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Path: xanth!nic.MR.NET!umn-d-ub!rutgers!mailrus!ulowell!page From: page@swan.ulowell.edu (Bob Page) Newsgroups: comp.sources.amiga Subject: v02i050: matlab - matrix laboratory, Part10/11 (doc01/02) Message-ID: <10025@swan.ulowell.edu> Date: 2 Nov 88 21:54:17 GMT Organization: University of Lowell, Computer Science Dept. Lines: 2320 Approved: page@swan.ulowell.edu Submitted-by: strovink%galaxy-43@afit-ab.arpa (Mark A. Strovink) Posting-number: Volume 2, Issue 50 Archive-name: applications/matlab/doc.1 # This is a shell archive. # Remove everything above and including the cut line. # Then run the rest of the file through sh. #----cut here-----cut here-----cut here-----cut here----# #!/bin/sh # shar: Shell Archiver # Run the following text with /bin/sh to create: # doc.1 # This archive created: Wed Nov 2 16:14:31 1988 cat << \SHAR_EOF > doc.1 6/24/81 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. CONTENTS 1. Elementary operations page 2 2. MATLAB functions 8 3. Rows, columns and submatrices 9 4. FOR, WHILE and IF 10 5. Commands, text, files and macros 12 6. Census example 13 7. Partial differential equation 19 8. Eigenvalue sensitivity example 23 9. Syntax diagrams 27 10. The parser-interpreter 31 11. The numerical algorithms 34 12. FLOP and CHOP 37 13. Communicating with other programs 41 Appendix. The HELP file 46 6/24/81 MATLAB Users' Guide November, 1980 Cleve Moler Department of Computer Science University of New Mexico 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. MATLAB, page 2 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 MATLAB, page 3 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, MATLAB, page 4 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, MATLAB, page 5 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 MATLAB, page 6 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 = MATLAB, page 7 -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 MATLAB, page 8 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. MATLAB, page 9 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 MATLAB, page 10 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 MATLAB, page 11 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 MATLAB, page 12 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. MATLAB, page 13 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 MATLAB, page 14 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 MATLAB, page 15 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 MATLAB, page 16 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 MATLAB, page 17 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 MATLAB, page 18 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. MATLAB, page 19 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. MATLAB, page 20 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". MATLAB, page 21 //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 MATLAB, page 22 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. MATLAB, page 23 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, MATLAB, page 24 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 MATLAB, page 25 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 MATLAB, page 26 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 MATLAB, page 27 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+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 -- A through Z, digit -- 0 through 9, MATLAB, page 28 char -- ( ) ; : + - * / \ = . , < > quote -- ' line |-----> statement >----| | | |-----> clause >-------| | | -------|-----> expr >---------|------> | | | | | |-----> command >------| | | | | | | |-> > >-> expr >-> < >-| | | | | | | |----------------------| | | | | |-< ; <-| | |--------| |---------| |-< , <-| statement |-> name >--------------------------------| | | | | | |--> : >---| | | | | | | | |-> ( >---|-> expr >-|---> ) >-| | | | | -----| |-----< , <----| |--> = >--> expr >---> | | | |--< , <---| | | | | | |-> < >---> name >---> > >----------------| MATLAB, page 29 clause |---> FOR >---> name >---> = >---> expr >--------------| | | | |-> WHILE >-| | |-| |-> expr >---------------------- | | |-> IF >-| | | | | | | | -----| < <= = <> >= > |----> | | | | | | | | | ----------------------> expr >--| | | |---> ELSE >--------------------------------------------| | | |---> END >--------------------------------------------| expr |-> + >-| | | -------|-------|-------> term >----------> | | | | |-> - >-| | |-< + <-| | | | | | |--|-< - <-|--| | | |-< : <-| term ---------------------> factor >----------------------> | | | |-< * <-| | | |-------| | | |-------| | |--| |--|-< / <-|--| |--| |-< . <-| | | |-< . <-| |-< \ <-| MATLAB, page 30 factor |----------------> number >---------------| | | |-> name >--------------------------------| | | | | | |--> : >---| | | | | | | | |-> ( >---|-> expr >-|---> ) >-| | | | | | |-----< , <----| | | | -----|------------> ( >-----> expr >-----> ) >-|-|-------|-----> | | | | | | |--------------| | |-> ' >-| | | | | | | |------------> < >-|---> expr >---|-> > >-| | | | | | | | |--< <---| | | | | | | | | |--< ; <---| | | | | | | | | |--< , <---| | | | | | |------------> > >-----> expr >-----> < >-| | | | | |-----> factor >---> ** >---> factor >----| | | | |------------> ' >-----> text >-----> ' >-------------| number |----------| |-> + >-| | | | | -----> int >-----> . >---> int >-----> E >---------> int >----> | | | | | | | | | |-> - >-| | | | | | |---------------------------------------------| int ------------> digit >-----------> | | |-----------| MATLAB, page 31 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. MATLAB, page 32 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 MATLAB, page 33 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 MATLAB, page 34 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. SHAR_EOF # End of shell archive exit 0 -- Bob Page, U of Lowell CS Dept. page@swan.ulowell.edu ulowell!page Have five nice days.