(X) produces a permutation matrix E , an % upper triangular R with decreasing diagonal elements and % a unitary Q so that X*E = Q*R . % By itself, QR(X) returns the output of CQRDC . TRIU(QR(X)) % is R . % % RAND Random numbers and matrices. RAND(N) is an N by N matrix % with random entries. RAND(M,N) is an M by N matrix with % random entries. RAND(A) is the same size as A . RAND % with no arguments is a scalar whose value changes each time % it is referenced. % Ordinarily, random numbers are uniformly distributed in % the interval (0.0,1.0) . RAND('NORMAL') switches to a % normal distribution with mean 0.0 and variance 1.0 . % RAND('UNIFORM') switches back to the uniform distribution. % RAND('SEED') returns the current value of the seed for the % generator. RAND('SEED',n) sets the seed to n . % RAND('SEED',0) resets the seed to 0, its value when MATLAB % is first entered. % % RANK Rank. K = RANK(X) is the number of singular values of X % that are larger than NORM(SIZE(X),'inf')*NORM(X)*EPS. % K = RANK(X,tol) is the number of singular values of X that % are larger than tol . % % RCOND RCOND(X) is an estimate for the reciprocal of the % condition of X in the 1-norm obtained by the LINPACK % condition estimator. If X is well conditioned, RCOND(X) % is near 1.0 . If X is badly conditioned, RCOND(X) is % near 0.0 . % = RCOND(A) sets R to RCOND(A) and also produces a % vector Z so that % NORM(A*Z,1) = R*NORM(A,1)*NORM(Z,1) % So, if RCOND(A) is small, then Z is an approximate null % vector. % % RAT An experimental function which attempts to remove the % roundoff error from results that should be "simple" % rational numbers. % RAT(X) approximates each element of X by a continued % fraction of the form % % a/b = d1 + 1/(d2 + 1/(d3 + ... + 1/dk)) % % with k <= len, integer di and abs(di) <= max . The default % values of the parameters are len = 5 and max = 100. % RAT(len,max) changes the default values. Increasing either % len or max increases the number of possible fractions. % = RAT(X) produces integer matrices A and B so that % % A ./ B = RAT(X) % % Some examples: % % long % T = hilb(6), X = inv(T) % = rat(X) % H = A ./ B, S = inv(H) % % short e % d = 1:8, e = ones(d), A = abs(d'*e - e'*d) % X = inv(A) % rat(X) % display(ans) % % % REAL REAL(X) is the real part of X . % % RETURN From the terminal, causes return to the operating system % or other program which invoked MATLAB. From inside an % EXEC, causes return to the invoking EXEC, or to the % terminal. % % RREF RREF(A) is the reduced row echelon form of the rectangular % matrix. RREF(A,B) is the same as RREF() . % % ROOTS Find polynomial roots. ROOTS(C) computes the roots of the % polynomial whose coefficients are the elements of the % vector C . If C has N+1 components, the polynomial is % C(1)*X**N + ... + C(N)*X + C(N+1) . See POLY. % % ROUND ROUND(X) rounds the elements of X to the nearest % integers. % % SAVE SAVE('file') stores all the current variables in a file. % SAVE('file',X) saves only X . See FILE . % The variables may be retrieved later by LOAD('file') or by % your own program using the following code for each matrix. % The lines involving XIMAG may be eliminated if everything % is known to be real. % % attach lunit to 'file' % REAL or DOUBLE PRECISION XREAL(MMAX,NMAX) % REAL or DOUBLE PRECISION XIMAG(MMAX,NMAX) % READ(lunit,101) ID,M,N,IMG % DO 10 J = 1, N % READ(lunit,102) (XREAL(I,J), I=1,M) % IF (IMG .NE. 0) READ(lunit,102) (XIMAG(I,J),I=1,M) % 10 CONTINUE % % The formats used are system dependent. The following are % typical. See SUBROUTINE SAVLOD in your local % implementation of MATLAB. % % 101 FORMAT(4A1,3I4) % 102 FORMAT(4Z18) % 102 FORMAT(4O20) % 102 FORMAT(4D25.18) % % SCHUR Schur decomposition. = SCHUR(X) produces an upper % triangular matrix T , with the eigenvalues of X on the % diagonal, and a unitary matrix U so that X = U*T*U' and % U'*U = EYE . By itself, SCHUR(X) returns T . % % SHORT See LONG . % % SEMI Semicolons at the end of lines will cause, rather than % suppress, printing. A second SEMI restores the initial % interpretation. % % SIN SIN(X) is the sine of X . See FUN . % SIZE If X is an M by N matrix, then SIZE(X) is . % Can also be used with a multiple assignment, % = SIZE(X) . % % SQRT SQRT(X) is the square root of X . See FUN . Complex % results are produced if X is not positive, or has % nonpositive eigenvalues. % % STOP Use EXIT instead. % % SUM SUM(X) is the sum of all the elements of X . % SUM(DIAG(X)) is the trace of X . % % SVD Singular value decomposition. = SVD(X) produces a % diagonal matrix S , of the same dimension as X and with % nonnegative diagonal elements in decreasing order, and % unitary matrices U and V so that X = U*S*V' . % By itself, SVD(X) returns a vector containing the singular % values. % = SVD(X,0) produces the "economy size" % decomposition. If X is m by n with m > n, then only the % first n columns of U are computed and S is n by n . % % TRIL Lower triangle. TRIL(X) is the lower triangular part of X. % TRIL(X,K) is the elements on and below the K-th diagonal of % X. K = 0 is the main diagonal, K > 0 is above the main % diagonal and K < 0 is below the main diagonal. % % TRIU Upper triangle. TRIU(X) is the upper triangular part of X. % TRIU(X,K) is the elements on and above the K-th diagonal of % X. K = 0 is the main diagonal, K > 0 is above the main % diagonal and K < 0 is below the main diagonal. % % USER Allows personal Fortran subroutines to be linked into % MATLAB . The subroutine should have the heading % % SUBROUTINE USER(A,M,N,S,T) % REAL or DOUBLE PRECISION A(M,N),S,T % % The MATLAB statement Y = USER(X,s,t) results in a call to % the subroutine with a copy of the matrix X stored in the % argument A , its column and row dimensions in M and N , % and the scalar parameters s and t stored in S and T % . If s and t are omitted, they are set to 0.0 . After % the return, A is stored in Y . The dimensions M and % N may be reset within the subroutine. The statement Y = % USER(K) results in a call with M = 1, N = 1 and A(1,1) = % FLOAT(K) . After the subroutine has been written, it must % be compiled and linked to the MATLAB object code within the % local operating system. % % WHAT Lists commands and functions currently available. % % WHILE Repeat statements an indefinite number of times. % WHILE expr rop expr, statement, ..., statement, END % where rop is =, <, >, <=, >=, or <> (not equal) . The END % at the end of a line may be omitted. The comma before the % END may also be omitted. The commas may be replaced by % semicolons to avoid printing. The statements are % repeatedly executed as long as the indicated comparison % between the real parts of the first components of the two % expressions is true. Example (assume a matrix A is % already defined). % E = 0*A; F = E + EYE; N = 1; % WHILE NORM(E+F-E,1) > 0, E = E + F; F = A*F/N; N = N + 1; % E % % WHO Lists current variables. % % WHY Provides succinct answers to any questions. % % // ##### SOURCE END ##### eca491f76e4243cdbdeb2ef47ed6b124 -->

LINPACK, Linear Equation Package

Wed, 01/24/2018 - 05:42

The ACM Special Interest Group on Programming Languages, SIGPLAN, expects to hold the fourth in a series of conferences on the History of Programming Languages in 2020, see HOPL-IV. The first drafts of papers are to be submitted by August, 2018. That long lead time gives me the opportunity to write a detailed history of MATLAB. I plan to write the paper in sections, which I'll post in this blog as they are available. This is the third such installment.

LINPACK and EISPACK are Fortran subroutine packages for matrix computation developed in the 1970's. They are a follow-up to the Algol 60 procedures that I described in my blog post about the Wilkinson and Reinsch Handbook. They led to the first version of MATLAB. This post is about LINPACK. My previous post was about EISPACK.

ContentsCart Before the Horse

Logically a package of subroutines for solving linear equations should come before a package for computing matrix eigenvalues. The tasks addressed and the algorithms involved are mathematically simpler. But chronologically EISPACK came before LINPACK. Actually, the first section of the Handbook is about linear equations, but we pretty much skipped over that section when we did EISPACK. The eigenvalue algorithms were new and we felt it was important to make Fortran translations widely available.

LINPACK Project

In 1975, as EISPACK was nearing completion, we proposed to the NSF another research project investigating methods for the development of mathematical software. A byproduct would be the software itself.

The four principal investigators, and eventual authors of the Users' Guide, were

  • Jack Dongarra, Argonne National Laboratory
  • Cleve Moler, University of New Mexico
  • James Bunch, University of California, San Diego
  • G. W. (Pete) Stewart, University of Maryland

The project was centered at Argonne. The three of us at universities worked at our home institutions during the academic year and we all got together at Argonne in the summer. Dongarra and I had previously worked on EISPACK.

Contents

Fortran subroutine names at the time were limited to six characters. We adopted a naming convention with five letter names of the form TXXYY. The first letter, T, indicates the matrix data type. Standard Fortran provides three such types and, although not standard, many systems provide a fourth type, double precision complex, COMPLEX*16. Unlike EISPACK which follows Algol and has separate arrays for the real and imaginary parts of a complex matrix, LINPACK takes full advantage of Fortran complex. The first letter can be

S Real D Double precision C Complex Z Complex*16

The next two letters, XX, indicate the form of the matrix or its decomposition. (A packed array is the upper half of a symmetric matrix stored with one linear subscript.)

GE General GB General band PO Positive definite PP Positive definite packed PB Positive definite band SI Symmetric indefinite SP Symmetric indefinite packed HI Hermitian indefinite HP Hermitian indefinite packed TR Triangular GT General tridiagonal PT Positive definite tridiagonal CH Cholesky decomposition QR Orthogonal triangular decomposition SV Singular value decomposition

The final two letters, YY, indicate the computation to be done.

FA Factor SL Solve CO Factor and estimate condition DI Determinant, inverse, inertia DC Decompose UD Update DD Downdate EX Exchange

This chart shows all 44 of the LINPACK subroutines. An initial S may be replaced by any of D, C, or Z. An initial C may be replaced by Z. For example, DGEFA factors a double precision matrix. This is probably the most important routine in the package.

CO FA SL DI SGE * * * * SGB * * * * SPO * * * * SPP * * * * SPB * * * * SSI * * * * SSP * * * * CHI * * * * CHP * * * * STR * * * SGT * SPT * DC SL UD DD EX SCH * * * * SQR * * SSV *

Following this naming convention the subroutine for computing the SVD, the singular value decomposition, fortuitously becomes SSVDC.

Programming Style

Fortran at the time was notorious for unstructured, unreadable, "spaghetti" code. We adopted a disciplined programming style and expected people as well as machines to read the codes. The scope of loops and if-then-else constructions were carefully shown by indenting. Go-to's and statement numbers were used only to exit blocks and properly handle possibly empty loops.

In fact, we, the authors, did not actually write the final programs, TAMPR did. TAMPR was a powerful software analysis tool developed by Jim Boyle and Ken Dritz at Argonne. It manipulates and formats Fortran programs to clarify their structure. It also generates the four data type variants of the programs. So our source code was the Z versions and the S, D, and C versions, as well as "clean" Z versions, were producing by TAMPR.

This wasn't just a text processing task. For example, in ZPOFA, the Cholesky factorization of a positive definite complex Hermitian matrix, the test for a real, positive diagonal is

if (s .le. 0.0d0 .or. dimag(a(j,j)) .ne. 0.0d0) go to 40

In the real version DPOFA there is no imaginary part, so this becomes simply

if (s .le. 0.0d0) go to 40BLAS

All of the inner loops are done by calls to the BLAS, the Basic Linear Algebra Subprograms, developed concurrently by Chuck Lawson and colleagues. On systems that did not have optimizing Fortran compilers, the BLAS could be implemented efficiently in machine language. On vector machines, like the CRAY-1, the loop is a single vector instruction.

The two most important BLAS are the inner product of two vectors, DDOT, and vector plus scalar times vector, DAXPY. All of the algorithms are column oriented to conform to Fortran storage of two-dimensional arrays and thereby provide locality of memory access.

Portability

The programs avoid all machine dependent quantities. There is no need for anything like the EISPACK parameter MACHEP. Only one of the algorithms, the SVD, is iterative and involves a convergence test. Convergence is detected as a special case of a negligible subdiagonal element. Here is the code for checking if the subdiagonal element e(l) is negligible compared to the two nearby diagonal elements s(l) and s(l+1). I have included enough of the code to show how TAMPR displays structure by indentation and loop exits by comments with an initial c.

do 390 ll = 1, m l = m - ll c ...exit if (l .eq. 0) go to 400 test = dabs(s(l)) + dabs(s(l+1)) ztest = test + dabs(e(l)) if (ztest .ne. test) go to 380 e(l) = 0.0d0 c ......exit go to 400 380 continue 390 continue 400 continuePublisher

No commercial book publisher would publish the LINPACK Users' Guide. It did not fit into their established categories; it was neither a textbook nor a research monograph and was neither mathematics nor computer science.

Only one nonprofit publisher, SIAM, was interested. I vividly remember sitting on a dock in Lake Mendota at the University of Wisconsin-Madison during a SIAM annual meeting, talking to SIAM's Executive Director Ed Block, and deciding that SIAM would take a chance. The Guide has turned out to be one of SIAM's all-time best sellers.

A few years later, in 1981, Beresford Parlett reviewed the publication in SIAM Review. He wrote

It may seem strange that SIAM should publish and review a users' guide for a set of Fortran programs. Yet history will show that SIAM is thereby helping to foster a new aspect of technology which currently rejoices in a name mathematical software. There is as yet no satisfying characterization of this activity, but it certainly concerns the strong effect that a computer system has on the efficiency of a program.

LINPACK Benchmark

Today, LINPACK is better known as a benchmark than as a subroutine library. I wrote a blog post about the LINPACK Benchmark in 2013.

In Retrospect

In a sense, the LINPACK and EISPACK projects were failures. We had proposed research projects to the NSF to "explore the methodology, costs, and resources required to produce, test, and disseminate high-quality mathematical software." We never wrote a report or paper addressing those objectives. We only produced software

References

J. Dongarra and G. W. Stewart, "LINPACK, A Package for Solving Linear Systems", chapter 2 in W. Cowell, 29 pages, Sources and Development of Mathematical Software, Prentice Hall, 1984, PDF available.

J. J. Dongarra, J. R.Bunch, C. B. Moler and G. W. Stewart, LINPACK User's Guide, SIAM, 1979, 344 pages, LINPACK User's Guide.

C. Lawson, R. Hanson, D. Kincaid and F. Krogh, "Basic Linear Algebra Subprograms for Fortran Usage, ACM Trans. Math. Software vol. 5, 308-371, 1979. Reprint available.

J. Boyle and K. Dritz, "An automated programming system to aid the development of quality mathematical software," IFIP Proceedings, North Holland, 1974.

\n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


Get the MATLAB code (requires JavaScript)

Published with MATLAB® R2018a

. The first drafts of % papers are to be submitted by August, 2018. That long lead time % gives me the opportunity to write a detailed history of MATLAB. % I plan to write the paper in sections, which I'll post in % this blog as they are available. This is the third such installment. % % LINPACK and EISPACK are Fortran subroutine packages for matrix % computation developed in the 1970's. They are a follow-up to % the Algol 60 procedures that I described in my blog post % about the % . They led to the first version % of MATLAB. This post is about LINPACK. My % was about EISPACK. % % <> %% Cart Before the Horse % Logically a package of subroutines for solving linear equations should % come before a package for computing matrix eigenvalues. The tasks % addressed and the algorithms involved are mathematically simpler. % But chronologically EISPACK came before LINPACK. Actually, the first % section of the _Handbook_ is about linear equations, but we pretty much % skipped over that section when we did EISPACK. The eigenvalue % algorithms were new and we felt it was important to make Fortran % translations widely available. %% LINPACK Project % In 1975, as EISPACK was nearing completion, we proposed to the NSF % another research project investigating methods for the development of % mathematical software. A byproduct would be the software itself. %% % The four principal investigators, and eventual authors of the Users' % Guide, were % % * Jack Dongarra, Argonne National Laboratory % * Cleve Moler, University of New Mexico % * James Bunch, University of California, San Diego % * G. W. (Pete) Stewart, University of Maryland %% % The project was centered at Argonne. The three of us at universities % worked at our home institutions during the academic year and we all % got together at Argonne in the summer. Dongarra and I had previously % worked on EISPACK. %% Contents % Fortran subroutine names at the time were limited to six characters. % We adopted a naming convention with five letter names of the % form TXXYY. The first letter, T, indicates the matrix data type. % Standard Fortran provides three such types and, although not standard, % many systems provide a fourth type, double precision complex, % COMPLEX*16. Unlike EISPACK which follows Algol and has separate arrays % for the real and imaginary parts of a complex matrix, LINPACK takes % full advantage of Fortran complex. The first letter can be % % S Real % D Double precision % C Complex % Z Complex*16 %% % The next two letters, XX, indicate the form of the matrix or its % decomposition. (A packed array is the upper half of a symmetric % matrix stored with one linear subscript.) % % GE General % GB General band % PO Positive definite % PP Positive definite packed % PB Positive definite band % SI Symmetric indefinite % SP Symmetric indefinite packed % HI Hermitian indefinite % HP Hermitian indefinite packed % TR Triangular % GT General tridiagonal % PT Positive definite tridiagonal % CH Cholesky decomposition % QR Orthogonal triangular decomposition % SV Singular value decomposition %% % The final two letters, YY, indicate the computation to be done. % % FA Factor % SL Solve % CO Factor and estimate condition % DI Determinant, inverse, inertia % DC Decompose % UD Update % DD Downdate % EX Exchange %% % This chart shows all 44 of the LINPACK subroutines. An initial % S may be replaced by any of D, C, or Z. An initial C may be % replaced by Z. For example, DGEFA factors a double precision matrix. % This is probably the most important routine in the package. % % CO FA SL DI % SGE * * * * % SGB * * * * % SPO * * * * % SPP * * * * % SPB * * * * % SSI * * * * % SSP * * * * % CHI * * * * % CHP * * * * % STR * * * % SGT * % SPT * % % DC SL UD DD EX % SCH * * * * % SQR * * % SSV * %% % Following this naming convention the subroutine for computing the % SVD, the singular value decomposition, fortuitously becomes SSVDC. %% Programming Style % Fortran at the time was notorious for unstructured, unreadable, % "spaghetti" code. We adopted a disciplined programming style and % expected people as well as machines to read the codes. The scope % of loops and if-then-else constructions were carefully shown by % indenting. Go-to's and statement numbers were used only to % exit blocks and properly handle possibly empty loops. %% % In fact, we, the authors, did not actually write the final programs, % TAMPR did. TAMPR was a powerful software analysis tool developed by % Jim Boyle and Ken Dritz at Argonne. It manipulates and formats Fortran % programs to clarify their structure. It also generates the four data % type variants of the programs. So our source code was the Z versions and % the S, D, and C versions, as well as "clean" Z versions, were producing % by TAMPR. %% % This wasn't just a text processing task. For example, in ZPOFA, % the Cholesky factorization of a positive definite complex Hermitian % matrix, the test for a real, positive diagonal is % % if (s .le. 0.0d0 .or. dimag(a(j,j)) .ne. 0.0d0) go to 40 % % In the real version DPOFA there is no imaginary part, so this becomes % simply % % if (s .le. 0.0d0) go to 40 %% BLAS % All of the inner loops are done by calls to the BLAS, the Basic % Linear Algebra Subprograms, developed concurrently by % and colleagues. On systems that did not have % optimizing Fortran compilers, the BLAS could be implemented efficiently % in machine language. On vector machines, like the CRAY-1, the % loop is a single vector instruction. %% % The two most important BLAS are the inner product of two vectors, DDOT, % and vector plus scalar times vector, DAXPY. All of the algorithms are % column oriented to conform to Fortran storage of two-dimensional arrays % and thereby provide locality of memory access. %% Portability % The programs avoid all machine dependent quantities. There is no need % for anything like the EISPACK parameter MACHEP. Only one of the % algorithms, the SVD, is iterative and involves a convergence test. % Convergence is detected as a special case of a negligible subdiagonal % element. Here is the code for checking if the subdiagonal element % |e(l)| is negligible compared to the two nearby diagonal elements % |s(l)| and |s(l+1)|. I have included enough of the code to show how % TAMPR displays structure by indentation and loop exits by comments with % an initial |c|. % % do 390 ll = 1, m % l = m - ll % c ...exit % if (l .eq. 0) go to 400 % test = dabs(s(l)) + dabs(s(l+1)) % ztest = test + dabs(e(l)) % if (ztest .ne. test) go to 380 % e(l) = 0.0d0 % c ......exit % go to 400 % 380 continue % 390 continue % 400 continue %% Publisher % No commercial book publisher would publish the _LINPACK Users' Guide_. % It did not fit into their established categories; it was neither % a textbook nor a research monograph and was neither mathematics nor % computer science. %% % Only one nonprofit publisher, SIAM, was interested. I vividly remember % sitting on a dock in Lake Mendota at the University of % Wisconsin-Madison during a SIAM annual meeting, talking to SIAM's % Executive Director Ed Block, and deciding that SIAM would take a chance. % The _Guide_ has turned out to be one of SIAM's all-time best sellers. %% % A few years later, in 1981, Beresford Parlett reviewed the publication % in . He wrote % % %

% It may seem strange that SIAM should publish and review a users' % guide for a set of Fortran programs. Yet history will show that % SIAM is thereby helping to foster a new aspect of technology % which currently rejoices in a name mathematical software. % There is as yet no satisfying characterization of this activity, % but it certainly concerns the strong effect that a computer % system has on the efficiency of a program. %

% %% LINPACK Benchmark % Today, LINPACK is better known as a benchmark than as a subroutine % library. I wrote a blog post about the % in 2013. %% In Retrospect % In a sense, the LINPACK and EISPACK projects were failures. We had % proposed research projects to the NSF to "explore the methodology, % costs, and resources required to produce, test, and disseminate % high-quality mathematical software." We never wrote a report or % paper addressing those objectives. We only produced software %% References % J. Dongarra and G. W. Stewart, % "LINPACK, A Package for Solving Linear Systems", % chapter 2 in W. Cowell, 29 pages, % _Sources and Development of Mathematical Software_, % Prentice Hall, 1984, % . % % J. J. Dongarra, J. R.Bunch, C. B. Moler and G. W. Stewart, % _LINPACK User's Guide_, % SIAM, 1979, 344 pages, % . % % C. Lawson, R. Hanson, D. Kincaid and F. Krogh, % "Basic Linear Algebra Subprograms for Fortran Usage, % _ACM Trans. Math. Software_ vol. 5, 308-371, 1979. % . % % J. Boyle and K. Dritz, % "An automated programming system to aid the development of quality % mathematical software," % _IFIP Proceedings_, North Holland, 1974. ##### SOURCE END ##### e86512150129490596d428f178e352a3 -->

EISPACK, Matrix Eigensystem Routines

Wed, 01/03/2018 - 00:52

The ACM Special Interest Group on Programming Languages, SIGPLAN, expects to hold the fourth in a series of conferences on the History of Programming Languages in 2020, see HOPL-IV. The first drafts of papers are to be submitted by August, 2018. That long lead time gives me the opportunity to write a detailed history of MATLAB. I plan to write the paper in sections, which I'll post in this blog as they are available. This is the second such installment.

