Mercurial > matrix-functions
changeset 0:8f23314345f4 draft
Create local repository for matrix toolboxes. Step #0 done.
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/blocking.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,73 @@ +function m = blocking(A,delta,showplot) +%BLOCKING Produce blocking pattern for block Parlett recurrence. +% M = BLOCKING(A, DELTA, SHOWPLOT) accepts an upper triangular matrix +% A and produces a blocking pattern, specified by the vector M, +% for the block Parlett recurrence. +% M(i) is the index of the block into which A(i,i) should be placed. +% DELTA is a gap parameter (default 0.1) used to determine the +% blocking. +% Setting SHOWPLOT nonzero produces a plot of the eigenvalues +% that indicates the blocking: +% - Black circles show a set of 1 eigenvalue. +% - Blue circles show a set of >1 eigenvalues. +% The lines connect eigenvalues in the same set. +% Red squares show the mean of each set. + +% For A coming from a real matrix it should be posible to take +% advantage of the symmetry about the real axis. This code does not. + +a = diag(A); n = length(a); +m = zeros(1,n); maxM = 0; + +if nargin < 2 | isempty(delta), delta = 0.1; end +if nargin < 3, showplot = 0; end + +if showplot, clf, hold on, end + +for i = 1:n + + if m(i) == 0 + m(i) = maxM + 1; % If a(i) hasn`t been assigned to a set + maxM = maxM + 1; % then make a new set and assign a(i) to it. + end + + for j = i+1:n + if m(i) ~= m(j) % If a(i) and a(j) are not in same set. + if abs(a(i)-a(j)) <= delta + if showplot + plot(real([a(i) a(j)]),imag([a(i) a(j)]),'o-') + end + + if m(j) == 0 + m(j) = m(i); % If a(j) hasn`t been assigned to a + % set, assign it to the same set as a(i). + else + p = max(m(i),m(j)); q = min(m(i),m(j)); + m(m==p) = q; % If a(j) has been assigned to a set + % place all the elements in the set + % containing a(j) into the set + % containing a(i) (or vice versa). + m(m>p) = m(m>p) -1; + maxM = maxM - 1; + % Tidying up. As we have deleted set + % p we reduce the index of the sets + % > p by 1. + end + end + end + end +end + +if showplot + for i = 1:max(m) + a_ind = a(m==i); + if length(a_ind) == 1 + plot(real(a_ind),imag(a_ind),'ok' ) + else +% plot(real(mean(a_ind)),imag(mean(a_ind)),'sr' ) + end + end + grid + hold off + box on +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/fun_cos.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,14 @@ +function f = fun_cos(x,k) +%FUN_COS + +if nargin < 2 | k == 0 + f = cos(x); +else + g = mod(ceil(k/2),2); + h = mod(k,2); + if h == 1 + f = sin(x)*(-1)^g; + else + f = cos(x)*(-1)^g; + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/fun_exp.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,4 @@ +function f = fun_exp(x,k) +%FUN_EXP + +f = exp(x);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/fun_log.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,5 @@ +function f = fun_log(x) +%FUN_LOG +% Only to be called for plain log evaluation. + +f = log(x);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/fun_sin.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,15 @@ +function f = fun_sin(x,k) +%FUN_SIN + +if nargin < 2 | k == 0 + f = sin(x); +else + k = k - 1; + g = mod(ceil(k/2),2); + h = mod(k,2); + if h == 1 + f = sin(x)*(-1)^g; + else + f = cos(x)*(-1)^g; + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/funm_atom.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,78 @@ +function [F,n_terms] = fun_atom(T,fun,tol,prnt) +%FUNM_ATOM Function of triangular matrix with nearly constant diagonal. +% [F, N_TERMS] = FUNM_ATOM(T, FUN, TOL, PRNT) evaluates function +% FUN at the upper triangular matrix T, where T has nearly constant +% diagonal. A Taylor series is used. +% FUN(X,K) must return the K'th derivative of +% the function represented by FUN evaluated at the vector X. +% TOL is a convergence tolerance for the Taylor series, +% defaulting to EPS. +% If PRNT ~= 0 trace information is printed. +% N_TERMS is the number of terms taken in the Taylor series. +% N_TERMS = -1 signals lack of convergence. + +if nargin < 3 | isempty(tol), tol = eps; end +if nargin < 4, prnt = 0; end + +if isequal(fun,@fun_log) % LOG is special case. + [F,n_terms] = logm_isst(T,prnt); + return +end + +itmax = 500; + +n = length(T); +if n == 1, F = feval(fun,T,0); n_terms = 1; return, end + +lambda = sum(diag(T))/n; +F = eye(n)*feval(fun,lambda,0); +f_deriv_max = zeros(itmax+n-1,1); +N = T - lambda*eye(n); +mu = norm( (eye(n)-abs(triu(T,1)))\ones(n,1),inf ); + +P = N; +max_d = 1; + +for k = 1:itmax + + f = feval(fun,lambda,k); + F_old = F; + F = F + P*f; + rel_diff = norm(F - F_old,inf)/(tol+norm(F_old,inf)); + if prnt + fprintf('%3.0f: coef = %5.0e', k, abs(f)/factorial(k)); + fprintf(' N^k/k! = %7.1e', norm(P,inf)); + fprintf(' rel_d = %5.0e',rel_diff); + fprintf(' abs_d = %5.0e',norm(F - F_old,inf)); + end + P = P*N/(k+1); + + if rel_diff <= tol + + % Approximate the maximum of derivatives in convex set containing + % eigenvalues by maximum of derivatives at eigenvalues. + for j = max_d:k+n-1 + f_deriv_max(j) = norm(feval(fun,diag(T),j),inf); + end + max_d = k+n; + omega = 0; + for j = 0:n-1 + omega = max(omega,f_deriv_max(k+j)/factorial(j)); + end + + trunc = norm(P,inf)*mu*omega; % norm(F) moved to RHS to avoid / 0. + if prnt + fprintf(' [trunc,test] = [%5.0e %5.0e]', ... + trunc, tol*norm(F,inf)) + end + if prnt == 5, trunc = 0; end % Force simple stopping test. + if trunc <= tol*norm(F,inf) + n_terms = k; + if prnt, fprintf('\n'), end, return + end + end + + if prnt, fprintf('\n'), end + +end +n_terms = -1;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/logm_isst.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,80 @@ +function [X, iter] = logm_isst(T, prnt) +%LOGM_ISST Log of triangular matrix by Schur-Pade method with scaling. +% X = LOGM_ISST(A) computes the logarithm of an upper triangular +% matrix A, for a matrix with no nonpositive real eigenvalues, +% using the inverse scaling and squaring method with Pade +% approximation. TOL is an error tolerance. +% [X, ITER] = LOGM_ISST(A, PRNT) returns the number ITER of square +% roots computed and prints this information if PRNT is nonzero. + +% References: +% S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, Approximating the +% logarithm of a matrix to specified accuracy, SIAM J. Matrix Anal. Appl., +% 22(4):1112-1125, 2001. +% N. J. Higham, Evaluating Pade approximants of the matrix logarithm, +% SIAM J. Matrix Anal. Appl., 22(4):1126-1135, 2001. + +if nargin < 2, prnt = 0; end +n = length(T); + +if any( imag(diag(T)) == 0 & real(diag(T)) <= 0 ) + error('A must not have nonpositive real eigenvalues!') +end + +if n == 1, X = log(T); iter = 0; return, end + +R = T; +maxlogiter = 50; + +for iter = 0:maxlogiter + + phi = norm(T-eye(n),'fro'); + + if phi <= 0.25 + if prnt, fprintf('LOGM_ISST computed %g square roots. \n', iter), end + break + end + if iter == maxlogiter, error('Too many square roots in LOGM_ISST.\n'), end + + % Compute upper triangular square root R of T, a column at a time. + for j=1:n + R(j,j) = sqrt(T(j,j)); + for i=j-1:-1:1 + R(i,j) = (T(i,j) - R(i,i+1:j-1)*R(i+1:j-1,j))/(R(i,i) + R(j,j)); + end + end + T = R; +end + +X = 2^(iter)*logm_pf(T-eye(n),8); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function S = logm_pf(A,m) +%LOGM_PF Pade approximation to matrix log by partial fraction expansion. +% Y = LOGM_PF(A,m) approximates LOG(I+A). + +[nodes,wts] = gauss_legendre(m); +% Convert from [-1,1] to [0,1]. +nodes = (nodes + 1)/2; +wts = wts/2; + +n = length(A); +S = zeros(n); + +for j=1:m + S = S + wts(j)*(A/(eye(n) + nodes(j)*A)); +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function [x,w] = gauss_legendre(n) +%GAUSS_LEGENDRE Nodes and weights for Gauss-Legendre quadrature. + +% Reference: +% G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature +% rules, Math. Comp., 23(106):221-230, 1969. + +i = 1:n-1; +v = i./sqrt((2*i).^2-1); +[V,D] = eig( diag(v,-1)+diag(v,1) ); +x = diag(D); +w = 2*(V(1,:)'.^2);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/myfunm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,96 @@ +function [F,n_swaps,n_calls,terms,ind,T] = myfunm(A,fun,delta,tol,prnt,m) +%MYFUNM Evaluate general matrix function. +% F = FUNM(A,FUN), for a square matrix argument A, evaluates the +% function FUN at the square matrix A. +% FUN(X,K) must return the K'th derivative of +% the function represented by FUN evaluated at the vector X. +% The MATLAB functions COS, SIN, EXP, LOG can be passed as FUN, +% i.e. FUNM(A,@COS), FUNM(A,@SIN), FUNM(@EXP), FUNM(A,@LOG). +% For matrix square roots use SQRTM(A) instead. +% For matrix exponentials, either of EXPM(A) and FUNM(A,@EXPM) +% may be the faster or the more accurate, depending on A. +% +% F = FUNM(A,FUN,DELTA,TOL,PRNT,M) specifies a tolerance +% DELTA used in determining the blocking (default 0.1), +% and a tolerance TOL used in a convergence test for evaluating the +% Taylor series (default EPS). +% If PRNT is nonzero then information describing the +% behaviour of the algorithm is printed. +% M, if supplied, defines a blocking. +% +% [F,N_SWAPS,N_CALLS,TERMS,IND,T] = FUNM(A,FUN,...) also returns +% N_SWAPS: the total number of swaps in the Schur re-ordering. +% N_CALLS: the total number of calls to ZTREXC for the re-ordering. +% TERMS(I): the number of Taylor series terms used when evaluating +% the I'th atomic triangular block. +% IND: a cell array specifying the blocking: the (I,J) block of +% the re-ordered Schur factor T is T(IND{I},IND{j}), +% T: the re-ordered Schur form. + +if isequal(fun,@cos) || isequal(fun,'cos'), fun = @fun_cos; end +if isequal(fun,@sin) || isequal(fun,'sin'), fun = @fun_sin; end +if isequal(fun,@exp) || isequal(fun,'exp'), fun = @fun_exp; end + +if nargin < 3 || isempty(delta), delta = 0.1; end +if nargin < 4 || isempty(tol), tol = eps; end +if nargin < 5 || isempty(prnt), prnt = 0; end +if nargin < 6, m = []; end + +n = length(A); + +% First form complex Schur form (if A not already upper triangular). +if isequal(A,triu(A)) + T = A; U = eye(n); +else + [U,T] = schur(A,'complex'); +end + +if isequal(T,tril(T)) % Handle special case of diagonal T. + F = U*diag(feval(fun,diag(T)))*U'; + n_swaps = 0; n_calls = 0; terms = 0; ind = {1:n}; + return +end + +% Determine reordering of Schur form into block form. +if isempty(m), m = blocking(T,delta,abs(prnt)>=3); end + +if prnt, fprintf('delta (blocking) = %9.2e, tol (TS) = %9.2e\n', delta, tol), +end + +[M,ind,n_swaps] = swapping(m); +n_calls = size(M,1); +if n_calls > 0 % If there are swaps to do... + [U,T] = swap(U,T,M); % MEX file +end + +m = length(ind); + +% Calculate F(T) +F = zeros(n); + +for col=1:m + j = ind{col}; + [F(j,j), n_terms] = funm_atom(T(j,j),fun,tol,abs(prnt)*(prnt ~= 1)); + terms(col) = n_terms; + + for row=col-1:-1:1 + i = ind{row}; + if length(i) == 1 && length(j) == 1 + % Scalar case. + k = i+1:j-1; + temp = T(i,j)*(F(i,i) - F(j,j)) + F(i,k)*T(k,j) - T(i,k)*F(k,j); + F(i,j) = temp/(T(i,i)-T(j,j)); + else + k = cat(2,ind{row+1:col-1}); + rhs = F(i,i)*T(i,j) - T(i,j)*F(j,j) + F(i,k)*T(k,j) - T(i,k)*F(k,j); + F(i,j) = sylv_tri(T(i,i),-T(j,j),rhs); + end + end +end + +F = U*F*U'; + +% As in FUNM: +if isreal(A) && norm(imag(F),1) <= 10*n*eps*norm(F,1) + F = real(F); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/swap.c Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,101 @@ +/* + * C mex file for MATLAB that implements LAPACK ctrexc_ for + * reordering the complex Schur decomposition A = UTU^* such that + * T is in block form. This program combines Cswap and Crealswap + * which deal with the complex and the real case respectively. + * + * Input matrices 'U' and 'T' are those from Schur decompositon + * + * Called by [U,T] = swap(U,T,M) + * + * The matrix M is produced using the m-files + * >> m = blocking(T,noplot,delta); where delta is some tolerance + * default: delta = 0.1; + * >> [n,M,ind,totalswaps] = swapping5(m); + * + * Output matrices 'U' and 'T' are the updated Schur decomposition + * + */ + +#include "mex.h" +#include "matrix.h" +#include "fort.h" /* defines mat2fort and fort2mat */ + +void mexFunction(int nlhs, mxArray *plhs[], int nrhs, const mxArray *prhs[]) +{ + + /* compq=V then the matrix of Schur vectors is updated */ + + char *compq = "V", msg[80]; + int n, ldt, ldq, i, info, ifst, ilst, lengthm; + double *t, *q, *m, *work; + mxArray *mm; + + /* expect 3 inputs and 2 outputs */ + if ((nrhs != 3) || (nlhs != 2)){ + mexErrMsgTxt("Expected 3 inputs and 2 outputs"); + } + + /* Sizes of stuff */ + + n = mxGetN( prhs[0] ); + ldt = n; + ldq = n; + lengthm = mxGetM( prhs[2] ); + + /* Copy input and output matrices to stuff */ + + mm = mxDuplicateArray( prhs[2] ); + m = mxGetPr(mm); + + if (mxIsComplex( prhs[1] )) { + + /* Convert complex input to complex FORTRAN format */ + + q = mat2fort( prhs[0], ldq, n ); + t = mat2fort( prhs[1], ldt, n ); } + + else { + + plhs[0] = mxCreateDoubleMatrix(n,n,mxREAL); + plhs[1] = mxCreateDoubleMatrix(n,n,mxREAL); + + plhs[1] = mxDuplicateArray( prhs[1] ); + t = mxGetPr(plhs[1]); + plhs[0] = mxDuplicateArray( prhs[0] ); + q = mxGetPr(plhs[0]); + } + + /* Allocate workspace */ + + work = (double *)mxCalloc(n,sizeof(double)); + + /* Do the loop */ + + for ( i = 0; i < lengthm; i++) { + info = 0; + ifst = m[lengthm + i]; + ilst = m[i]; + if (mxIsComplex( prhs[1] )) { + ztrexc(compq,&n,t,&ldt,q,&ldq,&ifst,&ilst,&info); } + else { + dtrexc(compq,&n,t,&ldt,q,&ldq,&ifst,&ilst,work,&info); + } + if (info < 0){ + sprintf(msg, "The routine DTREXC has detected an error"); + mexErrMsgTxt(msg); + } + } + + /* Convert output to MATLAB format */ + + if (mxIsComplex( prhs[1] )) { + plhs[0] = fort2mat( q, ldq, ldq, n ); + plhs[1] = fort2mat( t, ldt, ldt, n ); + } + + /* Free up memory */ + + mxFree(work); +} +