LINPACK and EISPACK are Fortran subroutine packages for matrix computation developed in the 1970's. They are a follow up to the Algol 60 procedures that I described in my previous blog post about the Wilkinson and Reinsch Handbook. They led to the first version of MATLAB. This post is about EISPACK. A post about LINPACK will follow.

Contents

Argonne

In 1970, even before the Handbook was published, a group at Argonne National Laboratory proposed to the U.S. National Science Foundation (NSF) to "explore the methodology, costs, and resources required to produce, test, and disseminate high-quality mathematical software and to test, certify, disseminate, and support packages of mathematical software in certain problem areas."

Argonne is a national research laboratory west of Chicago that has origins in the University of Chicago's work on the Manhattan project in the 1940s. In 1970 it was under the auspices of what was then called the Atomic Energy Commission (now the Department of Energy). As far as I know, this was the first time anybody at Argonne proposed a research project to the NSF.

At the time I was a summer consultant at Argonne and was spending a sabbatical year at Stanford. Yoshiko Ikebe, from the University of Texas, was also a participant. So initially we called the project NATS, for NSF, Argonne, Texas and Stanford, but that was soon changed to National Activity to Test Software.

NATS took on two projects. EISPACK, Matrix Eigensystem Package, translated the Algol procedures for eigenvalues from the second part of the Handbook to Fortran and worked extensively on testing and portability. FUNPACK, led by Jim Cody, developed rational approximations and software to compute special mathematical functions, including Bessel functions, gamma functions and the error function.

First Release

The Algol 60 procedures in the Wilkinson and Reinsch Handbook provide an excellent reference for the algorithms of matrix computation, but it was difficult to actually run the code because Algol compilers were not readily available.

In the 1970's the clear choice of a language for technical computation was Fortran. The EISPACK team rewrote the Algol in Fortran, being careful to retain the structure and the numerical properties. Wilkinson visited Argonne for a few weeks every summer. He didn't do any Fortran programming, but he provided advice about numerical behavior and testing procedures.

In 1971 the first release of the collection was ready to be sent to about two dozen universities and national laboratories where individuals had shown an interest in the project and agreed to test the software.

The Fortran subroutines in the first release were all derived from the Algol procedures in the second section of the Handbook, so the contents of the package was essentially the same as the contents I listed in my previous blog. Most of the subroutine names describe the underlying algorithm, not the problem being addressed. It is hard to tell from names like TRED1 and HQR2 what the subroutine actually does. There were 34 subroutines in the first release.

With seven authors of the users' guide for the first release, it took almost three years, until 1974, to complete the documentation. Since the project was considered to be language translation, the EISPACK Guide was published by Spring-Verlag, the publisher of the Handbook.

Second Release

By 1976 a second release was ready for public distribution. The number of subroutines nearly doubled, to 70. The additional capabilities included

  • The SVD algorithm of Golub, Kahan, and Reinsch for factoring rectangular matrices and solving overdetermined linear systems. The Algol procedures are in the first section of the Handbook.
  • The QZ algorithm of Moler and Stewart for the generalized eigenvalue problem, $Ax = \lambda Bx$. We developed this in Fortran at the same time as the rest of EISPACK, so there was never any Algol version.
  • A set of "driver routines" with names like RS, for real symmetric, and RSG, for real symmetric generalized, that reflect the problem being solved. These drivers collect the recommended subroutines from the underlying collection into one call.

A second Springer book, called the EISPACK Guide Extension, was published in 1976 to document the additional subroutines.

Floating Point

The hardware landscape that EISPACK faced initially was a diverse collection of mainframes from different manufacturers. The early 1970s was before the age of minicomputers. (The DEC VAX-11/780 was introduced in October, 1975.) It was also long before the IEEE 784 standard for floating point arithmetic. Each manufacturer had different word length and floating point format.

The most prevalent machines were from the IBM 360-370 family. These offered two floating point options. The 32-bit word with four bytes was designated REAL*4 and the 64-bit word was REAL*8. In contrast to today's floating point with the same word length, the 360 employed base 16. So all numbers between powers of 16, not powers of 2, were equally spaced. Jim Cody referred to this as "wobbling precision".

The Fortran distribution was available in several versions that differed in the setting of the machine accuracy parameter, MACHEP. Here are the different machines and corresponding values of MACHEP.

MACHINEMACHEP Burroughs 6700 2.**(-37) CDC 6000-7000 2.**(-47) DEC PDP-10 2.**(-26) Honeywell 6070 2.**(-26) Univac 2.**(-26) IBM 360-370, REAL*4 16.**(-5) IBM 360-370, REAL*8 16.D0**(-13)

Complex Arithmetic

Algol does not have a complex data type, so the derived subroutines do not use Fortran complex arithmetic . Real and imaginary parts of complex quantities are stored in separate arrays.

Certification

The distributed software does not have a copyright notice or involve any kind of license. The term "open source" was not yet widely used. The closest we came to a license agreement was a statement about "certification". Both guides listed the machines, operating systems, and compilers of the 15 test sites. About half of the sites were universities and half were government labs. One was from Canada and one from Sweden. Various models from six different manufactures were involved.

This list of test sites was followed by the following statement about support.

Certification implies the full support of the NATS project in the sense that reports of poor or incorrect performance on at least the computer systems listed above will be examined and any necessary corrections made. This support holds only when the software is obtained through the channels indicated below and has not been modified; it will continue to hold throughout the useful life of EISPACK or until, in the estimation of the developers, the package is superseded or incorporated into other supported, widely-available program libraries. The following individual serves as a contact about EISPACK, accepting reports from users concerning performance:

This was followed by Burt Garbow's mailing address and phone number at Argonne. As far as I know, Burt never received any serious bug reports or complaints.

References

J. Dongarra and C. Moler, "EISPACK, A Package for Solving Matrix Eigenvalue Problems", chapter 4 in W. Cowell, Sources and Development of Mathematical Software, Prentice Hall, 1984, 10 pages, PDF available.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide, 2nd Edition, vol. 6 in Lecture Notes in Computer Science, Springer, 1974, 551 pages, EISPACK Guide.

B. S. Garbow, J. M. Boyle, J. J. Dongarra and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide Extension, vol. 51 in Lecture Notes in Computer Science, Springer, 1977, 343 pages, EISPACK Guide Extension.

\n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


Get the MATLAB code (requires JavaScript)

Published with MATLAB® R2017a

. The first drafts of % papers are to be submitted by August, 2018. That long lead time % gives me the opportunity to write a detailed history of MATLAB. % I plan to write the paper in sections, which I'll post in % this blog as they are available. This is the second such installment. % % LINPACK and EISPACK are Fortran subroutine packages for matrix % computation developed in the 1970's. They are a follow up to % the Algol 60 procedures that I described in my previous blog post % about the % . They led to the first version % of MATLAB. This post is about EISPACK. A post about LINPACK % will follow. %% % % <> %% Argonne % In 1970, even before the _Handbook_ was published, a group % at Argonne National Laboratory proposed to the U.S. National % Science Foundation (NSF) to "explore the methodology, costs, and % resources required to produce, test, and disseminate high-quality % mathematical software and to test, certify, disseminate, and % support packages of mathematical software in certain problem % areas." % % Argonne is a national research laboratory west of Chicago % that has origins in the University of Chicago's work on the % Manhattan project in the 1940s. In 1970 it was under the % auspices of what was then called the Atomic Energy Commission % (now the Department of Energy). As far as I know, this was the % first time anybody at Argonne proposed a research project to the % NSF. % % At the time I was a summer consultant at Argonne and was spending % a sabbatical year at Stanford. Yoshiko Ikebe, from the University % of Texas, was also a participant. So initially we called the project % NATS, for NSF, Argonne, Texas and Stanford, but that was soon % changed to National Activity to Test Software. % % NATS took on two projects. EISPACK, Matrix Eigensystem Package, % translated the Algol procedures for eigenvalues from the second part % of the _Handbook_ to Fortran and worked extensively on testing and % portability. FUNPACK, led by Jim Cody, developed rational % approximations and software to compute special mathematical functions, % including Bessel functions, gamma functions and the error function. %% First Release % The Algol 60 procedures in the % Wilkinson and Reinsch _Handbook_ provide an excellent reference for % the algorithms of matrix computation, but it was difficult % to actually run the code because Algol compilers were not readily % available. % % In the 1970's the clear choice of a language for technical computation % was Fortran. The EISPACK team rewrote the Algol in Fortran, being % careful to retain the structure and the numerical properties. % Wilkinson visited Argonne for a few weeks every summer. % He didn't do any Fortran programming, but he provided advice about % numerical behavior and testing procedures. % % In 1971 the first release of the collection was ready to be sent to % about two dozen universities and national laboratories where individuals % had shown an interest in the project and agreed to test the software. % % The Fortran subroutines in the first release were all derived from % the Algol procedures in the second section of the _Handbook_, % so the contents % of the package was essentially the same as the contents I listed in % . Most of the subroutine names describe the % underlying algorithm, not the problem being addressed. It is hard % to tell from names like TRED1 and HQR2 what the subroutine actually % does. There were 34 subroutines in the first release. % % With seven authors of the users' guide for the first release, % it took almost three years, until 1974, to complete the documentation. % Since the project was considered to be language translation, % the _EISPACK Guide_ was published by Spring-Verlag, % the publisher of the _Handbook_. %% Second Release % By 1976 a second release was ready for public distribution. % The number of subroutines nearly doubled, to 70. % The additional capabilities included % % * The SVD algorithm of Golub, Kahan, and Reinsch for factoring % rectangular matrices and solving overdetermined linear systems. % The Algol procedures are in the first section of the _Handbook_. % % * The QZ algorithm of Moler and Stewart for the generalized % eigenvalue problem, $Ax = \lambda Bx$. We developed this in Fortran % at the same time as the rest of EISPACK, so there was never any Algol % version. % % * A set of "driver routines" with names like RS, for real symmetric, % and RSG, for real symmetric generalized, that reflect the % problem being solved. These drivers collect the recommended % subroutines from the underlying collection into one call. %% % A second Springer book, called the _EISPACK Guide Extension_, was % published in 1976 to document the additional subroutines. %% Floating Point % The hardware landscape that EISPACK faced initially was a diverse % collection of mainframes from different manufacturers. The early % 1970s was before the age of minicomputers. (The DEC VAX-11/780 was % introduced in October, 1975.) It was also long before the IEEE 784 % standard for floating point arithmetic. Each manufacturer had % different word length and floating point format. % % The most prevalent machines were from the IBM 360-370 family. These % offered two floating point options. The 32-bit word with four bytes % was designated REAL*4 and the 64-bit word was REAL*8. In contrast to % today's floating point with the same word length, the 360 employed % base 16. So all numbers between powers of 16, not powers of 2, were % equally spaced. Jim Cody referred to this as "wobbling precision". % % The Fortran distribution was available in several versions that % differed in the setting of the machine accuracy parameter, MACHEP. % Here are the different machines and corresponding values of MACHEP. % % % MACHINEMACHEP % Burroughs 6700 2.**(-37) % CDC 6000-7000 2.**(-47) % DEC PDP-10 2.**(-26) % Honeywell 6070 2.**(-26) % Univac 2.**(-26) % IBM 360-370, REAL*4 16.**(-5) % IBM 360-370, REAL*8 16.D0**(-13) % %% Complex Arithmetic % Algol does not have a complex data type, so the derived subroutines % do not use Fortran complex arithmetic . Real and imaginary parts % of complex quantities are stored in separate arrays. %% Certification % The distributed software does not have a copyright notice or involve % any kind of license. The term "open source" was not yet widely used. % The closest we came to a license agreement was a statement about % "certification". % Both guides listed the machines, operating systems, and compilers % of the 15 test sites. About half of the sites were universities % and half were government labs. One was from Canada and one from Sweden. % Various models from six different manufactures were involved. %% % This list of test sites was followed by the following statement about % support. % % %

% Certification implies the full support of the NATS project in the % sense that reports of poor or incorrect performance on at least the % computer systems listed above will be examined and any necessary % corrections made. This support holds only when the software is % obtained through the channels indicated below and has not been % modified; it will continue to hold throughout the useful life % of EISPACK or until, in the estimation of the developers, the % package is superseded or incorporated into other supported, % widely-available program libraries. The following individual serves % as a contact about EISPACK, accepting reports from users concerning % performance: %

% % % This was followed by Burt Garbow's mailing address and phone number % at Argonne. As far as I know, Burt never received any serious % bug reports or complaints. %% References % J. Dongarra and C. Moler, % "EISPACK, A Package for Solving Matrix Eigenvalue Problems", % chapter 4 in W. Cowell, % _Sources and Development of Mathematical Software_, % Prentice Hall, 1984, 10 pages, % . % % B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, % V. C. Klema, and C. B. Moler, % _Matrix Eigensystem Routines - EISPACK Guide, 2nd Edition_, % vol. 6 in % Lecture Notes in Computer Science, Springer, 1974, 551 pages, % . % % B. S. Garbow, J. M. Boyle, J. J. Dongarra and C. B. Moler, % _Matrix Eigensystem Routines - EISPACK Guide Extension_, % vol. 51 in % Lecture Notes in Computer Science, Springer, 1977, 343 pages, % . ##### SOURCE END ##### dec80a0ea4c64d958d1852f1a699360f -->

EISPACK, Matrix Eigensystem Routines

Wed, 01/03/2018 - 00:52

The ACM Special Interest Group on Programming Languages, SIGPLAN, expects to hold the fourth in a series of conferences on the History of Programming Languages in 2020, see HOPL-IV. The first drafts of papers are to be submitted by August, 2018. That long lead time gives me the opportunity to write a detailed history of MATLAB. I plan to write the paper in sections, which I'll post in this blog as they are available. This is the second such installment.

LINPACK and EISPACK are Fortran subroutine packages for matrix computation developed in the 1970's. They are a follow up to the Algol 60 procedures that I described in my previous blog post about the Wilkinson and Reinsch Handbook. They led to the first version of MATLAB. This post is about EISPACK. A post about LINPACK will follow.

Contents

Argonne

In 1970, even before the Handbook was published, a group at Argonne National Laboratory proposed to the U.S. National Science Foundation (NSF) to "explore the methodology, costs, and resources required to produce, test, and disseminate high-quality mathematical software and to test, certify, disseminate, and support packages of mathematical software in certain problem areas."

Argonne is a national research laboratory west of Chicago that has origins in the University of Chicago's work on the Manhattan project in the 1940s. In 1970 it was under the auspices of what was then called the Atomic Energy Commission (now the Department of Energy). As far as I know, this was the first time anybody at Argonne proposed a research project to the NSF.

At the time I was a summer consultant at Argonne and was spending a sabbatical year at Stanford. Yoshiko Ikebe, from the University of Texas, was also a participant. So initially we called the project NATS, for NSF, Argonne, Texas and Stanford, but that was soon changed to National Activity to Test Software.

NATS took on two projects. EISPACK, Matrix Eigensystem Package, translated the Algol procedures for eigenvalues from the second part of the Handbook to Fortran and worked extensively on testing and portability. FUNPACK, led by Jim Cody, developed rational approximations and software to compute special mathematical functions, including Bessel functions, gamma functions and the error function.

First Release

The Algol 60 procedures in the Wilkinson and Reinsch Handbook provide an excellent reference for the algorithms of matrix computation, but it was difficult to actually run the code because Algol compilers were not readily available.

In the 1970's the clear choice of a language for technical computation was Fortran. The EISPACK team rewrote the Algol in Fortran, being careful to retain the structure and the numerical properties. Wilkinson visited Argonne for a few weeks every summer. He didn't do any Fortran programming, but he provided advice about numerical behavior and testing procedures.

In 1971 the first release of the collection was ready to be sent to about two dozen universities and national laboratories where individuals had shown an interest in the project and agreed to test the software.

The Fortran subroutines in the first release were all derived from the Algol procedures in the second section of the Handbook, so the contents of the package was essentially the same as the contents I listed in my previous blog. Most of the subroutine names describe the underlying algorithm, not the problem being addressed. It is hard to tell from names like TRED1 and HQR2 what the subroutine actually does. There were 34 subroutines in the first release.

With seven authors of the users' guide for the first release, it took almost three years, until 1974, to complete the documentation. Since the project was considered to be language translation, the EISPACK Guide was published by Spring-Verlag, the publisher of the Handbook.

Second Release

By 1976 a second release was ready for public distribution. The number of subroutines nearly doubled, to 70. The additional capabilities included

  • The SVD algorithm of Golub, Kahan, and Reinsch for factoring rectangular matrices and solving overdetermined linear systems. The Algol procedures are in the first section of the Handbook.
  • The QZ algorithm of Moler and Stewart for the generalized eigenvalue problem, $Ax = \lambda Bx$. We developed this in Fortran at the same time as the rest of EISPACK, so there was never any Algol version.
  • A set of "driver routines" with names like RS, for real symmetric, and RSG, for real symmetric generalized, that reflect the problem being solved. These drivers collect the recommended subroutines from the underlying collection into one call.

A second Springer book, called the EISPACK Guide Extension, was published in 1976 to document the additional subroutines.

Floating Point

The hardware landscape that EISPACK faced initially was a diverse collection of mainframes from different manufacturers. The early 1970s was before the age of minicomputers. (The DEC VAX-11/780 was introduced in October, 1975.) It was also long before the IEEE 784 standard for floating point arithmetic. Each manufacturer had different word length and floating point format.

The most prevalent machines were from the IBM 360-370 family. These offered two floating point options. The 32-bit word with four bytes was designated REAL*4 and the 64-bit word was REAL*8. In contrast to today's floating point with the same word length, the 360 employed base 16. So all numbers between powers of 16, not powers of 2, were equally spaced. Jim Cody referred to this as "wobbling precision".

The Fortran distribution was available in several versions that differed in the setting of the machine accuracy parameter, MACHEP. Here are the different machines and corresponding values of MACHEP.

MACHINEMACHEP Burroughs 6700 2.**(-37) CDC 6000-7000 2.**(-47) DEC PDP-10 2.**(-26) Honeywell 6070 2.**(-26) Univac 2.**(-26) IBM 360-370, REAL*4 16.**(-5) IBM 360-370, REAL*8 16.D0**(-13)

Complex Arithmetic

Algol does not have a complex data type, so the derived subroutines do not use Fortran complex arithmetic . Real and imaginary parts of complex quantities are stored in separate arrays.