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/swapping.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,47 @@ +function [M,ind,n_swaps] = swapping(m) +%SWAPPING Confluent permutation by swapping adjacent elements. +% [ISWAP,IND,N_SWAPS] = SWAPPING(M) takes a vector M containing +% the integers 1:k (some repeated if K < LENGTH(M)) +% and constructs a swapping scheme that produces +% a confluent permutation, with elements ordered by ascending +% average position. The confluent permutation is obtained by using +% the LAPACK routine ZTREX to move m(ISWAP(i,2)) to m(ISWAP(i,1)) +% by swapping adjacent elements, for i = 1:SIZE(M,1). +% The cell array vector IND defines the resulting block form: +% IND{i} contains the indices of the i'th block in the permuted form. +% N_SWAPS is the total number of swaps required. + +mmax = max(m); M = []; ind = {}; h = zeros(1,mmax); +g = zeros(1,mmax); + +for i = 1:mmax + p = find(m==i); + h(i) = length(p); + g(i) = sum(p)/h(i); +end + +[x,y] = sort(g); +mdone = 1; + +for i = y + if any(m(mdone:mdone+h(i)-1) ~= i) + f = find(m==i); g = mdone:mdone+h(i)-1; + ff = f(f~=g); gg = g(f~=g); + + % Create vector v = mdone:f(end) with all elements of f deleted. + v = mdone-1 + find(m(mdone:f(end)) ~= i); + + % v = zeros(1,f(end)-g(1)+1); + % v(f-g(1)+1) = 1; v = g(1)-1 + find(v==0); + + M(end+1:end+length(gg),:) = [gg' ff']; + + m(g(end)+1:f(end)) = m(v); + m(g) = i*ones(1,h(i)); + ind = cat(2,ind,{mdone:mdone+h(i)-1}); mdone = mdone + h(i); + else + ind = cat(2,ind,{mdone:mdone+h(i)-1}); mdone = mdone + h(i); + end +end + +n_swaps = sum(abs(diff(M')));
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/funm_files/sylv_tri.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,13 @@ +function X = sylv_tri(T,U,B) +%SYLV_TRI Solves triangular Sylvester equation. +% x = SYLV_TRI(T,U,B) solves the Sylvester equation +% T*X + X*U = B, where T and U are square upper triangular matrices. + +m = length(T); +n = length(U); +X = zeros(m,n); + +% Forward substitution. +for i = 1:n + X(:,i) = (T + U(i,i)*eye(m)) \ (B(:,i) - X(:,1:i-1)*U(1:i-1,i)); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/Contents.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,67 @@ +% Matrix Computation Toolbox. +% Version 1.2 5-Sep-2002 +% Copyright (c) 2002 by N. J. Higham +% +% Demonstration +% mctdemo - Demonstration of Matrix Computation Toolbox. +% +% Test Matrices +% augment - Augmented system matrix. +% gfpp - Matrix giving maximal growth factor for Gaussian elimination +% with partial pivoting. +% makejcf - A matrix with specified Jordan canonical form. +% rschur - An upper quasi-triangular matrix. +% vand - Vandermonde matrix. +% vecperm - Vec-permutation matrix. +% +% Visualization +% fv - Field of values (or numerical range). +% gersh - Gershgorin disks. +% ps - Dot plot of a pseudospectrum. +% pscont - Contours and colour pictures of pseudospectra. +% see - Pictures of a matrix. +% +% Factorizations and Decompositions +% cholp - Cholesky factorization with pivoting of a positive semidefinite +% matrix. +% cod - Complete orthogonal decomposition. +% gep - Gaussian elimination with pivoting: none, complete, partial or +% rook. +% gj - Gauss-Jordan elimination with partial pivoting to solve Ax = b. +% gqr - Generalized QR factorization. +% gs_c - Classical Gram-Schmidt QR factorization. +% gs_m - Modified Gram-Schmidt QR factorization. +% ldlt_skew - Block LDL^T factorization for a skew-symmetric matrix. +% ldlt_symm - Block LDL^T factorization for a symmetric indefinite matrix. +% ldlt_sytr - Block LDL^T factorization for a symmetric tridiagonal +% matrix. +% matsignt - Matrix sign function of a triangular matrix. +% poldec - Polar decomposition. +% signm - Matrix sign decomposition. +% trap2tri - Unitary reduction of trapezoidal matrix to triangular form. +% +% Direct Search Optimization +% adsmax - Alternating directions method. +% mdsmax - Multidirectional search method. +% mmsmax - Nelder-Mead simplex method. +% +% Miscellaneous +% chop - Round matrix elements. +% cpltaxes - Determine suitable AXIS for plot of complex vector. +% dual - Dual vector with respect to Holder p-norm. +% lse - Solve the equality constrained least squares problem. +% matrix - Matrix Computation Toolbox information and matrix access by +% number. +% pnorm - Estimate of matrix p-norm (1 <= p <= inf). +% rootm - P'th root of a matrix. +% seqcheb - Sequence of points related to Chebyshev polynomials. +% seqm - Multiplicative sequence. +% show - Display signs of matrix elements. +% skewpart - Skew-symmetric (skew-Hermitian) part. +% sparsify - Randomly set matrix elements to zero. +% strassen - Strassen's fast matrix multiplication algorithm. +% strassenw - Strassen's fast matrix multiplication algorithm (Winograd +% variant). +% sub - Principal submatrix. +% symmpart - Symmetric (Hermitian) part. +% treshape - Reshape vector to or from (unit) triangular matrix.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/adsmax.