Certification

The distributed software does not have a copyright notice or involve any kind of license. The term "open source" was not yet widely used. The closest we came to a license agreement was a statement about "certification". Both guides listed the machines, operating systems, and compilers of the 15 test sites. About half of the sites were universities and half were government labs. One was from Canada and one from Sweden. Various models from six different manufactures were involved.

This list of test sites was followed by the following statement about support.

Certification implies the full support of the NATS project in the sense that reports of poor or incorrect performance on at least the computer systems listed above will be examined and any necessary corrections made. This support holds only when the software is obtained through the channels indicated below and has not been modified; it will continue to hold throughout the useful life of EISPACK or until, in the estimation of the developers, the package is superseded or incorporated into other supported, widely-available program libraries. The following individual serves as a contact about EISPACK, accepting reports from users concerning performance:

This was followed by Burt Garbow's mailing address and phone number at Argonne. As far as I know, Burt never received any serious bug reports or complaints.

References

J. Dongarra and C. Moler, "EISPACK, A Package for Solving Matrix Eigenvalue Problems", chapter 4 in W. Cowell, Sources and Development of Mathematical Software, Prentice Hall, 1984, 10 pages, PDF available.

B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, V. C. Klema, and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide, 2nd Edition, vol. 6 in Lecture Notes in Computer Science, Springer, 1974, 551 pages, EISPACK Guide.

B. S. Garbow, J. M. Boyle, J. J. Dongarra and C. B. Moler, Matrix Eigensystem Routines - EISPACK Guide Extension, vol. 51 in Lecture Notes in Computer Science, Springer, 1977, 343 pages, EISPACK Guide Extension.

\n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


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Published with MATLAB® R2017a

. The first drafts of % papers are to be submitted by August, 2018. That long lead time % gives me the opportunity to write a detailed history of MATLAB. % I plan to write the paper in sections, which I'll post in % this blog as they are available. This is the second such installment. % % LINPACK and EISPACK are Fortran subroutine packages for matrix % computation developed in the 1970's. They are a follow up to % the Algol 60 procedures that I described in my previous blog post % about the % . They led to the first version % of MATLAB. This post is about EISPACK. A post about LINPACK % will follow. %% % % <> %% Argonne % In 1970, even before the _Handbook_ was published, a group % at Argonne National Laboratory proposed to the U.S. National % Science Foundation (NSF) to "explore the methodology, costs, and % resources required to produce, test, and disseminate high-quality % mathematical software and to test, certify, disseminate, and % support packages of mathematical software in certain problem % areas." % % Argonne is a national research laboratory west of Chicago % that has origins in the University of Chicago's work on the % Manhattan project in the 1940s. In 1970 it was under the % auspices of what was then called the Atomic Energy Commission % (now the Department of Energy). As far as I know, this was the % first time anybody at Argonne proposed a research project to the % NSF. % % At the time I was a summer consultant at Argonne and was spending % a sabbatical year at Stanford. Yoshiko Ikebe, from the University % of Texas, was also a participant. So initially we called the project % NATS, for NSF, Argonne, Texas and Stanford, but that was soon % changed to National Activity to Test Software. % % NATS took on two projects. EISPACK, Matrix Eigensystem Package, % translated the Algol procedures for eigenvalues from the second part % of the _Handbook_ to Fortran and worked extensively on testing and % portability. FUNPACK, led by Jim Cody, developed rational % approximations and software to compute special mathematical functions, % including Bessel functions, gamma functions and the error function. %% First Release % The Algol 60 procedures in the % Wilkinson and Reinsch _Handbook_ provide an excellent reference for % the algorithms of matrix computation, but it was difficult % to actually run the code because Algol compilers were not readily % available. % % In the 1970's the clear choice of a language for technical computation % was Fortran. The EISPACK team rewrote the Algol in Fortran, being % careful to retain the structure and the numerical properties. % Wilkinson visited Argonne for a few weeks every summer. % He didn't do any Fortran programming, but he provided advice about % numerical behavior and testing procedures. % % In 1971 the first release of the collection was ready to be sent to % about two dozen universities and national laboratories where individuals % had shown an interest in the project and agreed to test the software. % % The Fortran subroutines in the first release were all derived from % the Algol procedures in the second section of the _Handbook_, % so the contents % of the package was essentially the same as the contents I listed in % . Most of the subroutine names describe the % underlying algorithm, not the problem being addressed. It is hard % to tell from names like TRED1 and HQR2 what the subroutine actually % does. There were 34 subroutines in the first release. % % With seven authors of the users' guide for the first release, % it took almost three years, until 1974, to complete the documentation. % Since the project was considered to be language translation, % the _EISPACK Guide_ was published by Spring-Verlag, % the publisher of the _Handbook_. %% Second Release % By 1976 a second release was ready for public distribution. % The number of subroutines nearly doubled, to 70. % The additional capabilities included % % * The SVD algorithm of Golub, Kahan, and Reinsch for factoring % rectangular matrices and solving overdetermined linear systems. % The Algol procedures are in the first section of the _Handbook_. % % * The QZ algorithm of Moler and Stewart for the generalized % eigenvalue problem, $Ax = \lambda Bx$. We developed this in Fortran % at the same time as the rest of EISPACK, so there was never any Algol % version. % % * A set of "driver routines" with names like RS, for real symmetric, % and RSG, for real symmetric generalized, that reflect the % problem being solved. These drivers collect the recommended % subroutines from the underlying collection into one call. %% % A second Springer book, called the _EISPACK Guide Extension_, was % published in 1976 to document the additional subroutines. %% Floating Point % The hardware landscape that EISPACK faced initially was a diverse % collection of mainframes from different manufacturers. The early % 1970s was before the age of minicomputers. (The DEC VAX-11/780 was % introduced in October, 1975.) It was also long before the IEEE 784 % standard for floating point arithmetic. Each manufacturer had % different word length and floating point format. % % The most prevalent machines were from the IBM 360-370 family. These % offered two floating point options. The 32-bit word with four bytes % was designated REAL*4 and the 64-bit word was REAL*8. In contrast to % today's floating point with the same word length, the 360 employed % base 16. So all numbers between powers of 16, not powers of 2, were % equally spaced. Jim Cody referred to this as "wobbling precision". % % The Fortran distribution was available in several versions that % differed in the setting of the machine accuracy parameter, MACHEP. % Here are the different machines and corresponding values of MACHEP. % % % MACHINEMACHEP % Burroughs 6700 2.**(-37) % CDC 6000-7000 2.**(-47) % DEC PDP-10 2.**(-26) % Honeywell 6070 2.**(-26) % Univac 2.**(-26) % IBM 360-370, REAL*4 16.**(-5) % IBM 360-370, REAL*8 16.D0**(-13) % %% Complex Arithmetic % Algol does not have a complex data type, so the derived subroutines % do not use Fortran complex arithmetic . Real and imaginary parts % of complex quantities are stored in separate arrays. %% Certification % The distributed software does not have a copyright notice or involve % any kind of license. The term "open source" was not yet widely used. % The closest we came to a license agreement was a statement about % "certification". % Both guides listed the machines, operating systems, and compilers % of the 15 test sites. About half of the sites were universities % and half were government labs. One was from Canada and one from Sweden. % Various models from six different manufactures were involved. %% % This list of test sites was followed by the following statement about % support. % % %

% Certification implies the full support of the NATS project in the % sense that reports of poor or incorrect performance on at least the % computer systems listed above will be examined and any necessary % corrections made. This support holds only when the software is % obtained through the channels indicated below and has not been % modified; it will continue to hold throughout the useful life % of EISPACK or until, in the estimation of the developers, the % package is superseded or incorporated into other supported, % widely-available program libraries. The following individual serves % as a contact about EISPACK, accepting reports from users concerning % performance: %

% % % This was followed by Burt Garbow's mailing address and phone number % at Argonne. As far as I know, Burt never received any serious % bug reports or complaints. %% References % J. Dongarra and C. Moler, % "EISPACK, A Package for Solving Matrix Eigenvalue Problems", % chapter 4 in W. Cowell, % _Sources and Development of Mathematical Software_, % Prentice Hall, 1984, 10 pages, % . % % B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, Y. Ikebe, % V. C. Klema, and C. B. Moler, % _Matrix Eigensystem Routines - EISPACK Guide, 2nd Edition_, % vol. 6 in % Lecture Notes in Computer Science, Springer, 1974, 551 pages, % . % % B. S. Garbow, J. M. Boyle, J. J. Dongarra and C. B. Moler, % _Matrix Eigensystem Routines - EISPACK Guide Extension_, % vol. 51 in % Lecture Notes in Computer Science, Springer, 1977, 343 pages, % . ##### SOURCE END ##### dec80a0ea4c64d958d1852f1a699360f -->

Bug in Half-Precision Floating Point Object

Thu, 12/21/2017 - 02:56

My post on May 8 was about "half-precision" and "quarter-precision" arithmetic. I also added code for objects fp16 and fp8 to Cleve's Laboratory. A few days ago I heard from Pierre Blanchard and my good friend Nick Higham at the University of Manchester about a serious bug in the constructors for those objects.

ContentsThe Bug

Let

format long e = eps(fp16(1)) e = 9.765625000000000e-04

The value of e is

1/1024 ans = 9.765625000000000e-04

This is the relative precision of half-precision floating point numbers, which is the spacing of half-precision numbers in the interval between 1 and 2. So in binary the next number after 1 is

disp(binary(1+e)) 0 01111 0000000001

And the last number before 2 is

disp(binary(2-e)) 0 01111 1111111111

The three fields displayed are the sign, which is one bit, the exponent, which has five bits and the fraction, which has ten bits.

So far, so good. The bug shows up when I try to convert any number between 2-e and 2 to half-precision. There aren't any more half-precision numbers between those limits. The values in the lower half of the interval should round down to 2-e and the values in the upper half should round up to 2. The round-to-even convention says that the midpoint, 2-e/2, should round to 2.

But I wasn't careful about how I did the rounding. I just used the MATLAB round function, which doesn't follow the round-to-even convention. Worse yet, I didn't check to see if the fraction rounded up into the exponent field. I tried to do everything in one statement.

dbtype oldfp16 48:49 48 u = bitxor(uint16(round(1024*f)), ... 49 bitshift(uint16(e+15),10));

For values between 2-e/2 and 2, the round(1024*f) is 1024, which requires eleven bits. The bitxor then clobbers the exponent field. I won't show the result here. If you have the May half-precision object on your machine, give it a try.

This doesn't just happen for values a little bit less than 2, it happens close to any power of 2.

The Fix

We need a proper round-to-nearest-even function.

dbtype fp16 31 31 rndevn = @(s) round(s-(rem(s,2)==0.5));

Then don't try to do it all at once.

dbtype fp16 50:56 50 % normal 51 t = uint16(rndevn(1024*f)); 52 if t==1024 53 t = uint16(0); 54 e = e+1; 55 end 56 u = bitxor(t, bitshift(uint16(e+15),10));

It turns out that the branch for denormals is OK, once round is replaced by rndeven. The exponent for denormals is all zeros, so when the fraction encroaches it produces the correct result.

A similar fix is required for the quarter-precision constructor, fp8.

Cleve's Laboratory

I am updating the code on the MATLAB Central File Exchange. Only @fp16/fp16 and @fp8/fp8 are affected. (Give me a few days to complete the update process.)

Thanks

Thanks to Pierre and Nick.

\n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


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Published with MATLAB® R2017a

was about "half-precision" and "quarter-precision" % arithmetic. I also added code for objects |fp16| and |fp8| to % . A few days ago I heard from Pierre Blanchard and my % good friend Nick Higham at the University of Manchester about a serious % bug in the constructors for those objects. %% The Bug % Let format long e = eps(fp16(1)) %% % The value of |e| is 1/1024 %% % This is the relative precision of half-precision floating point % numbers, which is the spacing of half-precision numbers in the interval % between 1 and 2. So in binary the next number after 1 is disp(binary(1+e)) %% % And the last number before 2 is disp(binary(2-e)) %% % The three fields displayed are the sign, which is one bit, the exponent, % which has five bits and the fraction, which has ten bits. %% % So far, so good. The bug shows up when I try to convert any % number between |2-e| and |2| to half-precision. There aren't any more % half-precision numbers between those limits. The values in the lower % half of the interval should round down to |2-e| and the values in the % upper half should round up to |2|. The round-to-even convention % says that the midpoint, |2-e/2|, should round to |2|. %% % But I wasn't careful about how I did the rounding. % I just used the MATLAB |round| function, which doesn't follow the % round-to-even convention. Worse yet, I didn't check to see if the % fraction rounded up into the exponent field. I tried to do % everything in one statement. dbtype oldfp16 48:49 %% % For values between |2-e/2| and |2|, the |round(1024*f)| is |1024|, which % requires eleven bits. The |bitxor| then clobbers the exponent field. % I won't show the result here. If you have the May half-precision object % on your machine, give it a try. %% % This doesn't just happen for values a little bit less than 2, it % happens close to any power of 2. %% The Fix % We need a proper round-to-nearest-even function. dbtype fp16 31 %% % Then don't try to do it all at once. dbtype fp16 50:56 %% % It turns out that the branch for denormals is OK, once |round| is % replaced by |rndeven|. The exponent for denormals is all zeros, so % when the fraction encroaches it produces the correct result. %% % A similar fix is required for the quarter-precision constructor, |fp8|. %% Cleve's Laboratory % I am updating the code on the % . Only |@fp16/fp16| and |@fp8/fp8| are % affected. (Give me a few days to complete the update process.) %% Thanks % Thanks to Pierre and Nick. ##### SOURCE END ##### 8d4defc65bb14c19941eaea060051666 -->

Bug in Half-Precision Floating Point Object

Thu, 12/21/2017 - 02:56

My post on May 8 was about "half-precision" and "quarter-precision" arithmetic. I also added code for objects fp16 and fp8 to Cleve's Laboratory. A few days ago I heard from Pierre Blanchard and my good friend Nick Higham at the University of Manchester about a serious bug in the constructors for those objects.

ContentsThe Bug

Let

format long e = eps(fp16(1)) e = 9.765625000000000e-04

The value of e is

1/1024 ans = 9.765625000000000e-04

This is the relative precision of half-precision floating point numbers, which is the spacing of half-precision numbers in the interval between 1 and 2. So in binary the next number after 1 is

disp(binary(1+e)) 0 01111 0000000001

And the last number before 2 is

disp(binary(2-e)) 0 01111 1111111111

The three fields displayed are the sign, which is one bit, the exponent, which has five bits and the fraction, which has ten bits.

So far, so good. The bug shows up when I try to convert any number between 2-e and 2 to half-precision. There aren't any more half-precision numbers between those limits. The values in the lower half of the interval should round down to 2-e and the values in the upper half should round up to 2. The round-to-even convention says that the midpoint, 2-e/2, should round to 2.

But I wasn't careful about how I did the rounding. I just used the MATLAB round function, which doesn't follow the round-to-even convention. Worse yet, I didn't check to see if the fraction rounded up into the exponent field. I tried to do everything in one statement.

dbtype oldfp16 48:49 48 u = bitxor(uint16(round(1024*f)), ... 49 bitshift(uint16(e+15),10));

For values between 2-e/2 and 2, the round(1024*f) is 1024, which requires eleven bits. The bitxor then clobbers the exponent field. I won't show the result here. If you have the May half-precision object on your machine, give it a try.

This doesn't just happen for values a little bit less than 2, it happens close to any power of 2.

The Fix

We need a proper round-to-nearest-even function.

dbtype fp16 31 31 rndevn = @(s) round(s-(rem(s,2)==0.5));

Then don't try to do it all at once.

dbtype fp16 50:56 50 % normal 51 t = uint16(rndevn(1024*f)); 52 if t==1024 53 t = uint16(0); 54 e = e+1; 55 end 56 u = bitxor(t, bitshift(uint16(e+15),10));

It turns out that the branch for denormals is OK, once round is replaced by rndeven. The exponent for denormals is all zeros, so when the fraction encroaches it produces the correct result.

A similar fix is required for the quarter-precision constructor, fp8.

Cleve's Laboratory

I am updating the code on the MATLAB Central File Exchange. Only @fp16/fp16 and @fp8/fp8 are affected. (Give me a few days to complete the update process.)

Thanks

Thanks to Pierre and Nick.

\n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


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Published with MATLAB® R2017a

was about "half-precision" and "quarter-precision" % arithmetic. I also added code for objects |fp16| and |fp8| to % . A few days ago I heard from Pierre Blanchard and my % good friend Nick Higham at the University of Manchester about a serious % bug in the constructors for those objects. %% The Bug % Let format long e = eps(fp16(1)) %% % The value of |e| is 1/1024 %% % This is the relative precision of half-precision floating point % numbers, which is the spacing of half-precision numbers in the interval % between 1 and 2. So in binary the next number after 1 is disp(binary(1+e)) %% % And the last number before 2 is disp(binary(2-e)) %% % The three fields displayed are the sign, which is one bit, the exponent, % which has five bits and the fraction, which has ten bits. %% % So far, so good. The bug shows up when I try to convert any % number between |2-e| and |2| to half-precision. There aren't any more % half-precision numbers between those limits. The values in the lower % half of the interval should round down to |2-e| and the values in the % upper half should round up to |2|. The round-to-even convention % says that the midpoint, |2-e/2|, should round to |2|. %% % But I wasn't careful about how I did the rounding. % I just used the MATLAB |round| function, which doesn't follow the % round-to-even convention. Worse yet, I didn't check to see if the % fraction rounded up into the exponent field. I tried to do % everything in one statement. dbtype oldfp16 48:49 %% % For values between |2-e/2| and |2|, the |round(1024*f)| is |1024|, which % requires eleven bits. The |bitxor| then clobbers the exponent field. % I won't show the result here. If you have the May half-precision object % on your machine, give it a try. %% % This doesn't just happen for values a little bit less than 2, it % happens close to any power of 2. %% The Fix % We need a proper round-to-nearest-even function. dbtype fp16 31 %% % Then don't try to do it all at once. dbtype fp16 50:56 %% % It turns out that the branch for denormals is OK, once |round| is % replaced by |rndeven|. The exponent for denormals is all zeros, so % when the fraction encroaches it produces the correct result. %% % A similar fix is required for the quarter-precision constructor, |fp8|. %% Cleve's Laboratory % I am updating the code on the % . Only |@fp16/fp16| and |@fp8/fp8| are % affected. (Give me a few days to complete the update process.) %% Thanks % Thanks to Pierre and Nick. ##### SOURCE END ##### 8d4defc65bb14c19941eaea060051666 -->

Wilkinson and Reinsch Handbook on Linear Algebra

Tue, 12/05/2017 - 05:00

The ACM Special Interest Group on Programming Languages, SIGPLAN, expects to hold the fourth in a series of conferences on the History of Programming Languages in 2020, see HOPL-IV. The first drafts of papers are to be submitted by August, 2018. That long lead time gives me the opportunity to write a detailed history of MATLAB. I plan to write the paper in sections, which I'll post in this blog as they are available.

The mathematical basis for the first version of MATLAB begins with a volume edited by J. H. Wilkinson and C. Reinsch, Handbook for Automatic Computation, Volume II, Linear Algebra, published by the eminent German publisher Springer-Verlag, in 1971.

ContentsEigenvalues

The mathematical term "eigenvalue" is a linguistic hybrid. In German the term is "eigenwert". One Web dictionary says the English translation is "intrinsic value". But after a few decades using terms like "characteristic value" and "latent root", mathematicians have given up trying to translate the entire word and generally translate only the second half.

The eigenvalue problem for a given matrix $A$ is the task of finding scalars $\lambda$ and, possibly, the corresponding vectors $x$ so that

$$Ax = \lambda x$$

It is important to distinguish the case where $A$ is symmetric. After the linear equation problem,

$$Ax = b$$

the eigenvalue problem is the most important computational task in numerical linear algebra.

The state of the art in numerical linear algebra 50 years ago, in the 1960's, did not yet provide reliable, efficient methods for solving matrix eigenvalue problems. Software libraries had a hodgepodge of methods, many of them based upon polynomial root finders.

Jim Wilkinson told a story about having two such subroutines, one using something called Bairstow's method and one using something called Laguerre's method. He couldn't decide which one was best, so he put them together in a program that first tried Bairstow's method, then printed a message and switched to Laguerre if Bairstow failed. After several months of frequent use at the National Physical Laboratory, he never saw a case where Bairstow failed. He wrote a paper with his recommendation for Bairstow's method and soon got a letter from a reader who claimed to have an example which Bairstow could not handle. Wilkinson tried the example with his program and found a bug. The program didn't print the message that it was switching to Laguerre.

Handbook for Automatic Computation

Over the period 1965-1970, Wilkinson and 18 of his colleagues published a series of papers in the Springer journal Numerische Mathematik. The papers offered procedures, written in Algol 60, for solving different versions of the linear equation problem and the eigenvalue problem. These were research papers presenting results about numerical stability, subtle details of implementation, and, in some cases, new methods.

In 1971, these papers were collected, sometimes with modifications, in the volume edited by Wilkinson and Reinsch. This book marks the first appearance is an organized library of the algorithms for the eigenvalue problem for dense, stored matrices that we still use in MATLAB today. The importance of using orthogonal transformations wherever possible was exposed by Wilkinson and the other authors. The effectiveness of the newly discovered QR algorithms of J. Francis and the related LR algorithm of H. Rutishauser had only recently been appreciated.

Contents

The Algol 60 procedures are the focus of each chapter. These codes remain a clear, readable reference for the important ideas of modern numerical linear algebra. Part I of the volume is about the linear equation problem; part II is about the eigenvalue problem. There are 40 procedures in part I and 43 in part II.

Here is a list of the procedures in Part II. A suffix 2 in the procedure name indicates that it computes both eigenvalues and eigenvectors. The "bak" procedures apply reduction transformations to eigenvectors.

Many of the procedures work with a reduced form, which is tridiagonal for symmetric or Hermitian matrices and Hessenberg (upper triangular plus one subdiagonal) for nonsymmetric matrices., Since Algol does not have a complex number data type, the complex arrays are represented by pairs of real arrays.

Symmetric matrices

Reduction to tridiagonal tred1, tred2, tred3, trbak1, trbak3 Orthogonal tridiagonalization. Band bandrd Tridiagonalization. symray Eigenvectors. bqr One eigenvalue. Tridiagonal imtql1,imtql2 All eigenvalues and vectors, implicit QR. tql1, tql2 All eigenvalues and vectors, explicit QR. ratqr Few eigenvalues, rational QR. bisect Few eigenvalues, bisection. tristurm Few eigenvalues and vectors. Few eigenvalues ritzit Simultaneous iteration. Jacobi method jacobi Jacobi method. Generalized problem, Ax = \lambda Bx reduc1, reduc2, rebaka, rebakb Symmetric A, positive definite B.

Nonsymmetric matrices

Reduction to Hessenberg balance, balbak Balance (scaling) dirhes, dirbak, dirtrans Elementary, accumulated innerprod. elmhes, elmbak, elmtrans Elementary. orthes, ortbak, ortrans Orthogonal. Band unsray Eigenvectors. Hessenberg hqr, hqr2 All eigenvalues and vectors, implicit QR. invit Few eigenvectors, inverse iteration. Norm reducing eigen Eigenvalues.

Complex matrices

comeig Norm reducing Jacobi. comhes, combak Reduce to Hessenberg form. comlr, comlr2 Complex LR algorithm. cxinvit Inverse iteration.

Preferred paths

The preferred path for finding all the eigenvalues of a real, symmetric matrix is tred1 followed by imtql1. The preferred path for finding all the eigenvalues and eigenvectors of a real, symmetric matrix is tred2 followed by imtql2.

The preferred path for finding all the eigenvalues of a real, nonsymmetric matrix is balanc, elmhes, and finally hqr. The preferred path for finding all the eigenvalues and eigenvectors of a real, nonsymmetric matrix is balanc, elmhes, elmtrans, hqr2, and finally balbak.

QR vs. QL

"QR" and "QL" are right- and left-handed, or forward and backward, versions of the same algorithm. The algorithm is usually described in terms of factoring a matrix into an orthogonal factor, Q, and an upper or right triangular factor, R. This leads to QR algorithms. But for reasons having to do with graded matrices and terminating a loop at 1 rather than n-1, the authors of the Handbook decided to use left triangular and QL algorithms.

Historical note

When the papers from Numerische Mathematik were collected to form the 1971 Handbook for Automatic Computation, Volume II, Linear Algebra, almost all of them where reprinted without change. But, despite what its footnote says, Contribution II/4, The Implicit QL Algorithm, never appeared in the journal. The paper is the merger of a half-page paper by Austin Dubrulle and earlier contributions by R.S. Martin and Wilkinson. Dubrulle was able to reduce the operation count in the inner loop.

The authors of Contribution II/4 of the Handbook are listed as A. Dubrulle, R.S.Martin, and J.H.Wilkinson, although the three of them never actually worked together. Proper credit is given, but I'm afraid that an interesting little bit of history has been lost.

Dubrulle's version of the implicit tridiagonal QR algorithm continues to be important today. In future posts I will describe how the Handbook led to EISPACK, LINPACK and LAPACK. In LAPACK the imtql1 and imtql2 functions are combined into one subroutine named DSTEQR.F The code for the inner loop is very similar to Austin's note.

Years after making this contribution, Austin returned to grad school and was one of my Ph.D. students at the University of New Mexico. His thesis was about the distribution of the singular value of the matrices derived from photographs.

\n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


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. The first drafts of % papers are to be submitted by August, 2018. That long lead time % gives me the opportunity to write a detailed history of MATLAB. % I plan to write the paper in sections, which I'll post in % this blog as they are available. % % The mathematical basis for the first version of MATLAB begins % with a volume edited by J. H. Wilkinson and C. Reinsch, % _Handbook for Automatic Computation, Volume II, Linear Algebra_, % published by the eminent German publisher Springer-Verlag, in 1971. % % <> %% Eigenvalues % The mathematical term "eigenvalue" is a linguistic hybrid. In German % the term is "eigenwert". One Web dictionary says the English translation % is "intrinsic value". But after a few decades using terms like % "characteristic value" and "latent root", mathematicians have given up % trying to translate the entire word and generally translate only the % second half. %% % The _eigenvalue problem_ for a given matrix $A$ is the task of finding % scalars $\lambda$ and, possibly, the corresponding vectors $x$ so that % % $$Ax = \lambda x$$ % % It is important to distinguish the case where $A$ is symmetric. % After the _linear equation problem_, % % $$Ax = b$$ % % the eigenvalue problem is the most important computational task % in numerical linear algebra. %% % The state of the art in numerical linear algebra 50 years ago, in the % 1960's, did not yet provide reliable, efficient methods for solving % matrix eigenvalue problems. Software libraries had a hodgepodge % of methods, many of them based upon polynomial root finders. %% % told a story about having two such subroutines, one using % something called Bairstow's method and one using something called % Laguerre's method. He couldn't decide which one was best, % so he put them together in a program that first tried % Bairstow's method, then printed a message and switched to Laguerre % if Bairstow failed. After several months of frequent use at the National % Physical Laboratory, he never saw a case where Bairstow failed. % He wrote a paper with his recommendation for Bairstow's method and % soon got a letter from a reader who claimed to have an example which % Bairstow could not handle. Wilkinson tried the example with his program % and found a bug. The program didn't print the message that it was % switching to Laguerre. %% Handbook for Automatic Computation % Over the period 1965-1970, Wilkinson and 18 of his colleagues % published a series of % papers in the Springer journal _Numerische Mathematik_. % The papers offered procedures, written in Algol 60, for solving different % versions of the linear equation problem and the eigenvalue problem. % These were _research_ papers presenting results about numerical stability, % subtle details of implementation, and, in some cases, new methods. %% % In 1971, these papers were collected, sometimes with modifications, % in the volume edited by Wilkinson and Reinsch. This book marks the % first appearance is an organized library of the algorithms for the % eigenvalue problem for dense, stored matrices that we still use in % MATLAB today. The importance of using orthogonal transformations % wherever possible was exposed by Wilkinson and the other authors. The % effectiveness of the newly discovered QR algorithms of J. Francis % and the related LR algorithm of H. Rutishauser had only recently been % appreciated. %% Contents % The Algol 60 procedures are the focus of each chapter. These codes % remain a clear, readable reference for the important ideas of modern % numerical linear algebra. Part I of the volume is about the linear % equation problem; part II is about the eigenvalue problem. % There are 40 procedures in part I and 43 in part II. %% % Here is a list of the procedures in Part II. A suffix 2 in the procedure % name indicates that it computes both eigenvalues and eigenvectors. % The "bak" procedures apply reduction transformations to eigenvectors. %% % Many of the procedures work with a reduced form, which is tridiagonal for % symmetric or Hermitian matrices and Hessenberg (upper triangular plus % one subdiagonal) for nonsymmetric matrices., % Since Algol does not have a complex number data type, the complex % arrays are represented by pairs of real arrays. %% Symmetric matrices % % % Reduction to tridiagonal % tred1, tred2, tred3, trbak1, trbak3 Orthogonal tridiagonalization. % Band % bandrd Tridiagonalization. % symray Eigenvectors. % bqr One eigenvalue. % Tridiagonal % imtql1,imtql2 All eigenvalues and vectors, implicit QR. % tql1, tql2 All eigenvalues and vectors, explicit QR. % ratqr Few eigenvalues, rational QR. % bisect Few eigenvalues, bisection. % tristurm Few eigenvalues and vectors. % Few eigenvalues % ritzit Simultaneous iteration. % Jacobi method % jacobi Jacobi method. % Generalized problem, Ax = \lambda Bx % reduc1, reduc2, rebaka, rebakb Symmetric A, positive definite B. % %% Nonsymmetric matrices % % % Reduction to Hessenberg % balance, balbak Balance (scaling) % dirhes, dirbak, dirtrans Elementary, accumulated innerprod. % elmhes, elmbak, elmtrans Elementary. % orthes, ortbak, ortrans Orthogonal. % Band % unsray Eigenvectors. % Hessenberg % hqr, hqr2 All eigenvalues and vectors, implicit QR. % invit Few eigenvectors, inverse iteration. % Norm reducing % eigen Eigenvalues. % %% Complex matrices % % % comeig Norm reducing Jacobi. % comhes, combak Reduce to Hessenberg form. % comlr, comlr2 Complex LR algorithm. % cxinvit Inverse iteration. % %% Preferred paths % The preferred path for finding all the eigenvalues of a real, symmetric % matrix is _tred1_ followed by _imtql1_. % The preferred path for finding all the eigenvalues and eigenvectors % of a real, symmetric matrix is _tred2_ followed by _imtql2_. % % The preferred path for finding all the eigenvalues of a real, nonsymmetric % matrix is _balanc_, _elmhes_, and finally _hqr_. % The preferred path for finding all the eigenvalues and eigenvectors % of a real, nonsymmetric matrix is _balanc_, _elmhes_, _elmtrans_, _hqr2_, % and finally _balbak_. %% QR vs. QL % "QR" and "QL" are right- and left-handed, or forward and backward, % versions of the same algorithm. The algorithm is usually described in % terms of factoring a matrix into an orthogonal factor, Q, and an upper % or right triangular factor, R. This leads to QR algorithms. But for % reasons having to do with graded matrices and terminating a loop at |1| % rather than |n-1|, the authors of the Handbook decided to use left % triangular and QL algorithms. %% Historical note % When the papers from _Numerische Mathematik_ were collected to form the % 1971 _Handbook for Automatic Computation, Volume II, Linear Algebra_, % almost all of them where reprinted without change. But, despite what % its footnote says, Contribution II/4, _The Implicit QL Algorithm_, % never appeared in the journal. The paper is the merger of a half-page % paper by % and earlier contributions by R.S. Martin and Wilkinson. % Dubrulle was able to reduce the operation count in the inner loop. %% % The authors of Contribution II/4 of the Handbook are listed as % A. Dubrulle, R.S.Martin, and J.H.Wilkinson, although the three of them % never actually worked together. % Proper credit is given, but I'm afraid that an interesting little bit % of history has been lost. % % Dubrulle's version of the implicit tridiagonal QR algorithm continues to % be important today. In future posts I will describe how the Handbook % led to EISPACK, LINPACK and LAPACK. % In LAPACK the _imtql1_ and _imtql2_ functions are combined into one % subroutine named % % The code for the inner loop is very similar to Austin's note. % % Years after making this contribution, Austin returned to grad school % and was one of my Ph.D. students at the University of New Mexico. % His thesis was about the distribution of the singular value of the % matrices derived from photographs. ##### SOURCE END ##### 26cefaedad044f33a20d375f994408b9 -->

Wilkinson and Reinsch Handbook on Linear Algebra

Tue, 12/05/2017 - 05:00

The ACM Special Interest Group on Programming Languages, SIGPLAN, expects to hold the fourth in a series of conferences on the History of Programming Languages in 2020, see HOPL-IV. The first drafts of papers are to be submitted by August, 2018. That long lead time gives me the opportunity to write a detailed history of MATLAB. I plan to write the paper in sections, which I'll post in this blog as they are available.

The mathematical basis for the first version of MATLAB begins with a volume edited by J. H. Wilkinson and C. Reinsch, Handbook for Automatic Computation, Volume II, Linear Algebra, published by the eminent German publisher Springer-Verlag, in 1971.

ContentsEigenvalues

The mathematical term "eigenvalue" is a linguistic hybrid. In German the term is "eigenwert". One Web dictionary says the English translation is "intrinsic value". But after a few decades using terms like "characteristic value" and "latent root", mathematicians have given up trying to translate the entire word and generally translate only the second half.

The eigenvalue problem for a given matrix $A$ is the task of finding scalars $\lambda$ and, possibly, the corresponding vectors $x$ so that

$$Ax = \lambda x$$

It is important to distinguish the case where $A$ is symmetric. After the linear equation problem,

$$Ax = b$$

the eigenvalue problem is the most important computational task in numerical linear algebra.

The state of the art in numerical linear algebra 50 years ago, in the 1960's, did not yet provide reliable, efficient methods for solving matrix eigenvalue problems. Software libraries had a hodgepodge of methods, many of them based upon polynomial root finders.

Jim Wilkinson told a story about having two such subroutines, one using something called Bairstow's method and one using something called Laguerre's method. He couldn't decide which one was best, so he put them together in a program that first tried Bairstow's method, then printed a message and switched to Laguerre if Bairstow failed. After several months of frequent use at the National Physical Laboratory, he never saw a case where Bairstow failed. He wrote a paper with his recommendation for Bairstow's method and soon got a letter from a reader who claimed to have an example which Bairstow could not handle. Wilkinson tried the example with his program and found a bug. The program didn't print the message that it was switching to Laguerre.

Handbook for Automatic Computation

Over the period 1965-1970, Wilkinson and 18 of his colleagues published a series of papers in the Springer journal Numerische Mathematik. The papers offered procedures, written in Algol 60, for solving different versions of the linear equation problem and the eigenvalue problem. These were research papers presenting results about numerical stability, subtle details of implementation, and, in some cases, new methods.

In 1971, these papers were collected, sometimes with modifications, in the volume edited by Wilkinson and Reinsch. This book marks the first appearance is an organized library of the algorithms for the eigenvalue problem for dense, stored matrices that we still use in MATLAB today. The importance of using orthogonal transformations wherever possible was exposed by Wilkinson and the other authors. The effectiveness of the newly discovered QR algorithms of J. Francis and the related LR algorithm of H. Rutishauser had only recently been appreciated.

Contents

The Algol 60 procedures are the focus of each chapter. These codes remain a clear, readable reference for the important ideas of modern numerical linear algebra. Part I of the volume is about the linear equation problem; part II is about the eigenvalue problem. There are 40 procedures in part I and 43 in part II.

Here is a list of the procedures in Part II. A suffix 2 in the procedure name indicates that it computes both eigenvalues and eigenvectors. The "bak" procedures apply reduction transformations to eigenvectors.

Many of the procedures work with a reduced form, which is tridiagonal for symmetric or Hermitian matrices and Hessenberg (upper triangular plus one subdiagonal) for nonsymmetric matrices., Since Algol does not have a complex number data type, the complex arrays are represented by pairs of real arrays.

Symmetric matrices

Reduction to tridiagonal tred1, tred2, tred3, trbak1, trbak3 Orthogonal tridiagonalization. Band bandrd Tridiagonalization. symray Eigenvectors. bqr One eigenvalue. Tridiagonal imtql1,imtql2 All eigenvalues and vectors, implicit QR. tql1, tql2 All eigenvalues and vectors, explicit QR. ratqr Few eigenvalues, rational QR. bisect Few eigenvalues, bisection. tristurm Few eigenvalues and vectors. Few eigenvalues ritzit Simultaneous iteration. Jacobi method jacobi Jacobi method. Generalized problem, Ax = \lambda Bx reduc1, reduc2, rebaka, rebakb Symmetric A, positive definite B.

Nonsymmetric matrices

Reduction to Hessenberg balance, balbak Balance (scaling) dirhes, dirbak, dirtrans Elementary, accumulated innerprod. elmhes, elmbak, elmtrans Elementary. orthes, ortbak, ortrans Orthogonal. Band unsray Eigenvectors. Hessenberg hqr, hqr2 All eigenvalues and vectors, implicit QR. invit Few eigenvectors, inverse iteration. Norm reducing eigen Eigenvalues.

Complex matrices

comeig Norm reducing Jacobi. comhes, combak Reduce to Hessenberg form. comlr, comlr2 Complex LR algorithm. cxinvit Inverse iteration.

Preferred paths

The preferred path for finding all the eigenvalues of a real, symmetric matrix is tred1 followed by imtql1. The preferred path for finding all the eigenvalues and eigenvectors of a real, symmetric matrix is tred2 followed by imtql2.

The preferred path for finding all the eigenvalues of a real, nonsymmetric matrix is balanc, elmhes, and finally hqr. The preferred path for finding all the eigenvalues and eigenvectors of a real, nonsymmetric matrix is balanc, elmhes, elmtrans, hqr2, and finally balbak.

QR vs. QL

"QR" and "QL" are right- and left-handed, or forward and backward, versions of the same algorithm. The algorithm is usually described in terms of factoring a matrix into an orthogonal factor, Q, and an upper or right triangular factor, R. This leads to QR algorithms. But for reasons having to do with graded matrices and terminating a loop at 1 rather than n-1, the authors of the Handbook decided to use left triangular and QL algorithms.

Historical note

When the papers from Numerische Mathematik were collected to form the 1971 Handbook for Automatic Computation, Volume II, Linear Algebra, almost all of them where reprinted without change. But, despite what its footnote says, Contribution II/4, The Implicit QL Algorithm, never appeared in the journal. The paper is the merger of a half-page paper by Austin Dubrulle and earlier contributions by R.S. Martin and Wilkinson. Dubrulle was able to reduce the operation count in the inner loop.

The authors of Contribution II/4 of the Handbook are listed as A. Dubrulle, R.S.Martin, and J.H.Wilkinson, although the three of them never actually worked together. Proper credit is given, but I'm afraid that an interesting little bit of history has been lost.

Dubrulle's version of the implicit tridiagonal QR algorithm continues to be important today. In future posts I will describe how the Handbook led to EISPACK, LINPACK and LAPACK. In LAPACK the imtql1 and imtql2 functions are combined into one subroutine named DSTEQR.F The code for the inner loop is very similar to Austin's note.

Years after making this contribution, Austin returned to grad school and was one of my Ph.D. students at the University of New Mexico. His thesis was about the distribution of the singular value of the matrices derived from photographs.