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,141 @@ +function [x, fmax, nf] = adsmax(f, x, stopit, savit, P, varargin) +%ADSMAX Alternating directions method for direct search optimization. +% [x, fmax, nf] = ADSMAX(FUN, x0, STOPIT, SAVIT, P) attempts to +% maximize the function FUN, using the starting vector x0. +% The alternating directions direct search method is used. +% Output arguments: +% x = vector yielding largest function value found, +% fmax = function value at x, +% nf = number of function evaluations. +% The iteration is terminated when either +% - the relative increase in function value between successive +% iterations is <= STOPIT(1) (default 1e-3), +% - STOPIT(2) function evaluations have been performed +% (default inf, i.e., no limit), or +% - a function value equals or exceeds STOPIT(3) +% (default inf, i.e., no test on function values). +% Progress of the iteration is not shown if STOPIT(5) = 0 (default 1). +% If a non-empty fourth parameter string SAVIT is present, then +% `SAVE SAVIT x fmax nf' is executed after each inner iteration. +% By default, the search directions are the co-ordinate directions. +% The columns of a fifth parameter matrix P specify alternative search +% directions (P = EYE is the default). +% NB: x0 can be a matrix. In the output argument, in SAVIT saves, +% and in function calls, x has the same shape as x0. +% ADSMAX(fun, x0, STOPIT, SAVIT, P, P1, P2,...) allows additional +% arguments to be passed to fun, via feval(fun,x,P1,P2,...). + +% Reference: +% N. J. Higham, Optimization by direct search in matrix computations, +% SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993. +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 20.5. + +x0 = x(:); % Work with column vector internally. +n = length(x0); + +mu = 1e-4; % Initial percentage change in components. +nstep = 25; % Max number of times to double or decrease h. + +% Set up convergence parameters. +if nargin < 3 | isempty(stopit), stopit(1) = 1e-3; end +tol = stopit(1); % Required rel. increase in function value over one iteration. +if length(stopit) == 1, stopit(2) = inf; end % Max no. of f-evaluations. +if length(stopit) == 2, stopit(3) = inf; end % Default target for f-values. +if length(stopit) < 5, stopit(5) = 1; end % Default: show progress. +trace = stopit(5); +if nargin < 4, savit = []; end % File name for snapshots. + +if nargin < 5 | isempty(P) + P = eye(n); % Matrix of search directions. +else + if ~isequal(size(P),[n n]) % Check for common error. + error('P must be of dimension the number of elements in x0.') + end +end + +fmax = feval(f,x,varargin{:}); nf = 1; +if trace, fprintf('f(x0) = %9.4e\n', fmax), end + +steps = zeros(n,1); +it = 0; y = x0; + +while 1 % Outer loop. +it = it+1; +if trace, fprintf('Iter %2.0f (nf = %2.0f)\n', it, nf), end +fmax_old = fmax; + +for i=1:n % Loop over search directions. + + pi = P(:,i); + flast = fmax; + yi = y; + h = sign(pi'*yi)*norm(pi.*yi)*mu; % Initial step size. + if h == 0, h = max(norm(yi,inf),1)*mu; end + y = yi + h*pi; + x(:) = y; fnew = feval(f,x,varargin{:}); nf = nf + 1; + if fnew > fmax + fmax = fnew; + if fmax >= stopit(3) + if trace + fprintf('Comp. = %2.0f, steps = %2.0f, f = %9.4e*\n', i,0,fmax) + fprintf('Exceeded target...quitting\n') + end + x(:) = y; return + end + h = 2*h; lim = nstep; k = 1; + else + h = -h; lim = nstep+1; k = 0; + end + + for j=1:lim + y = yi + h*pi; + x(:) = y; fnew = feval(f,x,varargin{:}); nf = nf + 1; + if fnew <= fmax, break, end + fmax = fnew; k = k + 1; + if fmax >= stopit(3) + if trace + fprintf('Comp. = %2.0f, steps = %2.0f, f = %9.4e*\n', i,j,fmax) + fprintf('Exceeded target...quitting\n') + end + x(:) = y; return + end + h = 2*h; + end + + steps(i) = k; + y = yi + 0.5*h*pi; + if k == 0, y = yi; end + + if trace + fprintf('Comp. = %2.0f, steps = %2.0f, f = %9.4e', i, k, fmax) + fprintf(' (%2.1f%%)\n', 100*(fmax-flast)/(abs(flast)+eps)) + end + + + if nf >= stopit(2) + if trace + fprintf('Max no. of function evaluations exceeded...quitting\n') + end + x(:) = y; return + end + + if fmax > flast & ~isempty(savit) + x(:) = y; + eval(['save ' savit ' x fmax nf']) + end + +end % Loop over search directions. + +if isequal(steps,zeros(n,1)) + if trace, fprintf('Stagnated...quitting\n'), end + x(:) = y; return +end + +if fmax-fmax_old <= tol*abs(fmax_old) + if trace, fprintf('Function values ''converged''...quitting\n'), end + x(:) = y; return +end + +end %%%%%% Of outer loop.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/augment.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,40 @@ +function C = augment(A, alpha) +%AUGMENT Augmented system matrix. +% AUGMENT(A, ALPHA) is the square matrix +% [ALPHA*EYE(m) A; A' ZEROS(n)] of dimension m+n, where A is m-by-n. +% It is the symmetric and indefinite coefficient matrix of the +% augmented system associated with a least squares problem +% minimize NORM(A*x-b). ALPHA defaults to 1. +% Special case: if A is a scalar, n say, then AUGMENT(A) is the +% same as AUGMENT(RANDN(p,q)) where n = p+q and +% p = ROUND(n/2), that is, a random augmented matrix +% of dimension n is produced. +% The eigenvalues of AUGMENT(A,ALPHA) are given in terms of the +% singular values s(i) of A (where m>n) by +% ALPHA/2 +/- SQRT( s(i)^2*ALPHA^2 + 1/4 ), i=1:n (2n eigenvalues), +% ALPHA, (m-n eigenvalues). +% If m < n then the first expression provides 2m eigenvalues and the +% remaining n-m eigenvalues are zero. +% +% See also SPAUGMENT. + +% References: +% G. H. Golub and C. F. Van Loan, Matrix Computations, third +% Edition, Johns Hopkins University Press, Baltimore, Maryland, +% 1996; sec. 5.6.4. +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 20.5. + +[m, n] = size(A); +if nargin < 2, alpha = 1; end + +if max(m,n) == 1 + n = A; + p = round(n/2); + q = n - p; + A = randn(p,q); + m = p; n = q; +end + +C = [alpha*eye(m) A; A' zeros(n)];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/cholp.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,78 @@ +function [R, P, I] = cholp(A, piv) +%CHOLP Cholesky factorization with pivoting of a positive semidefinite matrix. +% [R, P] = CHOLP(A) returns an upper triangular matrix R and a +% permutation matrix P such that R'*R = P'*A*P. Only the upper +% triangular part of A is used. If A is not positive semidefinite, +% an error message is printed. +% +% [R, P, I] = CHOLP(A) never produces an error message. +% If A is positive semidefinite then I = 0 and R is the Cholesky factor. +% If A is not positive semidefinite then I is positive and +% R is (I-1)-by-N with P'*A*P - R'*R zero in columns 1:I-1 and +% rows 1:I-1. +% [R, I] = CHOLP(A, 0) forces P = EYE(SIZE(A)), and therefore behaves +% like [R, I] = CHOL(A). + +% This routine is based on the LINPACK routine CCHDC. It works +% for both real and complex matrices. +% +% Reference: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 10.3. + +if nargin == 1, piv = 1; end + +n = length(A); +pp = 1:n; +I = 0; + +for k = 1:n + + if piv + d = diag(A); + [big, m] = max( d(k:n) ); + m = m+k-1; + else + big = A(k,k); m = k; + end + if big < 0, I = k; break, end + +% Symmetric row/column permutations. + if m ~= k + A(:, [k m]) = A(:, [m k]); + A([k m], :) = A([m k], :); + pp( [k m] ) = pp( [m k] ); + end + + if big == 0 + if norm(A(k+1:n,k)) ~= 0 + I = k; break + else + continue + end + end + + A(k,k) = sqrt( A(k,k) ); + if k == n, break, end + A(k, k+1:n) = A(k, k+1:n) / A(k,k); + +% For simplicity update the whole of the remaining submatrix (rather +% than just the upper triangle). + + j = k+1:n; + A(j,j) = A(j,j) - A(k,j)'*A(k,j); + +end + +R = triu(A); +if I > 0 + if nargout < 3, error('Matrix must be positive semidefinite.'), end + R = R(1:I-1,:); +end + +if piv == 0 + P = I; +else + P = eye(n); P = P(:,pp); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/chop.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,18 @@ +function c = chop(x, t) +%CHOP Round matrix elements. +% CHOP(X, t) is the matrix obtained by rounding the elements of X +% to t significant binary places. +% Default is t = 24, corresponding to IEEE single precision. + +if nargin < 2, t = 24; end +[m, n] = size(x); + +% Use the representation: +% x(i,j) = 2^e(i,j) * .d(1)d(2)...d(s) * sign(x(i,j)) + +% On the next line `+(x==0)' avoids passing a zero argument to LOG, which +% would cause a warning message to be generated. + +y = abs(x) + (x==0); +e = floor(log2(y) + 1); +c = pow2(round( pow2(x, t-e) ), e-t);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/cod.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,38 @@ +function [U, R, V] = cod(A, tol) +%COD Complete orthogonal decomposition. +% [U, R, V] = COD(A, TOL) computes a decomposition A = U*T*V, +% where U and V are unitary, T = [R 0; 0 0] has the same dimensions as +% A, and R is upper triangular and nonsingular of dimension rank(A). +% Rank decisions are made using TOL, which defaults to approximately +% LENGTH(A)*NORM(A)*EPS. +% By itself, COD(A, TOL) returns R. + +% Reference: +% G. H. Golub and C. F. Van Loan, Matrix Computations, third +% edition, Johns Hopkins University Press, Baltimore, Maryland, +% 1996; sec. 5.4.2. + +[m, n] = size(A); + +% QR decomposition. +[U, R, P] = qr(A); % AP = UR +V = P'; % A = URV; +if nargin == 1, tol = max(m,n)*eps*abs(R(1,1)); end % |R(1,1)| approx NORM(A). + +% Determine r = effective rank. +r = sum(abs(diag(R)) > tol); +r = r(1); % Fix for case where R is vector. +R = R(1:r,:); % Throw away negligible rows (incl. all zero rows, m>n). + +if r ~= n + + % Reduce nxr R' = r [L] to lower triangular form: QR' = [Lbar]. + % n-r [M] [0] + + [Q, R] = trap2tri(R'); + V = Q*V; + R = R'; + +end + +if nargout <= 1, U = R; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/cpltaxes.