\n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


Get the MATLAB code (requires JavaScript)

Published with MATLAB® R2017a

. The first drafts of % papers are to be submitted by August, 2018. That long lead time % gives me the opportunity to write a detailed history of MATLAB. % I plan to write the paper in sections, which I'll post in % this blog as they are available. % % The mathematical basis for the first version of MATLAB begins % with a volume edited by J. H. Wilkinson and C. Reinsch, % _Handbook for Automatic Computation, Volume II, Linear Algebra_, % published by the eminent German publisher Springer-Verlag, in 1971. % % <> %% Eigenvalues % The mathematical term "eigenvalue" is a linguistic hybrid. In German % the term is "eigenwert". One Web dictionary says the English translation % is "intrinsic value". But after a few decades using terms like % "characteristic value" and "latent root", mathematicians have given up % trying to translate the entire word and generally translate only the % second half. %% % The _eigenvalue problem_ for a given matrix $A$ is the task of finding % scalars $\lambda$ and, possibly, the corresponding vectors $x$ so that % % $$Ax = \lambda x$$ % % It is important to distinguish the case where $A$ is symmetric. % After the _linear equation problem_, % % $$Ax = b$$ % % the eigenvalue problem is the most important computational task % in numerical linear algebra. %% % The state of the art in numerical linear algebra 50 years ago, in the % 1960's, did not yet provide reliable, efficient methods for solving % matrix eigenvalue problems. Software libraries had a hodgepodge % of methods, many of them based upon polynomial root finders. %% % told a story about having two such subroutines, one using % something called Bairstow's method and one using something called % Laguerre's method. He couldn't decide which one was best, % so he put them together in a program that first tried % Bairstow's method, then printed a message and switched to Laguerre % if Bairstow failed. After several months of frequent use at the National % Physical Laboratory, he never saw a case where Bairstow failed. % He wrote a paper with his recommendation for Bairstow's method and % soon got a letter from a reader who claimed to have an example which % Bairstow could not handle. Wilkinson tried the example with his program % and found a bug. The program didn't print the message that it was % switching to Laguerre. %% Handbook for Automatic Computation % Over the period 1965-1970, Wilkinson and 18 of his colleagues % published a series of % papers in the Springer journal _Numerische Mathematik_. % The papers offered procedures, written in Algol 60, for solving different % versions of the linear equation problem and the eigenvalue problem. % These were _research_ papers presenting results about numerical stability, % subtle details of implementation, and, in some cases, new methods. %% % In 1971, these papers were collected, sometimes with modifications, % in the volume edited by Wilkinson and Reinsch. This book marks the % first appearance is an organized library of the algorithms for the % eigenvalue problem for dense, stored matrices that we still use in % MATLAB today. The importance of using orthogonal transformations % wherever possible was exposed by Wilkinson and the other authors. The % effectiveness of the newly discovered QR algorithms of J. Francis % and the related LR algorithm of H. Rutishauser had only recently been % appreciated. %% Contents % The Algol 60 procedures are the focus of each chapter. These codes % remain a clear, readable reference for the important ideas of modern % numerical linear algebra. Part I of the volume is about the linear % equation problem; part II is about the eigenvalue problem. % There are 40 procedures in part I and 43 in part II. %% % Here is a list of the procedures in Part II. A suffix 2 in the procedure % name indicates that it computes both eigenvalues and eigenvectors. % The "bak" procedures apply reduction transformations to eigenvectors. %% % Many of the procedures work with a reduced form, which is tridiagonal for % symmetric or Hermitian matrices and Hessenberg (upper triangular plus % one subdiagonal) for nonsymmetric matrices., % Since Algol does not have a complex number data type, the complex % arrays are represented by pairs of real arrays. %% Symmetric matrices % % % Reduction to tridiagonal % tred1, tred2, tred3, trbak1, trbak3 Orthogonal tridiagonalization. % Band % bandrd Tridiagonalization. % symray Eigenvectors. % bqr One eigenvalue. % Tridiagonal % imtql1,imtql2 All eigenvalues and vectors, implicit QR. % tql1, tql2 All eigenvalues and vectors, explicit QR. % ratqr Few eigenvalues, rational QR. % bisect Few eigenvalues, bisection. % tristurm Few eigenvalues and vectors. % Few eigenvalues % ritzit Simultaneous iteration. % Jacobi method % jacobi Jacobi method. % Generalized problem, Ax = \lambda Bx % reduc1, reduc2, rebaka, rebakb Symmetric A, positive definite B. % %% Nonsymmetric matrices % % % Reduction to Hessenberg % balance, balbak Balance (scaling) % dirhes, dirbak, dirtrans Elementary, accumulated innerprod. % elmhes, elmbak, elmtrans Elementary. % orthes, ortbak, ortrans Orthogonal. % Band % unsray Eigenvectors. % Hessenberg % hqr, hqr2 All eigenvalues and vectors, implicit QR. % invit Few eigenvectors, inverse iteration. % Norm reducing % eigen Eigenvalues. % %% Complex matrices % % % comeig Norm reducing Jacobi. % comhes, combak Reduce to Hessenberg form. % comlr, comlr2 Complex LR algorithm. % cxinvit Inverse iteration. % %% Preferred paths % The preferred path for finding all the eigenvalues of a real, symmetric % matrix is _tred1_ followed by _imtql1_. % The preferred path for finding all the eigenvalues and eigenvectors % of a real, symmetric matrix is _tred2_ followed by _imtql2_. % % The preferred path for finding all the eigenvalues of a real, nonsymmetric % matrix is _balanc_, _elmhes_, and finally _hqr_. % The preferred path for finding all the eigenvalues and eigenvectors % of a real, nonsymmetric matrix is _balanc_, _elmhes_, _elmtrans_, _hqr2_, % and finally _balbak_. %% QR vs. QL % "QR" and "QL" are right- and left-handed, or forward and backward, % versions of the same algorithm. The algorithm is usually described in % terms of factoring a matrix into an orthogonal factor, Q, and an upper % or right triangular factor, R. This leads to QR algorithms. But for % reasons having to do with graded matrices and terminating a loop at |1| % rather than |n-1|, the authors of the Handbook decided to use left % triangular and QL algorithms. %% Historical note % When the papers from _Numerische Mathematik_ were collected to form the % 1971 _Handbook for Automatic Computation, Volume II, Linear Algebra_, % almost all of them where reprinted without change. But, despite what % its footnote says, Contribution II/4, _The Implicit QL Algorithm_, % never appeared in the journal. The paper is the merger of a half-page % paper by % and earlier contributions by R.S. Martin and Wilkinson. % Dubrulle was able to reduce the operation count in the inner loop. %% % The authors of Contribution II/4 of the Handbook are listed as % A. Dubrulle, R.S.Martin, and J.H.Wilkinson, although the three of them % never actually worked together. % Proper credit is given, but I'm afraid that an interesting little bit % of history has been lost. % % Dubrulle's version of the implicit tridiagonal QR algorithm continues to % be important today. In future posts I will describe how the Handbook % led to EISPACK, LINPACK and LAPACK. % In LAPACK the _imtql1_ and _imtql2_ functions are combined into one % subroutine named % % The code for the inner loop is very similar to Austin's note. % % Years after making this contribution, Austin returned to grad school % and was one of my Ph.D. students at the University of New Mexico. % His thesis was about the distribution of the singular value of the % matrices derived from photographs. ##### SOURCE END ##### 26cefaedad044f33a20d375f994408b9 -->

Leslie Fox

Mon, 11/20/2017 - 22:30

Leslie Fox (1918-1992) is a British numerical analyst who was a contemporary of three men who played such an important role in my education and early professional life, John Todd, George Forsythe, and Jim Wilkinson. Although I saw less of Fox than I did of the others, he was still an important influence in my life.

ContentsNPL

The National Physical Laboratory, in Teddington, England, just west of London, is the British equivalent of the U.S. agency that was originally called the National Bureau of Standards and is now known as the National Institute of Science and Technology. Fox worked at NPL from 1945 until 1956. Wilkinson worked there his entire professional life. Despite the fact that Fox was from Oxford and Wilkinson was from Cambridge, they became close friends.

In the first few years after World War II, both NPL and NBS sponsored the development of first generation stored program electronic computers. An initial design for the NPL machine, the Automatic Computing Engine, was proposed by Alan Turing. But the design was based in part on secret work that Turing had done during the War for breaking German military codes. The management of NPL at the time was not allowed access to Turing's secret work and judged the machine to be impractical with current technology. Disillusioned, Turing left NPL and went to the University of Manchester, where they were also building a computer.

It fell to one of Turing's colleagues, Jim Wilkinson, to take over the project and build a simplified version of the ACE, the Pilot Ace. Wilkinson went on to become the world's leading authority on matrix computation. He was my inspiration and friend. His work provided the mathematical basis for the first MATLAB. Fox was less involved in the hardware development, but he was certainly an early user of the Pilot Ace.

NBS

Meanwhile in U.S., NBS was sponsoring the development of two early machines, one at its headquarters in Maryland, the Standards Eastern Automatic Computer, or SEAC, and one in southern California, at UCLA, the Standards Western Automatic Computer, or SWAC.

NBS also established a research group at UCLA, the Institute for Numerical Analysis, or INA, to contribute to the work on the SWAC. Two members of the INA were John Todd, who a few years later as a faculty member at Caltech, would introduce me to numerical analysis and computing, and George Forsythe, who would go on to establish the Computer Science Department at Stanford and also be my Ph.D. thesis advisor.

The groups at NPL in England and INA in California were in close contact with each other and often exchanged visits. As a result, the four men -- Fox, Wilkinson, Todd, and Forsythe -- became friends and colleagues and leading figures in the budding field that today would be called computational science.

Modern Computing Methods

When I took my first numerical analysis course from Todd at Caltech in 1959, we didn't have a regular textbook. There weren't many to choose from. Todd was in the process of writing what would become his Survey of Numerical Analysis. He passed out purple dittoed copies of an early draft of the book. (Today, Amazon has 19 used copies available for $8.23 with free shipping.)

In lieu of a formal textbook, Todd also referred to this 130-page paper bound pamphlet. Its title says "Modern", but it was first published 60 years ago, in 1957. Here is a link to the hardcover second addition, Modern Computing Methods. (Amazon has 3 used copies available for $7.21, plus $3.99 shipping.)

The publisher of the first edition is listed as the picturesque Her Majesty's Stationery Office. But there are no authors listed. The source is just the National Physical Laboratory. For many years I believed the authors were Fox and Wilkinson. Now, while writing this post, I have come across this review by Charles Saltzer in SIAM Review. The authors are revealed to be "E. T. Goodwin et al". Goodwin was another member of the NPL group. So, I suspect that Fox and Wilkinson were that " et al ".

Oxford

Fox resigned from NPL in 1956 and, after a year in the U.S. visiting U. C. Berkeley, joined the faculty at his old school, Oxford. He remained there the rest of his life. He established the Oxford Computing Laboratory. In 1963 he was appointed Professor of Numerical Analysis and Fellow of Balliol College. (My good friend Nick Trefethen now holds both of those appointments.)

Fox was a prolific author. The bibliography in Leslie Fox booklet lists eight books. He was also a popular professor. The same booklet lists 18 D. Phil. students that he supervised and 71 D. Phil. students where he a member of the committee.

NAG

The Numerical Algorithms Group is a British company, founded in 1970 and with headquarters in Oxford, which markets mathematical software, consulting, and related services. Fox helped the company get started and served on its Board of Directors for many years.

Householder VIII

Fox hosted the eighth Householder Symposium at Oxford in 1981. Here is the group photo. Fox is in the center, in the brown suit. (I am in the second row, second from left, in the blue shirt.)

Fox Prize

The Leslie Fox Prize for Numerical Analysis is awarded every two years by the IMA, the British Institute of Mathematics. The prize honors "young numerical analysts worldwide", that is anyone who is less than 31 years old. One or two first prize winners and usually several second prize winners are announced at symposia in the UK in June of odd numbered years. Eighteen meetings and prize announcements have been held since the award was established in 1985. The 2017 call for papers is IMA Leslie Fox Prize.

The Prize Winners for 2017 were announced in a meeting in Glasgow on June 26. The first prize was awarded to Nicole Spillain of Ecole Polytechnique. Her winning paper was An Adaptive Multipreconditioned Conjugate Gradient Algorithm.

Blue Plaque

Wikipedia explains that "a blue plaque is a permanent sign installed in a public place in the United Kingdom and elsewhere to commemorate a link between that location and a famous person or event, serving as a historical marker." In June, two blue plaques were unveiled in the Dewsbury Railroad Station, in West Yorkshire, about 40 miles west of Manchester. (Google maps tells me that today trains to Manchester leave every half hour and usually can make the trip in 40 minites.)

Photo credit: Huddersfield Daily Examiner

The plaques honor two computer scientists, Leslie Fox and Tom Kilburn. Here is a local newspaper story covering the unveiling. Huddersfield Daily Examiner. It turns out the two were school mates in the local grammar school.

Tom Kilburn worked with Freddie Williams at the University of Manchester to develop the Williams-Kilburn Tube, an early random access memory employing the static charge left by a grid of dots on a cathode ray tube. The scheme provided fast access time, but required constant tuning.

Kilburn and Williams needed a stored program digital computer to test their storage device. So they built a relatively simple machine that came to be known as "Manchester Baby". One day in June, 1948, on the train from Dewsbury to Manchester, Kilburn wrote a program to find the prime factors of an integer. The code was probably only a few dozen machine instructions. When the team got the machine in shape to actually run the program it worked correctly the first time. This was the world's first working computer program, and first instance of mathematical software.

Kilburn became the Professor of Computer Science at Manchester about the same time as Fox became the Professor of Numerical Analysis at Oxford.

Further Reading

Two articles about Leslie Fox are available on the Web at Wikipedia and at the MacTutor History of Mathematics Archive.

The booklet prepared by Oxford for his memorial service is available at Leslie Fox booklet.

\n'); d.write(code_string); // Add copyright line at the bottom if specified. if (copyright.length > 0) { d.writeln(''); d.writeln('%%'); if (copyright.length > 0) { d.writeln('% _' + copyright + '_'); } } d.write('\n'); d.title = title + ' (MATLAB code)'; d.close(); } -->


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Published with MATLAB® R2017a

, % , and % . % Although I saw less of Fox than I did of the others, he was still % an important influence in my life. % % <> %% NPL % The National Physical Laboratory, in Teddington, England, just west % of London, is the British equivalent of the U.S. agency that was % originally called the National Bureau of Standards and is now known as % the National Institute of Science and Technology. Fox worked at % NPL from 1945 until 1956. Wilkinson worked there his entire % professional life. Despite the fact that Fox was from Oxford and % Wilkinson was from Cambridge, they became close friends. %% % In the first few years after World War II, both NPL and NBS sponsored % the development of first generation stored program electronic computers. % An initial design for the NPL machine, the % , was proposed by Alan Turing. But the % design was based in part on secret work that Turing had done during the % War for breaking German military codes. The management of NPL at % the time was not allowed access to Turing's secret work and judged the % machine to be impractical with current technology. Disillusioned, Turing % left NPL and went to the University of Manchester, where they were also % building a computer. %% % It fell to one of Turing's colleagues, Jim Wilkinson, to take over the % project and build a simplified version of the ACE, the % . Wilkinson went % on to become the world's leading authority on matrix computation. % He was my inspiration and friend. His work provided the mathematical % basis for the first MATLAB. Fox was less involved in the hardware % development, but he was certainly an early user of the Pilot Ace. %% NBS % Meanwhile in U.S., NBS was sponsoring the development of two early % machines, one at its headquarters in Maryland, the Standards Eastern % Automatic Computer, or % , % and one in southern California, at UCLA, the Standards Western % Automatic Computer, or % . %% % NBS also established a research group at UCLA, the Institute for % Numerical Analysis, or % , to contribute to the work on the SWAC. % Two members of the INA were John Todd, who a few years later as a % faculty member at Caltech, would introduce me to numerical analysis % and computing, and George Forsythe, who would go on to establish % the Computer Science Department at Stanford and also be my Ph.D. thesis % advisor. %% % The groups at NPL in England and INA in California were in close % contact with each other and often exchanged visits. As a result, the % four men REPLACE_WITH_DASH_DASH Fox, Wilkinson, Todd, and Forsythe REPLACE_WITH_DASH_DASH became friends and % colleagues and leading figures in the budding field that today would % be called computational science. %% Modern Computing Methods % When I took my first numerical analysis course from Todd at Caltech % in 1959, we didn't have a regular textbook. There weren't many to % choose from. Todd was in the process of writing what would become his % . He passed out purple dittoed copies of % an early draft of the book. (Today, Amazon has 19 used copies available % for $8.23 with free shipping.) %% % In lieu of a formal textbook, Todd also referred to this 130-page % paper bound pamphlet. Its title says "Modern", but it was first % published 60 years ago, in 1957. % Here is a link to the hardcover second addition, % . (Amazon has 3 used copies available for % $7.21, plus $3.99 shipping.) % % <> %% % The publisher of the first edition is listed as the picturesque % _Her Majesty's Stationery Office_. But there are no authors listed. % The source is just the National Physical Laboratory. For many years % I believed the authors were Fox and Wilkinson. Now, while writing this % post, I have come across % by % Charles Saltzer in _SIAM Review_. The authors are revealed to be % "E. T. Goodwin _et al_". Goodwin was another member of the NPL group. % So, I suspect that Fox and Wilkinson were that " _et al_ ". %% Oxford % Fox resigned from NPL in 1956 and, after a year in the U.S. visiting % U. C. Berkeley, joined the faculty at his old school, Oxford. % He remained there the rest of his life. He established the % Oxford Computing Laboratory. In 1963 he was appointed Professor of % Numerical Analysis and Fellow of Balliol College. (My good friend % Nick Trefethen now holds both of those appointments.) %% % Fox was a prolific author. The bibliography in % lists eight books. % He was also a popular professor. The same booklet lists 18 D. Phil. % students that he supervised and 71 D. Phil. students where he a % member of the committee. %% NAG % The is a British % company, founded in 1970 and with headquarters in Oxford, which markets % mathematical software, consulting, and related services. Fox helped % the company get started and served on its Board of Directors for many % years. %% Householder VIII % Fox hosted the eighth Householder Symposium at Oxford in 1981. % Here is the group photo. Fox is in the center, in the brown suit. % (I am in the second row, second from left, in the blue shirt.) % % <> % %% Fox Prize % The % is awarded every two years % by the IMA, the British Institute of Mathematics. The prize honors % "young numerical analysts worldwide", that is anyone who is less than % 31 years old. One or two first prize winners and usually several second % prize winners are announced at symposia in the UK in June of odd numbered % years. Eighteen meetings and prize announcements have been held since % the award was established in 1985. The 2017 call for papers is % . %% % The were announced in a meeting in Glasgow % on June 26. The first prize was awarded to % of Ecole Polytechnique. % Her winning paper was % . %% Blue Plaque % Wikipedia explains that "a is a permanent sign installed in a public place in the % United Kingdom and elsewhere to commemorate a link between that location % and a famous person or event, serving as a historical marker." % In June, two blue plaques were unveiled in the Dewsbury Railroad % Station, in West Yorkshire, about 40 miles west of Manchester. % (Google maps tells me that today trains to Manchester leave every half % hour and usually can make the trip in 40 minites.) % % <> % % Photo credit: Huddersfield Daily Examiner %% % The plaques honor two computer scientists, Leslie Fox and Tom Kilburn. % Here is a local newspaper story covering the unveiling. % . It turns out the two were school mates in % the local grammar school. %% % worked with % Freddie Williams at the University of Manchester to develop the % , % an early random access memory employing the static charge left by % a grid of dots on a cathode ray tube. The scheme provided fast access % time, but required constant tuning. % % Kilburn and Williams needed a stored program digital computer to test % their storage device. So they built a relatively simple machine that % came to be known as % . % One day in June, 1948, on the train from Dewsbury to Manchester, % Kilburn wrote a program to find the prime factors of an integer. % The code was probably only a few dozen machine instructions. % When the team got the machine in shape to actually run the program % it worked correctly the first time. This was the world's first working % computer program, and first instance of mathematical software. % % Kilburn became the Professor of Computer Science at Manchester about % the same time as Fox became the Professor of Numerical Analysis at % Oxford. %% Further Reading % Two articles about Leslie Fox are available on the Web at % and at the % . % % The booklet prepared by Oxford for his memorial service is available at % . ##### SOURCE END ##### ed1184c94b7b4a1da68df57a87340027 -->

Leslie Fox

Mon, 11/20/2017 - 22:30

Leslie Fox (1918-1992) is a British numerical analyst who was a contemporary of three men who played such an important role in my education and early professional life, John Todd, George Forsythe, and Jim Wilkinson. Although I saw less of Fox than I did of the others, he was still an important influence in my life.

ContentsNPL

The National Physical Laboratory, in Teddington, England, just west of London, is the British equivalent of the U.S. agency that was originally called the National Bureau of Standards and is now known as the National Institute of Science and Technology. Fox worked at NPL from 1945 until 1956. Wilkinson worked there his entire professional life. Despite the fact that Fox was from Oxford and Wilkinson was from Cambridge, they became close friends.

In the first few years after World War II, both NPL and NBS sponsored the development of first generation stored program electronic computers. An initial design for the NPL machine, the Automatic Computing Engine, was proposed by Alan Turing. But the design was based in part on secret work that Turing had done during the War for breaking German military codes. The management of NPL at the time was not allowed access to Turing's secret work and judged the machine to be impractical with current technology. Disillusioned, Turing left NPL and went to the University of Manchester, where they were also building a computer.

It fell to one of Turing's colleagues, Jim Wilkinson, to take over the project and build a simplified version of the ACE, the Pilot Ace. Wilkinson went on to become the world's leading authority on matrix computation. He was my inspiration and friend. His work provided the mathematical basis for the first MATLAB. Fox was less involved in the hardware development, but he was certainly an early user of the Pilot Ace.

NBS

Meanwhile in U.S., NBS was sponsoring the development of two early machines, one at its headquarters in Maryland, the Standards Eastern Automatic Computer, or SEAC, and one in southern California, at UCLA, the Standards Western Automatic Computer, or SWAC.

NBS also established a research group at UCLA, the Institute for Numerical Analysis, or INA, to contribute to the work on the SWAC. Two members of the INA were John Todd, who a few years later as a faculty member at Caltech, would introduce me to numerical analysis and computing, and George Forsythe, who would go on to establish the Computer Science Department at Stanford and also be my Ph.D. thesis advisor.

The groups at NPL in England and INA in California were in close contact with each other and often exchanged visits. As a result, the four men -- Fox, Wilkinson, Todd, and Forsythe -- became friends and colleagues and leading figures in the budding field that today would be called computational science.