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,37 @@ +function x = cpltaxes(z) +%CPLTAXES Determine suitable AXIS for plot of complex vector. +% X = CPLTAXES(Z), where Z is a complex vector, +% determines a 4-vector X such that AXIS(X) sets axes for a plot +% of Z that has axes of equal length and leaves a reasonable amount +% of space around the edge of the plot. + +% Called by FV, GERSH, PS and PSCONT. + +% Set x and y axis ranges so both have the same length. + +xmin = min(real(z)); xmax = max(real(z)); +ymin = min(imag(z)); ymax = max(imag(z)); + +% Fix for rare case of `trivial data'. +if xmin == xmax, xmin = xmin - 1/2; xmax = xmax + 1/2; end +if ymin == ymax, ymin = ymin - 1/2; ymax = ymax + 1/2; end + +if xmax-xmin >= ymax-ymin + ymid = (ymin + ymax)/2; + ymin = ymid - (xmax-xmin)/2; ymax = ymid + (xmax-xmin)/2; +else + xmid = (xmin + xmax)/2; + xmin = xmid - (ymax-ymin)/2; xmax = xmid + (ymax-ymin)/2; +end +axis('square') + +% Scale ranges by 1+2*alpha to give extra space around edges of plot. + +alpha = 0.1; +x(1) = xmin - alpha*(xmax-xmin); +x(2) = xmax + alpha*(xmax-xmin); +x(3) = ymin - alpha*(ymax-ymin); +x(4) = ymax + alpha*(ymax-ymin); + +if x(1) == x(2), x(2) = x(2) + 0.1; end +if x(3) == x(4), x(4) = x(3) + 0.1; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/dual.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,46 @@ +function y = dual(x, p) +%DUAL Dual vector with respect to Holder p-norm. +% Y = DUAL(X, p), where 1 <= p <= inf, is a vector of unit q-norm +% that is dual to X with respect to the p-norm, that is, +% norm(Y, q) = 1 where 1/p + 1/q = 1 and there is +% equality in the Holder inequality: X'*Y = norm(X, p)*norm(Y, q). +% Special case: DUAL(X), where X >= 1 is a scalar, returns Y such +% that 1/X + 1/Y = 1. + +% Called by PNORM. + +warns = warning; +warning('off') + +if nargin == 1 + if length(x) == 1 + y = 1/(1-1/x); + return + else + error('Second argument missing.') + end +end + +q = 1/(1-1/p); + +if norm(x,inf) == 0, y = x; return, end + +if p == 1 + + y = sign(x) + (x == 0); % y(i) = +1 or -1 (if x(i) real). + +elseif p == inf + + [xmax, k] = max(abs(x)); + f = find(abs(x)==xmax); k = f(1); + y = zeros(size(x)); + y(k) = sign(x(k)); % y is a multiple of unit vector e_k. + +else % 1 < p < inf. Dual is unique in this case. + + x = x/norm(x,inf); % This scaling helps to avoid under/over-flow. + y = abs(x).^(p-1) .* ( sign(x) + (x==0) ); + y = y / norm(y,q); % Normalize to unit q-norm. + +end +warning(warns)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/fv.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,101 @@ +function [f, e] = fv(B, nk, thmax, noplot) +%FV Field of values (or numerical range). +% FV(A, NK, THMAX) evaluates and plots the field of values of the +% NK largest leading principal submatrices of A, using THMAX +% equally spaced angles in the complex plane. +% The defaults are NK = 1 and THMAX = 16. +% (For a `publication quality' picture, set THMAX higher, say 32.) +% The eigenvalues of A are displayed as `x'. +% Alternative usage: [F, E] = FV(A, NK, THMAX, 1) suppresses the +% plot and returns the field of values plot data in F, with A's +% eigenvalues in E. Note that NORM(F,INF) approximates the +% numerical radius, +% max {abs(z): z is in the field of values of A}. + +% Theory: +% Field of values FV(A) = set of all Rayleigh quotients. FV(A) is a +% convex set containing the eigenvalues of A. When A is normal FV(A) is +% the convex hull of the eigenvalues of A (but not vice versa). +% z = x'Ax/(x'x), z' = x'A'x/(x'x) +% => REAL(z) = x'Hx/(x'x), H = (A+A')/2 +% so MIN(EIG(H)) <= REAL(z) <= MAX(EIG(H)), +% with equality for x = corresponding eigenvectors of H. For these x, +% RQ(A,x) is on the boundary of FV(A). +% +% Based on an original routine by A. Ruhe. +% +% References: +% R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge +% University Press, 1991; sec. 1.5. +% A. S. Householder, The Theory of Matrices in Numerical Analysis, +% Blaisdell, New York, 1964; sec. 3.3. +% C. R. Johnson, Numerical determination of the field of values of a +% general complex matrix, SIAM J. Numer. Anal., 15 (1978), +% pp. 595-602. + +if nargin < 2 | isempty(nk), nk = 1; end +if nargin < 3 | isempty(thmax), thmax = 16; end +thmax = thmax - 1; % Because code below uses thmax + 1 angles. + +iu = sqrt(-1); +[n, p] = size(B); +if n ~= p, error('Matrix must be square.'), end +f = []; +z = zeros(2*thmax+1,1); +e = eig(B); + +% Filter out cases where B is Hermitian or skew-Hermitian, for efficiency. +if isequal(B,B') + + f = [min(e) max(e)]; + +elseif isequal(B,-B') + + e = imag(e); + f = [min(e) max(e)]; + e = iu*e; f = iu*f; + +else + +for m = 1:nk + + ns = n+1-m; + A = B(1:ns, 1:ns); + + for i = 0:thmax + th = i/thmax*pi; + Ath = exp(iu*th)*A; % Rotate A through angle th. + H = 0.5*(Ath + Ath'); % Hermitian part of rotated A. + [X, D] = eig(H); + [lmbh, k] = sort(real(diag(D))); + z(1+i) = rq(A,X(:,k(1))); % RQ's of A corr. to eigenvalues of H + z(1+i+thmax) = rq(A,X(:,k(ns))); % with smallest/largest real part. + end + + f = [f; z]; + +end +% Next line ensures boundary is `joined up' (needed for orthogonal matrices). +f = [f; f(1,:)]; + +end +if thmax == 0; f = e; end + +if nargin < 4 + + ax = cpltaxes(f); + plot(real(f), imag(f)) % Plot the field of values + axis(ax); + axis('square'); + + hold on + plot(real(e), imag(e), 'x') % Plot the eigenvalues too. + hold off + +end + +function z = rq(A,x) +%RQ Rayleigh quotient. +% RQ(A,x) is the Rayleigh quotient of A and x, x'*A*x/(x'*x). + +z = x'*A*x/(x'*x);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/gep.