Modern Computing Methods

When I took my first numerical analysis course from Todd at Caltech in 1959, we didn't have a regular textbook. There weren't many to choose from. Todd was in the process of writing what would become his Survey of Numerical Analysis. He passed out purple dittoed copies of an early draft of the book. (Today, Amazon has 19 used copies available for $8.23 with free shipping.)

In lieu of a formal textbook, Todd also referred to this 130-page paper bound pamphlet. Its title says "Modern", but it was first published 60 years ago, in 1957. Here is a link to the hardcover second addition, Modern Computing Methods. (Amazon has 3 used copies available for $7.21, plus $3.99 shipping.)

The publisher of the first edition is listed as the picturesque Her Majesty's Stationery Office. But there are no authors listed. The source is just the National Physical Laboratory. For many years I believed the authors were Fox and Wilkinson. Now, while writing this post, I have come across this review by Charles Saltzer in SIAM Review. The authors are revealed to be "E. T. Goodwin et al". Goodwin was another member of the NPL group. So, I suspect that Fox and Wilkinson were that " et al ".

Oxford

Fox resigned from NPL in 1956 and, after a year in the U.S. visiting U. C. Berkeley, joined the faculty at his old school, Oxford. He remained there the rest of his life. He established the Oxford Computing Laboratory. In 1963 he was appointed Professor of Numerical Analysis and Fellow of Balliol College. (My good friend Nick Trefethen now holds both of those appointments.)

Fox was a prolific author. The bibliography in Leslie Fox booklet lists eight books. He was also a popular professor. The same booklet lists 18 D. Phil. students that he supervised and 71 D. Phil. students where he a member of the committee.

NAG

The Numerical Algorithms Group is a British company, founded in 1970 and with headquarters in Oxford, which markets mathematical software, consulting, and related services. Fox helped the company get started and served on its Board of Directors for many years.

Householder VIII

Fox hosted the eighth Householder Symposium at Oxford in 1981. Here is the group photo. Fox is in the center, in the brown suit. (I am in the second row, second from left, in the blue shirt.)

Fox Prize

The Leslie Fox Prize for Numerical Analysis is awarded every two years by the IMA, the British Institute of Mathematics. The prize honors "young numerical analysts worldwide", that is anyone who is less than 31 years old. One or two first prize winners and usually several second prize winners are announced at symposia in the UK in June of odd numbered years. Eighteen meetings and prize announcements have been held since the award was established in 1985. The 2017 call for papers is IMA Leslie Fox Prize.

The Prize Winners for 2017 were announced in a meeting in Glasgow on June 26. The first prize was awarded to Nicole Spillain of Ecole Polytechnique. Her winning paper was An Adaptive Multipreconditioned Conjugate Gradient Algorithm.

Blue Plaque

Wikipedia explains that "a blue plaque is a permanent sign installed in a public place in the United Kingdom and elsewhere to commemorate a link between that location and a famous person or event, serving as a historical marker." In June, two blue plaques were unveiled in the Dewsbury Railroad Station, in West Yorkshire, about 40 miles west of Manchester. (Google maps tells me that today trains to Manchester leave every half hour and usually can make the trip in 40 minites.)

Photo credit: Huddersfield Daily Examiner

The plaques honor two computer scientists, Leslie Fox and Tom Kilburn. Here is a local newspaper story covering the unveiling. Huddersfield Daily Examiner. It turns out the two were school mates in the local grammar school.

Tom Kilburn worked with Freddie Williams at the University of Manchester to develop the Williams-Kilburn Tube, an early random access memory employing the static charge left by a grid of dots on a cathode ray tube. The scheme provided fast access time, but required constant tuning.

Kilburn and Williams needed a stored program digital computer to test their storage device. So they built a relatively simple machine that came to be known as "Manchester Baby". One day in June, 1948, on the train from Dewsbury to Manchester, Kilburn wrote a program to find the prime factors of an integer. The code was probably only a few dozen machine instructions. When the team got the machine in shape to actually run the program it worked correctly the first time. This was the world's first working computer program, and first instance of mathematical software.

Kilburn became the Professor of Computer Science at Manchester about the same time as Fox became the Professor of Numerical Analysis at Oxford.

Further Reading

Two articles about Leslie Fox are available on the Web at Wikipedia and at the MacTutor History of Mathematics Archive.

The booklet prepared by Oxford for his memorial service is available at Leslie Fox booklet.

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, % , and % . % Although I saw less of Fox than I did of the others, he was still % an important influence in my life. % % <> %% NPL % The National Physical Laboratory, in Teddington, England, just west % of London, is the British equivalent of the U.S. agency that was % originally called the National Bureau of Standards and is now known as % the National Institute of Science and Technology. Fox worked at % NPL from 1945 until 1956. Wilkinson worked there his entire % professional life. Despite the fact that Fox was from Oxford and % Wilkinson was from Cambridge, they became close friends. %% % In the first few years after World War II, both NPL and NBS sponsored % the development of first generation stored program electronic computers. % An initial design for the NPL machine, the % , was proposed by Alan Turing. But the % design was based in part on secret work that Turing had done during the % War for breaking German military codes. The management of NPL at % the time was not allowed access to Turing's secret work and judged the % machine to be impractical with current technology. Disillusioned, Turing % left NPL and went to the University of Manchester, where they were also % building a computer. %% % It fell to one of Turing's colleagues, Jim Wilkinson, to take over the % project and build a simplified version of the ACE, the % . Wilkinson went % on to become the world's leading authority on matrix computation. % He was my inspiration and friend. His work provided the mathematical % basis for the first MATLAB. Fox was less involved in the hardware % development, but he was certainly an early user of the Pilot Ace. %% NBS % Meanwhile in U.S., NBS was sponsoring the development of two early % machines, one at its headquarters in Maryland, the Standards Eastern % Automatic Computer, or % , % and one in southern California, at UCLA, the Standards Western % Automatic Computer, or % . %% % NBS also established a research group at UCLA, the Institute for % Numerical Analysis, or % , to contribute to the work on the SWAC. % Two members of the INA were John Todd, who a few years later as a % faculty member at Caltech, would introduce me to numerical analysis % and computing, and George Forsythe, who would go on to establish % the Computer Science Department at Stanford and also be my Ph.D. thesis % advisor. %% % The groups at NPL in England and INA in California were in close % contact with each other and often exchanged visits. As a result, the % four men REPLACE_WITH_DASH_DASH Fox, Wilkinson, Todd, and Forsythe REPLACE_WITH_DASH_DASH became friends and % colleagues and leading figures in the budding field that today would % be called computational science. %% Modern Computing Methods % When I took my first numerical analysis course from Todd at Caltech % in 1959, we didn't have a regular textbook. There weren't many to % choose from. Todd was in the process of writing what would become his % . He passed out purple dittoed copies of % an early draft of the book. (Today, Amazon has 19 used copies available % for $8.23 with free shipping.) %% % In lieu of a formal textbook, Todd also referred to this 130-page % paper bound pamphlet. Its title says "Modern", but it was first % published 60 years ago, in 1957. % Here is a link to the hardcover second addition, % . (Amazon has 3 used copies available for % $7.21, plus $3.99 shipping.) % % <> %% % The publisher of the first edition is listed as the picturesque % _Her Majesty's Stationery Office_. But there are no authors listed. % The source is just the National Physical Laboratory. For many years % I believed the authors were Fox and Wilkinson. Now, while writing this % post, I have come across % by % Charles Saltzer in _SIAM Review_. The authors are revealed to be % "E. T. Goodwin _et al_". Goodwin was another member of the NPL group. % So, I suspect that Fox and Wilkinson were that " _et al_ ". %% Oxford % Fox resigned from NPL in 1956 and, after a year in the U.S. visiting % U. C. Berkeley, joined the faculty at his old school, Oxford. % He remained there the rest of his life. He established the % Oxford Computing Laboratory. In 1963 he was appointed Professor of % Numerical Analysis and Fellow of Balliol College. (My good friend % Nick Trefethen now holds both of those appointments.) %% % Fox was a prolific author. The bibliography in % lists eight books. % He was also a popular professor. The same booklet lists 18 D. Phil. % students that he supervised and 71 D. Phil. students where he a % member of the committee. %% NAG % The is a British % company, founded in 1970 and with headquarters in Oxford, which markets % mathematical software, consulting, and related services. Fox helped % the company get started and served on its Board of Directors for many % years. %% Householder VIII % Fox hosted the eighth Householder Symposium at Oxford in 1981. % Here is the group photo. Fox is in the center, in the brown suit. % (I am in the second row, second from left, in the blue shirt.) % % <> % %% Fox Prize % The % is awarded every two years % by the IMA, the British Institute of Mathematics. The prize honors % "young numerical analysts worldwide", that is anyone who is less than % 31 years old. One or two first prize winners and usually several second % prize winners are announced at symposia in the UK in June of odd numbered % years. Eighteen meetings and prize announcements have been held since % the award was established in 1985. The 2017 call for papers is % . %% % The were announced in a meeting in Glasgow % on June 26. The first prize was awarded to % of Ecole Polytechnique. % Her winning paper was % . %% Blue Plaque % Wikipedia explains that "a is a permanent sign installed in a public place in the % United Kingdom and elsewhere to commemorate a link between that location % and a famous person or event, serving as a historical marker." % In June, two blue plaques were unveiled in the Dewsbury Railroad % Station, in West Yorkshire, about 40 miles west of Manchester. % (Google maps tells me that today trains to Manchester leave every half % hour and usually can make the trip in 40 minites.) % % <> % % Photo credit: Huddersfield Daily Examiner %% % The plaques honor two computer scientists, Leslie Fox and Tom Kilburn. % Here is a local newspaper story covering the unveiling. % . It turns out the two were school mates in % the local grammar school. %% % worked with % Freddie Williams at the University of Manchester to develop the % , % an early random access memory employing the static charge left by % a grid of dots on a cathode ray tube. The scheme provided fast access % time, but required constant tuning. % % Kilburn and Williams needed a stored program digital computer to test % their storage device. So they built a relatively simple machine that % came to be known as % . % One day in June, 1948, on the train from Dewsbury to Manchester, % Kilburn wrote a program to find the prime factors of an integer. % The code was probably only a few dozen machine instructions. % When the team got the machine in shape to actually run the program % it worked correctly the first time. This was the world's first working % computer program, and first instance of mathematical software. % % Kilburn became the Professor of Computer Science at Manchester about % the same time as Fox became the Professor of Numerical Analysis at % Oxford. %% Further Reading % Two articles about Leslie Fox are available on the Web at % and at the % . % % The booklet prepared by Oxford for his memorial service is available at % . ##### SOURCE END ##### ed1184c94b7b4a1da68df57a87340027 -->

Three-Term Recurrence Relations and Bessel Functions

Mon, 11/06/2017 - 22:30

Three-term recurrence relations are the basis for computing Bessel functions.

ContentsA Familiar Three-Term Recurrence

Start with two large integers, say 514229 and 317811. For reasons that will soon become clear, I'll label them $f_{30}$ and $f_{29}$.

clear n = 30 f(30) = 514229; f(29) = 317811 n = 30 f = Columns 1 through 6 0 0 0 0 0 0 Columns 7 through 12 0 0 0 0 0 0 Columns 13 through 18 0 0 0 0 0 0 Columns 19 through 24 0 0 0 0 0 0 Columns 25 through 30 0 0 0 0 317811 514229

Now take repeated differences, and count backwards.

$$ f_{n-1} = f_{n+1} - f_n, \ n = 29, 28, ..., 2, 1 $$

for n = 29:-1:2 f(n-1) = f(n+1) - f(n); end n f n = 2 f = Columns 1 through 6 0 1 1 2 3 5 Columns 7 through 12 8 13 21 34 55 89 Columns 13 through 18 144 233 377 610 987 1597 Columns 19 through 24 2584 4181 6765 10946 17711 28657 Columns 25 through 30 46368 75025 121393 196418 317811 514229

Recognize those integers? They're the venerable Fibonacci numbers. I've just reversed the usual procedure for computing them. And, I didn't pick the first two values at random. In fact, their choice is very delicate. Try picking any other six-digit integers, and see if you can take 30 differences before you hit a negative value. Of course, we usually start with $f_0 = 0$, $f_1 = 1$ and go forward with additions.

The relation

$$ f_n = f_{n+1} - f_{n-1} $$

is an example of a three-term recurrence. I've written it in such a way that I can go either forward or backward.

Friedrich Bessel

Friedrich Bessel was a German astronomer, mathematician and physicist who lived from 1784 until 1846. He was a contemporary of Gauss, but the two had some sort of falling out. Although he did not have a university education, he made major contributions to precise measurements of planetary orbits and distances to stars. He has a crater on the Moon and an asteroid named after him.

In mathematics, the special functions that bear his name were actually discovered by someone else, Daniel Bernoulli, and christened after Bessel's death. They play the same role in polar coordinates that the trig functions play in Cartesian coordinates. They are applicable in a wide range of fields, from physics to finance.

Bessel Functions

Bessel functions are defined by an ordinary differential equation that generalizes the harmonic oscillator to polar coordinates. The equation contains a parameter $n$, the order.

$$ x^2 y'' + x y' + (x^2 - n^2) y = 0 $$

Solutions to this equation are also sometimes called cylindrical harmonics.

For my purposes here, their most important property is the three-term recurrence relation:

$$ \frac{2n}{x} J_n(x) = J_{n-1}(x) + J_{n+1}(x) $$

This has a variable coefficient that depends upon the two parameters, the order $n$ and the argument $x$.

To complete the specification, it would be necessary to supply two initial conditions that depend upon $x$. But, as we shall see, using the recurrence in the forward direction is disastrous numerically. Alternatively, if we could somehow supply two end conditions, we could use the recurrence in the reverse direction. This the idea behind a classical method known as Miller's algorithm.

Miller's Algorithm

Miller's algorithm also employs this interesting identity, which I'll call the normalizer.

$$ J_0(x) + 2 J_2(x) + 2 J_4(x) + 2 J_6(x) + \ ... = 1 $$

It's not obvious where this identity comes from and I won't try to derive it here.

We don't know the end conditions, but we can pick arbitrary values, run the recurrence backwards, and then use the normalizer to rescale.

Step 1. Pick a large $N$ and temporarily let

$$ \tilde{J}_N = 0 $$

$$ \tilde{J}_{N-1} = 1 $$

Then for $n = N-1, ..., 1$ compute

$$ \tilde{J}_{n-1} = 2n/x \tilde{J}_n \ - \tilde{J}_{n+1} $$

Step 2. Normalize:

$$ \sigma = \tilde{J}_0 + 2 \tilde{J}_2 + 2 \tilde{J}_4 + ... $$

$$ J_n = \tilde{J}_n/\sigma, \ n = 0, ..., N $$

I'll show some results in a moment, after I introduce an alternative formulation.

Matrix Formulation

You know that I like to describe algorithms in matrix terms. The three-term recurrence generates a tridiagonal matrix. But which three diagonals? I can put the three below the diagonal, above the diagonal, or centered on the diagonal. This is accomplished with the second argument of the sparse matrix generator, spdiags. Here is the code.

type blt function B = blt(n,x,loc) % BesseL Three term recurence. % B = blt(n,x,loc) is an n-by-n sparse tridiagonal coefficent matrix % that generates the Bessel functions besselj((0:n-1)',x). % loc specifies the location of the diagonals. % loc = 'center', the default, centers the three diagonals. % loc = 'lower' puts the three diagonals in the lower triangle. % loc = 'upper' puts the three diagonals in the upper triangle. if nargin == 2 loc = 'center'; end switch loc case 'center', locd = -1:1; case 'lower', locd = -2:0; case 'upper', locd = 0:2; end % Three term recurrence: 2*n/x * J_n(x) = J_(n-1)(x) + J_n+1(x). d = -2*(0:n-1)'/x; e = ones(n,1); B = spdiags([e d e],locd,n,n); end Lower

If there were no roundoff error, and if we knew the values of the first two Bessel functions, $J_0(x)$ and $J_1(x)$, we could use backslash, the linear equation solver, with the diagonals in the lower triangle to compute $J_n(x)$ for $n > 1$. Here, for example, is $x = 1$.

format long J = @(n,x) besselj(n,x); n = 8 x = 1.0 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs n = 8 x = 1 v = 0.765197686557967 0.440050585744934 0.114903484931901 0.019563353982668 0.002476638964110 0.000249757730213 0.000020938338022 0.000001502326053

These values are accurate to almost all the digits shown. Except for the last one. It's beginning to show the effects of severe numerical instability. Let's try a larger value of n.

n = 18 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs; semilogy(0:n-1,v,'o-') title('J_n(1)') xlabel('n') n = 18

The values beyond about n = 9 are suspect and, in fact, totally wrong. The forward elimination that backslash uses to solve this lower triangular system is just the three-term recurrence for increasing n. And that recurrence has two solutions -- the one we're trying to compute and a second one corresponding to $Y_n(x)$, the Bessel function of second kind. $Y_n(x)$ is stimulated by roundoff error and completely takes over for $n > 9$.

So the lower triangular option -- the forward recurrence -- is unsatisfactory for two reasons: it requires two Bessel values to get started and it is unstable.

Upper

We need to make one modification to the upper triangular coefficient matrix.

n = 30; B = blt(n,x,'upper'); B(n-1,n) = 0;

Now the last two rows and columns are a 2-by-2 identity.

full(B(n-1:n,n-1:n)) ans = 1 0 0 1

The back substitution process just runs the three-term recursion backwards. We need to start with $J_{n-1}(x)$ and $J_{n-2}(x)$.

rhs = zeros(n,1); rhs(n-1:n) = J((n-2:n-1)',x); format long e v = B\rhs v = 7.651976865579656e-01 4.400505857449330e-01 1.149034849319004e-01 1.956335398266838e-02 2.476638964109952e-03 2.497577302112342e-04 2.093833800238925e-05 1.502325817436807e-06 9.422344172604491e-08 5.249250179911870e-09 2.630615123687451e-10 1.198006746303136e-11 4.999718179448401e-13 1.925616764480172e-14 6.885408200044221e-16 2.297531532210343e-17 7.186396586807488e-19 2.115375568053260e-20 5.880344573595754e-22 1.548478441211652e-23 3.873503008524655e-25 9.227621982096665e-27 2.098223955943776e-28 4.563424055950103e-30 9.511097932712488e-32 1.902951751891381e-33 3.660826744416801e-35 6.781552053554108e-37 1.211364502417112e-38 2.089159981718163e-40

These values are accurate to almost all the significant digits shown. The backwards recursion suppresses the unwanted $Y_n(x)$ and the process is stable.

Center

Can we avoid having to know those two starting values? Yes, by using a matrix formulation of the Miller algorithm. Insert the normalization condition in the first row. Now $B$ is tridiagonal, except for first row, which is

[1 0 2 0 2 0 2 ... ]

Other rows have $1$ s on the super- and subdiagonal and $-2n/x$ on the diagonal of the $(n+1)$ -st row.

n = 30; B = blt(n,x); B(1,1:2) = [1 0]; B(1,3:2:n) = 2; spy(B) title('B')

The beauty of this approach is that it does not require any values of the Bessel function a priori. The right hand side is simply a unit vector.

rhs = eye(n,1); format long e v = B\rhs semilogy(0:n-1,v,'o-') xlabel('n') title('J_n(1)') v = 7.651976865579665e-01 4.400505857449335e-01 1.149034849319005e-01 1.956335398266840e-02 2.476638964109954e-03 2.497577302112343e-04 2.093833800238926e-05 1.502325817436807e-06 9.422344172604494e-08 5.249250179911872e-09 2.630615123687451e-10 1.198006746303136e-11 4.999718179448401e-13 1.925616764480171e-14 6.885408200044220e-16 2.297531532210343e-17 7.186396586807487e-19 2.115375568053260e-20 5.880344573595753e-22 1.548478441211652e-23 3.873503008524655e-25 9.227621982096663e-27 2.098223955943775e-28 4.563424055950102e-30 9.511097932712487e-32 1.902951751891381e-33 3.660826744416762e-35 6.781552053355331e-37 1.211364395117367e-38 2.088559301926495e-40 Relative error w = J((0:n-1)',x); relerr = (v - w)./w; semilogy(0:n-1,abs(relerr) + eps*(v==w),'o-') xlabel('n') title('relative error')

The relative error deteriorates for the last few values of n. This is not unexpected because we have effectively truncated a semi-infinite matrix.

Triangular Factors

The LU decomposition of $B$ shows no pivoting for this value of $x$, but there is fill-in. The normalization condition is mixing in with the recurrence.

[L,U] = lu(B); spy(L) title('L') snapnow spy(U) title('U') References

J. C. P. Miller, "On the choice of standard solutions for a homogeneous linear equation of the second order", Quart. J. Mech. Appl. Math. 3, 1950, 225-235.

Walter Gautschi, "Computational Aspects of Three-Term Recurrence Relations", SIAM Review 9, 1967, 24-82.

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. I've just reversed the usual procedure for % computing them. And, I didn't pick the first two values % at random. In fact, their choice is very delicate. Try picking % any other six-digit integers, and see if you can take 30 differences % before you hit a negative value. Of course, we usually start with % $f_0 = 0$, $f_1 = 1$ and go forward with additions. %% % The relation % % $$ f_n = f_{n+1} - f_{n-1} $$ % % is an example of a _three-term recurrence_. I've written it in such % a way that I can go either forward or backward. %% Friedrich Bessel % Friedrich Bessel was a German astronomer, mathematician and physicist % who lived from 1784 until 1846. % He was a contemporary of Gauss, but the two had some sort of falling % out. Although he did not have a university education, he made major % contributions to precise measurements of planetary orbits and % distances to stars. He has a crater on the Moon and an asteroid % named after him. %% % In mathematics, the special functions that bear his name were actually % discovered by someone else, Daniel Bernoulli, and christened after % Bessel's death. They play the same role in polar coordinates that the % trig functions play in Cartesian coordinates. They are applicable % in a wide range of fields, from physics to finance. %% Bessel Functions % Bessel functions are defined by an ordinary differential % equation that generalizes the harmonic oscillator to polar % coordinates. The equation contains a parameter $n$, the _order_. % % $$ x^2 y'' + x y' + (x^2 - n^2) y = 0 $$ % % Solutions to this equation are also sometimes called % _cylindrical harmonics_. %% % For my purposes here, their most important property is the % three-term recurrence relation: % % $$ \frac{2n}{x} J_n(x) = J_{n-1}(x) + J_{n+1}(x) $$ % % This has a variable coefficient that depends upon the two parameters, % the order $n$ and the argument $x$. %% % To complete the specification, it would be necessary to supply two % initial conditions that depend upon $x$. % But, as we shall see, using the recurrence in the forward direction % is disastrous numerically. % Alternatively, if we could somehow supply two end conditions, % we could use the recurrence in the reverse direction. % This the idea behind a classical method known as Miller's algorithm. %% Miller's Algorithm % Miller's algorithm also employs this interesting identity, % which I'll call the _normalizer_. % % $$ J_0(x) + 2 J_2(x) + 2 J_4(x) + 2 J_6(x) + \ ... = 1 $$ % % It's not obvious where this identity comes from and I won't try to % derive it here. %% % We don't know the end conditions, but we can pick arbitrary values, % run the recurrence backwards, and then use the normalizer to rescale. %% % Step 1. Pick a large $N$ and temporarily let % % $$ \tilde{J}_N = 0 $$ % % $$ \tilde{J}_{N-1} = 1 $$ % % Then for $n = N-1, ..., 1$ compute % % $$ \tilde{J}_{n-1} = 2n/x \tilde{J}_n \ - \tilde{J}_{n+1} $$ % %% % Step 2. Normalize: % % $$ \sigma = \tilde{J}_0 + 2 \tilde{J}_2 + 2 \tilde{J}_4 + ... $$ % % $$ J_n = \tilde{J}_n/\sigma, \ n = 0, ..., N $$ %% % I'll show some results in a moment, after I introduce an % alternative formulation. %% Matrix Formulation % You know that I like to describe algorithms in matrix terms. % The three-term recurrence generates a tridiagonal matrix. % But which three diagonals? I can put the three below the diagonal, % above the diagonal, or centered on the diagonal. This is accomplished % with the second argument of the sparse matrix generator, |spdiags|. % Here is the code. type blt %% Lower % If there were no roundoff error, and if we knew the values of the % first two Bessel functions, $J_0(x)$ and $J_1(x)$, we could use % backslash, the linear equation solver, with the diagonals in the % lower triangle to compute $J_n(x)$ for $n > 1$. Here, for example, % is $x = 1$. format long J = @(n,x) besselj(n,x); n = 8 x = 1.0 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs %% % These values are accurate to almost all the digits shown. % Except for the last one. It's beginning to show the effects of % severe numerical instability. Let's try a larger value of |n|. n = 18 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs; semilogy(0:n-1,v,'o-') title('J_n(1)') xlabel('n') %% % The values beyond about |n = 9| are suspect and, in fact, totally % wrong. The forward elimination that backslash uses to solve this % lower triangular system is just the three-term recurrence for % increasing |n|. And that recurrence has two solutions REPLACE_WITH_DASH_DASH the one % we're trying to compute and a second one corresponding to $Y_n(x)$, % the _Bessel function of second kind_. $Y_n(x)$ is stimulated % by roundoff error and completely takes over for $n > 9$. %% % So the lower triangular option REPLACE_WITH_DASH_DASH the forward recurrence REPLACE_WITH_DASH_DASH is % unsatisfactory for two reasons: it requires two Bessel values to get % started and it is unstable. %% Upper % We need to make one modification to the upper triangular coefficient % matrix. n = 30; B = blt(n,x,'upper'); B(n-1,n) = 0; %% % Now the last two rows and columns are a 2-by-2 identity. full(B(n-1:n,n-1:n)) %% % The back substitution process just runs the three-term recursion % backwards. We need to start with $J_{n-1}(x)$ and $J_{n-2}(x)$. rhs = zeros(n,1); rhs(n-1:n) = J((n-2:n-1)',x); format long e v = B\rhs %% % These values are accurate to almost all the significant digits % shown. The backwards recursion suppresses the unwanted $Y_n(x)$ and % the process is stable. %% Center % Can we avoid having to know those two starting values? % Yes, by using a matrix formulation of the Miller algorithm. % Insert the normalization condition in the first row. % Now $B$ is tridiagonal, except for first row, which is % % [1 0 2 0 2 0 2 ... ] % % Other rows have $1$ s on the super- and subdiagonal and $-2n/x$ % on the diagonal of the $(n+1)$ -st row. n = 30; B = blt(n,x); B(1,1:2) = [1 0]; B(1,3:2:n) = 2; spy(B) title('B') %% % The beauty of this approach is that it does not require any % values of the Bessel function _a priori_. The right hand side % is simply a unit vector. rhs = eye(n,1); format long e v = B\rhs semilogy(0:n-1,v,'o-') xlabel('n') title('J_n(1)') %% Relative error w = J((0:n-1)',x); relerr = (v - w)./w; semilogy(0:n-1,abs(relerr) + eps*(v==w),'o-') xlabel('n') title('relative error') %% % The relative error deteriorates for the last few values of |n|. % This is not unexpected because we have effectively truncated % a semi-infinite matrix. %% Triangular Factors % The LU decomposition of $B$ shows no pivoting for this value of $x$, % but there is fill-in. The normalization condition is mixing in % with the recurrence. [L,U] = lu(B); spy(L) title('L') snapnow spy(U) title('U') %% References % J. C. P. Miller, "On the choice of standard solutions for a % homogeneous linear equation of the second order", % _Quart. J. Mech. Appl. Math._ 3, 1950, 225-235. % % Walter Gautschi, "Computational Aspects of Three-Term Recurrence % Relations", _SIAM Review_ 9, 1967, 24-82. % ##### SOURCE END ##### 5da534de2d7a43c0ad4c7a4c28cb0e39 -->

Three-Term Recurrence Relations and Bessel Functions

Mon, 11/06/2017 - 22:30
d

Three-term recurrence relations are the basis for computing Bessel functions.

ContentsA Familiar Three-Term Recurrence

Start with two large integers, say 514229 and 317811. For reasons that will soon become clear, I'll label them $f_{30}$ and $f_{29}$.

clear n = 30 f(30) = 514229; f(29) = 317811 n = 30 f = Columns 1 through 6 0 0 0 0 0 0 Columns 7 through 12 0 0 0 0 0 0 Columns 13 through 18 0 0 0 0 0 0 Columns 19 through 24 0 0 0 0 0 0 Columns 25 through 30 0 0 0 0 317811 514229

Now take repeated differences, and count backwards.

$$ f_{n-1} = f_{n+1} - f_n, \ n = 29, 28, ..., 2, 1 $$

for n = 29:-1:2 f(n-1) = f(n+1) - f(n); end n f n = 2 f = Columns 1 through 6 0 1 1 2 3 5 Columns 7 through 12 8 13 21 34 55 89 Columns 13 through 18 144 233 377 610 987 1597 Columns 19 through 24 2584 4181 6765 10946 17711 28657 Columns 25 through 30 46368 75025 121393 196418 317811 514229

Recognize those integers? They're the venerable Fibonacci numbers. I've just reversed the usual procedure for computing them. And, I didn't pick the first two values at random. In fact, their choice is very delicate. Try picking any other six-digit integers, and see if you can take 30 differences before you hit a negative value. Of course, we usually start with $f_0 = 0$, $f_1 = 1$ and go forward with additions.

The relation

$$ f_n = f_{n+1} - f_{n-1} $$

is an example of a three-term recurrence. I've written it in such a way that I can go either forward or backward.

Friedrich Bessel

Friedrich Bessel was a German astronomer, mathematician and physicist who lived from 1784 until 1846. He was a contemporary of Gauss, but the two had some sort of falling out. Although he did not have a university education, he made major contributions to precise measurements of planetary orbits and distances to stars. He has a crater on the Moon and an asteroid named after him.

In mathematics, the special functions that bear his name were actually discovered by someone else, Daniel Bernoulli, and christened after Bessel's death. They play the same role in polar coordinates that the trig functions play in Cartesian coordinates. They are applicable in a wide range of fields, from physics to finance.

Bessel Functions

Bessel functions are defined by an ordinary differential equation that generalizes the harmonic oscillator to polar coordinates. The equation contains a parameter $n$, the order.

$$ x^2 y'' + x y' + (x^2 - n^2) y = 0 $$

Solutions to this equation are also sometimes called cylindrical harmonics.

For my purposes here, their most important property is the three-term recurrence relation:

$$ \frac{2n}{x} J_n(x) = J_{n-1}(x) + J_{n+1}(x) $$

This has a variable coefficient that depends upon the two parameters, the order $n$ and the argument $x$.

To complete the specification, it would be necessary to supply two initial conditions that depend upon $x$. But, as we shall see, using the recurrence in the forward direction is disastrous numerically. Alternatively, if we could somehow supply two end conditions, we could use the recurrence in the reverse direction. This the idea behind a classical method known as Miller's algorithm.

Miller's Algorithm

Miller's algorithm also employs this interesting identity, which I'll call the normalizer.

$$ J_0(x) + 2 J_2(x) + 2 J_4(x) + 2 J_6(x) + \ ... = 1 $$

It's not obvious where this identity comes from and I won't try to derive it here.

We don't know the end conditions, but we can pick arbitrary values, run the recurrence backwards, and then use the normalizer to rescale.

Step 1. Pick a large $N$ and temporarily let

$$ \tilde{J}_N = 0 $$

$$ \tilde{J}_{N-1} = 1 $$

Then for $n = N-1, ..., 1$ compute

$$ \tilde{J}_{n-1} = 2n/x \tilde{J}_n \ - \tilde{J}_{n+1} $$

Step 2. Normalize:

$$ \sigma = \tilde{J}_0 + 2 \tilde{J}_2 + 2 \tilde{J}_4 + ... $$

$$ J_n = \tilde{J}_n/\sigma, \ n = 0, ..., N $$

I'll show some results in a moment, after I introduce an alternative formulation.

Matrix Formulation

You know that I like to describe algorithms in matrix terms. The three-term recurrence generates a tridiagonal matrix. But which three diagonals? I can put the three below the diagonal, above the diagonal, or centered on the diagonal. This is accomplished with the second argument of the sparse matrix generator, spdiags. Here is the code.

type blt function B = blt(n,x,loc) % BesseL Three term recurence. % B = blt(n,x,loc) is an n-by-n sparse tridiagonal coefficent matrix % that generates the Bessel functions besselj((0:n-1)',x). % loc specifies the location of the diagonals. % loc = 'center', the default, centers the three diagonals. % loc = 'lower' puts the three diagonals in the lower triangle. % loc = 'upper' puts the three diagonals in the upper triangle. if nargin == 2 loc = 'center'; end switch loc case 'center', locd = -1:1; case 'lower', locd = -2:0; case 'upper', locd = 0:2; end % Three term recurrence: 2*n/x * J_n(x) = J_(n-1)(x) + J_n+1(x). d = -2*(0:n-1)'/x; e = ones(n,1); B = spdiags([e d e],locd,n,n); end Lower

If there were no roundoff error, and if we knew the values of the first two Bessel functions, $J_0(x)$ and $J_1(x)$, we could use backslash, the linear equation solver, with the diagonals in the lower triangle to compute $J_n(x)$ for $n > 1$. Here, for example, is $x = 1$.

format long J = @(n,x) besselj(n,x); n = 8 x = 1.0 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs n = 8 x = 1 v = 0.765197686557967 0.440050585744934 0.114903484931901 0.019563353982668 0.002476638964110 0.000249757730213 0.000020938338022 0.000001502326053

These values are accurate to almost all the digits shown. Except for the last one. It's beginning to show the effects of severe numerical instability. Let's try a larger value of n.

n = 18 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs; semilogy(0:n-1,v,'o-') title('J_n(1)') xlabel('n') n = 18

The values beyond about n = 9 are suspect and, in fact, totally wrong. The forward elimination that backslash uses to solve this lower triangular system is just the three-term recurrence for increasing n. And that recurrence has two solutions -- the one we're trying to compute and a second one corresponding to $Y_n(x)$, the Bessel function of second kind. $Y_n(x)$ is stimulated by roundoff error and completely takes over for $n > 9$.

So the lower triangular option -- the forward recurrence -- is unsatisfactory for two reasons: it requires two Bessel values to get started and it is unstable.

Upper

We need to make one modification to the upper triangular coefficient matrix.

n = 30; B = blt(n,x,'upper'); B(n-1,n) = 0;

Now the last two rows and columns are a 2-by-2 identity.

full(B(n-1:n,n-1:n)) ans = 1 0 0 1

The back substitution process just runs the three-term recursion backwards. We need to start with $J_{n-1}(x)$ and $J_{n-2}(x)$.

rhs = zeros(n,1); rhs(n-1:n) = J((n-2:n-1)',x); format long e v = B\rhs v = 7.651976865579656e-01 4.400505857449330e-01 1.149034849319004e-01 1.956335398266838e-02 2.476638964109952e-03 2.497577302112342e-04 2.093833800238925e-05 1.502325817436807e-06 9.422344172604491e-08 5.249250179911870e-09 2.630615123687451e-10 1.198006746303136e-11 4.999718179448401e-13 1.925616764480172e-14 6.885408200044221e-16 2.297531532210343e-17 7.186396586807488e-19 2.115375568053260e-20 5.880344573595754e-22 1.548478441211652e-23 3.873503008524655e-25 9.227621982096665e-27 2.098223955943776e-28 4.563424055950103e-30 9.511097932712488e-32 1.902951751891381e-33 3.660826744416801e-35 6.781552053554108e-37 1.211364502417112e-38 2.089159981718163e-40

These values are accurate to almost all the significant digits shown. The backwards recursion suppresses the unwanted $Y_n(x)$ and the process is stable.

Center

Can we avoid having to know those two starting values? Yes, by using a matrix formulation of the Miller algorithm. Insert the normalization condition in the first row. Now $B$ is tridiagonal, except for first row, which is

[1 0 2 0 2 0 2 ... ]

Other rows have $1$ s on the super- and subdiagonal and $-2n/x$ on the diagonal of the $(n+1)$ -st row.

n = 30; B = blt(n,x); B(1,1:2) = [1 0]; B(1,3:2:n) = 2; spy(B) title('B')

The beauty of this approach is that it does not require any values of the Bessel function a priori. The right hand side is simply a unit vector.

rhs = eye(n,1); format long e v = B\rhs semilogy(0:n-1,v,'o-') xlabel('n') title('J_n(1)') v = 7.651976865579665e-01 4.400505857449335e-01 1.149034849319005e-01 1.956335398266840e-02 2.476638964109954e-03 2.497577302112343e-04 2.093833800238926e-05 1.502325817436807e-06 9.422344172604494e-08 5.249250179911872e-09 2.630615123687451e-10 1.198006746303136e-11 4.999718179448401e-13 1.925616764480171e-14 6.885408200044220e-16 2.297531532210343e-17 7.186396586807487e-19 2.115375568053260e-20 5.880344573595753e-22 1.548478441211652e-23 3.873503008524655e-25 9.227621982096663e-27 2.098223955943775e-28 4.563424055950102e-30 9.511097932712487e-32 1.902951751891381e-33 3.660826744416762e-35 6.781552053355331e-37 1.211364395117367e-38 2.088559301926495e-40 Relative error w = J((0:n-1)',x); relerr = (v - w)./w; semilogy(0:n-1,abs(relerr) + eps*(v==w),'o-') xlabel('n') title('relative error')

The relative error deteriorates for the last few values of n. This is not unexpected because we have effectively truncated a semi-infinite matrix.

Triangular Factors

The LU decomposition of $B$ shows no pivoting for this value of $x$, but there is fill-in. The normalization condition is mixing in with the recurrence.

[L,U] = lu(B); spy(L) title('L') snapnow spy(U) title('U') References

J. C. P. Miller, "On the choice of standard solutions for a homogeneous linear equation of the second order", Quart. J. Mech. Appl. Math. 3, 1950, 225-235.

Walter Gautschi, "Computational Aspects of Three-Term Recurrence Relations", SIAM Review 9, 1967, 24-82.

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. I've just reversed the usual procedure for % computing them. And, I didn't pick the first two values % at random. In fact, their choice is very delicate. Try picking % any other six-digit integers, and see if you can take 30 differences % before you hit a negative value. Of course, we usually start with % $f_0 = 0$, $f_1 = 1$ and go forward with additions. %% % The relation % % $$ f_n = f_{n+1} - f_{n-1} $$ % % is an example of a _three-term recurrence_. I've written it in such % a way that I can go either forward or backward. %% Friedrich Bessel % Friedrich Bessel was a German astronomer, mathematician and physicist % who lived from 1784 until 1846. % He was a contemporary of Gauss, but the two had some sort of falling % out. Although he did not have a university education, he made major % contributions to precise measurements of planetary orbits and % distances to stars. He has a crater on the Moon and an asteroid % named after him. %% % In mathematics, the special functions that bear his name were actually % discovered by someone else, Daniel Bernoulli, and christened after % Bessel's death. They play the same role in polar coordinates that the % trig functions play in Cartesian coordinates. They are applicable % in a wide range of fields, from physics to finance. %% Bessel Functions % Bessel functions are defined by an ordinary differential % equation that generalizes the harmonic oscillator to polar % coordinates. The equation contains a parameter $n$, the _order_. % % $$ x^2 y'' + x y' + (x^2 - n^2) y = 0 $$ % % Solutions to this equation are also sometimes called % _cylindrical harmonics_. %% % For my purposes here, their most important property is the % three-term recurrence relation: % % $$ \frac{2n}{x} J_n(x) = J_{n-1}(x) + J_{n+1}(x) $$ % % This has a variable coefficient that depends upon the two parameters, % the order $n$ and the argument $x$. %% % To complete the specification, it would be necessary to supply two % initial conditions that depend upon $x$. % But, as we shall see, using the recurrence in the forward direction % is disastrous numerically. % Alternatively, if we could somehow supply two end conditions, % we could use the recurrence in the reverse direction. % This the idea behind a classical method known as Miller's algorithm. %% Miller's Algorithm % Miller's algorithm also employs this interesting identity, % which I'll call the _normalizer_. % % $$ J_0(x) + 2 J_2(x) + 2 J_4(x) + 2 J_6(x) + \ ... = 1 $$ % % It's not obvious where this identity comes from and I won't try to % derive it here. %% % We don't know the end conditions, but we can pick arbitrary values, % run the recurrence backwards, and then use the normalizer to rescale. %% % Step 1. Pick a large $N$ and temporarily let % % $$ \tilde{J}_N = 0 $$ % % $$ \tilde{J}_{N-1} = 1 $$ % % Then for $n = N-1, ..., 1$ compute % % $$ \tilde{J}_{n-1} = 2n/x \tilde{J}_n \ - \tilde{J}_{n+1} $$ % %% % Step 2. Normalize: % % $$ \sigma = \tilde{J}_0 + 2 \tilde{J}_2 + 2 \tilde{J}_4 + ... $$ % % $$ J_n = \tilde{J}_n/\sigma, \ n = 0, ..., N $$ %% % I'll show some results in a moment, after I introduce an % alternative formulation. %% Matrix Formulation % You know that I like to describe algorithms in matrix terms. % The three-term recurrence generates a tridiagonal matrix. % But which three diagonals? I can put the three below the diagonal, % above the diagonal, or centered on the diagonal. This is accomplished % with the second argument of the sparse matrix generator, |spdiags|. % Here is the code. type blt %% Lower % If there were no roundoff error, and if we knew the values of the % first two Bessel functions, $J_0(x)$ and $J_1(x)$, we could use % backslash, the linear equation solver, with the diagonals in the % lower triangle to compute $J_n(x)$ for $n > 1$. Here, for example, % is $x = 1$. format long J = @(n,x) besselj(n,x); n = 8 x = 1.0 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs %% % These values are accurate to almost all the digits shown. % Except for the last one. It's beginning to show the effects of % severe numerical instability. Let's try a larger value of |n|. n = 18 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs; semilogy(0:n-1,v,'o-') title('J_n(1)') xlabel('n') %% % The values beyond about |n = 9| are suspect and, in fact, totally % wrong. The forward elimination that backslash uses to solve this % lower triangular system is just the three-term recurrence for % increasing |n|. And that recurrence has two solutions REPLACE_WITH_DASH_DASH the one % we're trying to compute and a second one corresponding to $Y_n(x)$, % the _Bessel function of second kind_. $Y_n(x)$ is stimulated % by roundoff error and completely takes over for $n > 9$. %% % So the lower triangular option REPLACE_WITH_DASH_DASH the forward recurrence REPLACE_WITH_DASH_DASH is % unsatisfactory for two reasons: it requires two Bessel values to get % started and it is unstable. %% Upper % We need to make one modification to the upper triangular coefficient % matrix. n = 30; B = blt(n,x,'upper'); B(n-1,n) = 0; %% % Now the last two rows and columns are a 2-by-2 identity. full(B(n-1:n,n-1:n)) %% % The back substitution process just runs the three-term recursion % backwards. We need to start with $J_{n-1}(x)$ and $J_{n-2}(x)$. rhs = zeros(n,1); rhs(n-1:n) = J((n-2:n-1)',x); format long e v = B\rhs %% % These values are accurate to almost all the significant digits % shown. The backwards recursion suppresses the unwanted $Y_n(x)$ and % the process is stable. %% Center % Can we avoid having to know those two starting values? % Yes, by using a matrix formulation of the Miller algorithm. % Insert the normalization condition in the first row. % Now $B$ is tridiagonal, except for first row, which is % % [1 0 2 0 2 0 2 ... ] % % Other rows have $1$ s on the super- and subdiagonal and $-2n/x$ % on the diagonal of the $(n+1)$ -st row. n = 30; B = blt(n,x); B(1,1:2) = [1 0]; B(1,3:2:n) = 2; spy(B) title('B') %% % The beauty of this approach is that it does not require any % values of the Bessel function _a priori_. The right hand side % is simply a unit vector. rhs = eye(n,1); format long e v = B\rhs semilogy(0:n-1,v,'o-') xlabel('n') title('J_n(1)') %% Relative error w = J((0:n-1)',x); relerr = (v - w)./w; semilogy(0:n-1,abs(relerr) + eps*(v==w),'o-') xlabel('n') title('relative error') %% % The relative error deteriorates for the last few values of |n|. % This is not unexpected because we have effectively truncated % a semi-infinite matrix. %% Triangular Factors % The LU decomposition of $B$ shows no pivoting for this value of $x$, % but there is fill-in. The normalization condition is mixing in % with the recurrence. [L,U] = lu(B); spy(L) title('L') snapnow spy(U) title('U') %% References % J. C. P. Miller, "On the choice of standard solutions for a % homogeneous linear equation of the second order", % _Quart. J. Mech. Appl. Math._ 3, 1950, 225-235. % % Walter Gautschi, "Computational Aspects of Three-Term Recurrence % Relations", _SIAM Review_ 9, 1967, 24-82. % ##### SOURCE END ##### eac8550e3f7a4d2ea24b0951cd0803d1 -->

Three-Term Recurrence Relations and Bessel Functions

Mon, 11/06/2017 - 22:30
d

Three-term recurrence relations are the basis for computing Bessel functions.

ContentsA Familiar Three-Term Recurrence

Start with two large integers, say 514229 and 317811. For reasons that will soon become clear, I'll label them $f_{30}$ and $f_{29}$.

clear n = 30 f(30) = 514229; f(29) = 317811 n = 30 f = Columns 1 through 6 0 0 0 0 0 0 Columns 7 through 12 0 0 0 0 0 0 Columns 13 through 18 0 0 0 0 0 0 Columns 19 through 24 0 0 0 0 0 0 Columns 25 through 30 0 0 0 0 317811 514229

Now take repeated differences, and count backwards.

$$ f_{n-1} = f_{n+1} - f_n, \ n = 29, 28, ..., 2, 1 $$

for n = 29:-1:2 f(n-1) = f(n+1) - f(n); end n f n = 2 f = Columns 1 through 6 0 1 1 2 3 5 Columns 7 through 12 8 13 21 34 55 89 Columns 13 through 18 144 233 377 610 987 1597 Columns 19 through 24 2584 4181 6765 10946 17711 28657 Columns 25 through 30 46368 75025 121393 196418 317811 514229

Recognize those integers? They're the venerable Fibonacci numbers. I've just reversed the usual procedure for computing them. And, I didn't pick the first two values at random. In fact, their choice is very delicate. Try picking any other six-digit integers, and see if you can take 30 differences before you hit a negative value. Of course, we usually start with $f_0 = 0$, $f_1 = 1$ and go forward with additions.

The relation

$$ f_n = f_{n+1} - f_{n-1} $$

is an example of a three-term recurrence. I've written it in such a way that I can go either forward or backward.

Friedrich Bessel

Friedrich Bessel was a German astronomer, mathematician and physicist who lived from 1784 until 1846. He was a contemporary of Gauss, but the two had some sort of falling out. Although he did not have a university education, he made major contributions to precise measurements of planetary orbits and distances to stars. He has a crater on the Moon and an asteroid named after him.

In mathematics, the special functions that bear his name were actually discovered by someone else, Daniel Bernoulli, and christened after Bessel's death. They play the same role in polar coordinates that the trig functions play in Cartesian coordinates. They are applicable in a wide range of fields, from physics to finance.

Bessel Functions

Bessel functions are defined by an ordinary differential equation that generalizes the harmonic oscillator to polar coordinates. The equation contains a parameter $n$, the order.

$$ x^2 y'' + x y' + (x^2 - n^2) y = 0 $$

Solutions to this equation are also sometimes called cylindrical harmonics.

For my purposes here, their most important property is the three-term recurrence relation:

$$ \frac{2n}{x} J_n(x) = J_{n-1}(x) + J_{n+1}(x) $$

This has a variable coefficient that depends upon the two parameters, the order $n$ and the argument $x$.

To complete the specification, it would be necessary to supply two initial conditions that depend upon $x$. But, as we shall see, using the recurrence in the forward direction is disastrous numerically. Alternatively, if we could somehow supply two end conditions, we could use the recurrence in the reverse direction. This the idea behind a classical method known as Miller's algorithm.

Miller's Algorithm

Miller's algorithm also employs this interesting identity, which I'll call the normalizer.

$$ J_0(x) + 2 J_2(x) + 2 J_4(x) + 2 J_6(x) + \ ... = 1 $$

It's not obvious where this identity comes from and I won't try to derive it here.

We don't know the end conditions, but we can pick arbitrary values, run the recurrence backwards, and then use the normalizer to rescale.

Step 1. Pick a large $N$ and temporarily let

$$ \tilde{J}_N = 0 $$

$$ \tilde{J}_{N-1} = 1 $$

Then for $n = N-1, ..., 1$ compute

$$ \tilde{J}_{n-1} = 2n/x \tilde{J}_n \ - \tilde{J}_{n+1} $$

Step 2. Normalize:

$$ \sigma = \tilde{J}_0 + 2 \tilde{J}_2 + 2 \tilde{J}_4 + ... $$

$$ J_n = \tilde{J}_n/\sigma, \ n = 0, ..., N $$

I'll show some results in a moment, after I introduce an alternative formulation.

Matrix Formulation

You know that I like to describe algorithms in matrix terms. The three-term recurrence generates a tridiagonal matrix. But which three diagonals? I can put the three below the diagonal, above the diagonal, or centered on the diagonal. This is accomplished with the second argument of the sparse matrix generator, spdiags. Here is the code.

type blt function B = blt(n,x,loc) % BesseL Three term recurence. % B = blt(n,x,loc) is an n-by-n sparse tridiagonal coefficent matrix % that generates the Bessel functions besselj((0:n-1)',x). % loc specifies the location of the diagonals. % loc = 'center', the default, centers the three diagonals. % loc = 'lower' puts the three diagonals in the lower triangle. % loc = 'upper' puts the three diagonals in the upper triangle. if nargin == 2 loc = 'center'; end switch loc case 'center', locd = -1:1; case 'lower', locd = -2:0; case 'upper', locd = 0:2; end % Three term recurrence: 2*n/x * J_n(x) = J_(n-1)(x) + J_n+1(x). d = -2*(0:n-1)'/x; e = ones(n,1); B = spdiags([e d e],locd,n,n); end Lower

If there were no roundoff error, and if we knew the values of the first two Bessel functions, $J_0(x)$ and $J_1(x)$, we could use backslash, the linear equation solver, with the diagonals in the lower triangle to compute $J_n(x)$ for $n > 1$. Here, for example, is $x = 1$.

format long J = @(n,x) besselj(n,x); n = 8 x = 1.0 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs n = 8 x = 1 v = 0.765197686557967 0.440050585744934 0.114903484931901 0.019563353982668 0.002476638964110 0.000249757730213 0.000020938338022 0.000001502326053

These values are accurate to almost all the digits shown. Except for the last one. It's beginning to show the effects of severe numerical instability. Let's try a larger value of n.

n = 18 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs; semilogy(0:n-1,v,'o-') title('J_n(1)') xlabel('n') n = 18

The values beyond about n = 9 are suspect and, in fact, totally wrong. The forward elimination that backslash uses to solve this lower triangular system is just the three-term recurrence for increasing n. And that recurrence has two solutions -- the one we're trying to compute and a second one corresponding to $Y_n(x)$, the Bessel function of second kind. $Y_n(x)$ is stimulated by roundoff error and completely takes over for $n > 9$.

So the lower triangular option -- the forward recurrence -- is unsatisfactory for two reasons: it requires two Bessel values to get started and it is unstable.

Upper

We need to make one modification to the upper triangular coefficient matrix.

n = 30; B = blt(n,x,'upper'); B(n-1,n) = 0;

Now the last two rows and columns are a 2-by-2 identity.

full(B(n-1:n,n-1:n)) ans = 1 0 0 1

The back substitution process just runs the three-term recursion backwards. We need to start with $J_{n-1}(x)$ and $J_{n-2}(x)$.

rhs = zeros(n,1); rhs(n-1:n) = J((n-2:n-1)',x); format long e v = B\rhs v = 7.651976865579656e-01 4.400505857449330e-01 1.149034849319004e-01 1.956335398266838e-02 2.476638964109952e-03 2.497577302112342e-04 2.093833800238925e-05 1.502325817436807e-06 9.422344172604491e-08 5.249250179911870e-09 2.630615123687451e-10 1.198006746303136e-11 4.999718179448401e-13 1.925616764480172e-14 6.885408200044221e-16 2.297531532210343e-17 7.186396586807488e-19 2.115375568053260e-20 5.880344573595754e-22 1.548478441211652e-23 3.873503008524655e-25 9.227621982096665e-27 2.098223955943776e-28 4.563424055950103e-30 9.511097932712488e-32 1.902951751891381e-33 3.660826744416801e-35 6.781552053554108e-37 1.211364502417112e-38 2.089159981718163e-40

These values are accurate to almost all the significant digits shown. The backwards recursion suppresses the unwanted $Y_n(x)$ and the process is stable.

Center

Can we avoid having to know those two starting values? Yes, by using a matrix formulation of the Miller algorithm. Insert the normalization condition in the first row. Now $B$ is tridiagonal, except for first row, which is

[1 0 2 0 2 0 2 ... ]

Other rows have $1$ s on the super- and subdiagonal and $-2n/x$ on the diagonal of the $(n+1)$ -st row.

n = 30; B = blt(n,x); B(1,1:2) = [1 0]; B(1,3:2:n) = 2; spy(B) title('B')

The beauty of this approach is that it does not require any values of the Bessel function a priori. The right hand side is simply a unit vector.

rhs = eye(n,1); format long e v = B\rhs semilogy(0:n-1,v,'o-') xlabel('n') title('J_n(1)') v = 7.651976865579665e-01 4.400505857449335e-01 1.149034849319005e-01 1.956335398266840e-02 2.476638964109954e-03 2.497577302112343e-04 2.093833800238926e-05 1.502325817436807e-06 9.422344172604494e-08 5.249250179911872e-09 2.630615123687451e-10 1.198006746303136e-11 4.999718179448401e-13 1.925616764480171e-14 6.885408200044220e-16 2.297531532210343e-17 7.186396586807487e-19 2.115375568053260e-20 5.880344573595753e-22 1.548478441211652e-23 3.873503008524655e-25 9.227621982096663e-27 2.098223955943775e-28 4.563424055950102e-30 9.511097932712487e-32 1.902951751891381e-33 3.660826744416762e-35 6.781552053355331e-37 1.211364395117367e-38 2.088559301926495e-40 Relative error w = J((0:n-1)',x); relerr = (v - w)./w; semilogy(0:n-1,abs(relerr) + eps*(v==w),'o-') xlabel('n') title('relative error')

The relative error deteriorates for the last few values of n. This is not unexpected because we have effectively truncated a semi-infinite matrix.

Triangular Factors

The LU decomposition of $B$ shows no pivoting for this value of $x$, but there is fill-in. The normalization condition is mixing in with the recurrence.

[L,U] = lu(B); spy(L) title('L') snapnow spy(U) title('U') References

J. C. P. Miller, "On the choice of standard solutions for a homogeneous linear equation of the second order", Quart. J. Mech. Appl. Math. 3, 1950, 225-235.

Walter Gautschi, "Computational Aspects of Three-Term Recurrence Relations", SIAM Review 9, 1967, 24-82.

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. I've just reversed the usual procedure for % computing them. And, I didn't pick the first two values % at random. In fact, their choice is very delicate. Try picking % any other six-digit integers, and see if you can take 30 differences % before you hit a negative value. Of course, we usually start with % $f_0 = 0$, $f_1 = 1$ and go forward with additions. %% % The relation % % $$ f_n = f_{n+1} - f_{n-1} $$ % % is an example of a _three-term recurrence_. I've written it in such % a way that I can go either forward or backward. %% Friedrich Bessel % Friedrich Bessel was a German astronomer, mathematician and physicist % who lived from 1784 until 1846. % He was a contemporary of Gauss, but the two had some sort of falling % out. Although he did not have a university education, he made major % contributions to precise measurements of planetary orbits and % distances to stars. He has a crater on the Moon and an asteroid % named after him. %% % In mathematics, the special functions that bear his name were actually % discovered by someone else, Daniel Bernoulli, and christened after % Bessel's death. They play the same role in polar coordinates that the % trig functions play in Cartesian coordinates. They are applicable % in a wide range of fields, from physics to finance. %% Bessel Functions % Bessel functions are defined by an ordinary differential % equation that generalizes the harmonic oscillator to polar % coordinates. The equation contains a parameter $n$, the _order_. % % $$ x^2 y'' + x y' + (x^2 - n^2) y = 0 $$ % % Solutions to this equation are also sometimes called % _cylindrical harmonics_. %% % For my purposes here, their most important property is the % three-term recurrence relation: % % $$ \frac{2n}{x} J_n(x) = J_{n-1}(x) + J_{n+1}(x) $$ % % This has a variable coefficient that depends upon the two parameters, % the order $n$ and the argument $x$. %% % To complete the specification, it would be necessary to supply two % initial conditions that depend upon $x$. % But, as we shall see, using the recurrence in the forward direction % is disastrous numerically. % Alternatively, if we could somehow supply two end conditions, % we could use the recurrence in the reverse direction. % This the idea behind a classical method known as Miller's algorithm. %% Miller's Algorithm % Miller's algorithm also employs this interesting identity, % which I'll call the _normalizer_. % % $$ J_0(x) + 2 J_2(x) + 2 J_4(x) + 2 J_6(x) + \ ... = 1 $$ % % It's not obvious where this identity comes from and I won't try to % derive it here. %% % We don't know the end conditions, but we can pick arbitrary values, % run the recurrence backwards, and then use the normalizer to rescale. %% % Step 1. Pick a large $N$ and temporarily let % % $$ \tilde{J}_N = 0 $$ % % $$ \tilde{J}_{N-1} = 1 $$ % % Then for $n = N-1, ..., 1$ compute % % $$ \tilde{J}_{n-1} = 2n/x \tilde{J}_n \ - \tilde{J}_{n+1} $$ % %% % Step 2. Normalize: % % $$ \sigma = \tilde{J}_0 + 2 \tilde{J}_2 + 2 \tilde{J}_4 + ... $$ % % $$ J_n = \tilde{J}_n/\sigma, \ n = 0, ..., N $$ %% % I'll show some results in a moment, after I introduce an % alternative formulation. %% Matrix Formulation % You know that I like to describe algorithms in matrix terms. % The three-term recurrence generates a tridiagonal matrix. % But which three diagonals? I can put the three below the diagonal, % above the diagonal, or centered on the diagonal. This is accomplished % with the second argument of the sparse matrix generator, |spdiags|. % Here is the code. type blt %% Lower % If there were no roundoff error, and if we knew the values of the % first two Bessel functions, $J_0(x)$ and $J_1(x)$, we could use % backslash, the linear equation solver, with the diagonals in the % lower triangle to compute $J_n(x)$ for $n > 1$. Here, for example, % is $x = 1$. format long J = @(n,x) besselj(n,x); n = 8 x = 1.0 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs %% % These values are accurate to almost all the digits shown. % Except for the last one. It's beginning to show the effects of % severe numerical instability. Let's try a larger value of |n|. n = 18 B = blt(n,x,'lower'); rhs = zeros(n,1); rhs(1:2) = J((0:1)',x); v = B\rhs; semilogy(0:n-1,v,'o-') title('J_n(1)') xlabel('n') %% % The values beyond about |n = 9| are suspect and, in fact, totally % wrong. The forward elimination that backslash uses to solve this % lower triangular system is just the three-term recurrence for % increasing |n|. And that recurrence has two solutions REPLACE_WITH_DASH_DASH the one % we're trying to compute and a second one corresponding to $Y_n(x)$, % the _Bessel function of second kind_. $Y_n(x)$ is stimulated % by roundoff error and completely takes over for $n > 9$. %% % So the lower triangular option REPLACE_WITH_DASH_DASH the forward recurrence REPLACE_WITH_DASH_DASH is % unsatisfactory for two reasons: it requires two Bessel values to get % started and it is unstable. %% Upper % We need to make one modification to the upper triangular coefficient % matrix. n = 30; B = blt(n,x,'upper'); B(n-1,n) = 0; %% % Now the last two rows and columns are a 2-by-2 identity. full(B(n-1:n,n-1:n)) %% % The back substitution process just runs the three-term recursion % backwards. We need to start with $J_{n-1}(x)$ and $J_{n-2}(x)$. rhs = zeros(n,1); rhs(n-1:n) = J((n-2:n-1)',x); format long e v = B\rhs %% % These values are accurate to almost all the significant digits % shown. The backwards recursion suppresses the unwanted $Y_n(x)$ and % the process is stable. %% Center % Can we avoid having to know those two starting values? % Yes, by using a matrix formulation of the Miller algorithm. % Insert the normalization condition in the first row. % Now $B$ is tridiagonal, except for first row, which is % % [1 0 2 0 2 0 2 ... ] % % Other rows have $1$ s on the super- and subdiagonal and $-2n/x$ % on the diagonal of the $(n+1)$ -st row. n = 30; B = blt(n,x); B(1,1:2) = [1 0]; B(1,3:2:n) = 2; spy(B) title('B') %% % The beauty of this approach is that it does not require any % values of the Bessel function _a priori_. The right hand side % is simply a unit vector. rhs = eye(n,1); format long e v = B\rhs semilogy(0:n-1,v,'o-') xlabel('n') title('J_n(1)') %% Relative error w = J((0:n-1)',x); relerr = (v - w)./w; semilogy(0:n-1,abs(relerr) + eps*(v==w),'o-') xlabel('n') title('relative error') %% % The relative error deteriorates for the last few values of |n|. % This is not unexpected because we have effectively truncated % a semi-infinite matrix. %% Triangular Factors % The LU decomposition of $B$ shows no pivoting for this value of $x$, % but there is fill-in. The normalization condition is mixing in % with the recurrence. [L,U] = lu(B); spy(L) title('L') snapnow spy(U) title('U') %% References % J. C. P. Miller, "On the choice of standard solutions for a % homogeneous linear equation of the second order", % _Quart. J. Mech. Appl. Math._ 3, 1950, 225-235. % % Walter Gautschi, "Computational Aspects of Three-Term Recurrence % Relations", _SIAM Review_ 9, 1967, 24-82. % ##### SOURCE END ##### eac8550e3f7a4d2ea24b0951cd0803d1 -->