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,119 @@ +function [L, U, P, Q, rho, ncomp] = gep(A, piv) +%GEP Gaussian elimination with pivoting: none, complete, partial or rook. +% [L, U, P, Q, RHO] = GEP(A, piv) computes the factorization P*A*Q = L*U +% of the m-by-n matrix A, where m >= n, +% where L is m-by-n unit lower triangular, U is n-by-n upper triangular, +% and P and Q are permutation matrices. RHO is the growth factor. +% PIV controls the pivoting strategy: +% PIV = 'c': complete pivoting, +% PIV = 'p': partial pivoting, +% PIV = 'r': rook pivoting. +% The default is no pivoting (PIV = ''). +% For PIV = 'r' only, NCOMP is the total number of comparisons. +% +% By itself, GEP(A) returns the final reduced matrix from the +% elimination containing both L and U. + +% Reference: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; chap. 9. + +[m, n] = size(A); +if m < n, error('Matrix must be m-by-n with m >= n.'), end +if nargin < 2, piv = ''; end +pp = 1:m; qq = 1:n; +if nargout >= 5 + maxA = norm(A(:), inf); + rho = maxA; +end +ncomp = 0; + +for k = 1:min(m-1,n) + + if findstr(piv, 'cpr') + if strcmp(piv, 'c') + + % Find largest element in remaining square submatrix. + % Note: when tie for max, no guarantee which element is chosen. + [colmaxima, rowindices] = max( abs(A(k:m, k:n)) ); + [biggest, colindex] = max(colmaxima); + row = rowindices(colindex)+k-1; col = colindex+k-1; + + elseif strcmp(piv, 'p') + + % Find largest element in k'th column. + [colmaxima, rowindices] = max( abs(A(k:m, k)) ); + row = rowindices(1)+k-1; col = k; + + elseif strcmp(piv, 'r') + + % Find element that is largest in its row and its column. + col_last = k; + for it = 1:inf + [colmaxima, rowindices] = max( abs(A(k:m, col_last)) ); + ncomp = ncomp + m-k; + row = rowindices(1)+k-1; + new_abs = abs(A(row,col_last)); + if it > 1 + if new_abs == last_abs + row = row_last; + break + end + end + last_abs = new_abs; + row_last = row; + [rowmaxima, colindices] = max( abs(A(row, k:n)) ); + ncomp = ncomp + n-k; + col = colindices(1)+k-1; + new_abs = abs(A(row,col)); + if new_abs == last_abs + col = col_last; + break + end + last_abs = new_abs; + col_last = col; + end + + end + + % Permute largest element into pivot position. + A( [k, row], : ) = A( [row, k], : ); + A( :, [k, col] ) = A( :, [col, k] ); + pp( [k, row] ) = pp( [row, k] ); qq( [k, col] ) = qq( [col, k] ); + end + + if A(k,k) == 0 + if findstr(piv, 'c') + break + elseif strcmp(piv, '') % Zero pivot is problem only for no pivoting. + error('Elimination breaks down with zero pivot. Quitting...') + end + end + + i = k+1:m; + if A(k,k) ~= 0 % Simplest way to handle zero pivot for partial and rook. + A(i,k) = A(i,k)/A(k,k); % Multipliers. + end + + if k+1 <= n + % Elimination + j = k+1:n; + A(i,j) = A(i,j) - A(i,k) * A(k,j); + if nargout >= 5, rho = max( rho, max(max(abs(A(i,j)))) ); end + end + +end + +if nargout <= 1 + L = A; + return +end + +L = tril(A,-1) + eye(m,n); +U = triu(A); +U = U(1:n,:); + +if nargout >= 3, P = eye(m); P = P(pp,:); end +if nargout >= 4, Q = eye(n); Q = Q(:,qq); end +if nargout >= 5, rho = rho/maxA; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/gersh.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,38 @@ +function [G, e] = gersh(A, noplot) +%GERSH Gershgorin disks. +% GERSH(A) draws the Gershgorin disks for the square matrix A. +% The eigenvalues are plotted as crosses `x'. +% Alternative usage: [G, E] = GERSH(A, 1) suppresses the plot +% and returns the data in G, with A's eigenvalues in E. +% +% Try GERSH(GALLERY('LESP',N)) and GERSH(GALLERY('SMOKE',N)). + +if diff(size(A)), error('Matrix must be square.'), end + +n = length(A); +m = 40; +G = zeros(m,n); + +d = diag(A); +r = sum( abs( A-diag(d) )' )'; +e = eig(A); + +radvec = exp(i * linspace(0,2*pi,m)'); + +for j=1:n + G(:,j) = d(j)*ones(m,1) + r(j)*radvec; +end + +if nargin < 2 + + ax = cpltaxes(G(:)); + for j=1:n + plot(real(G(:,j)), imag(G(:,j)),'-') % Plot the disks. + hold on + end + plot(real(e), imag(e), 'x') % Plot the eigenvalues too. + axis(ax) + axis('square') + hold off + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/gfpp.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,45 @@ +function A = gfpp(T, c) +%GFPP Matrix giving maximal growth factor for Gaussian elim. with pivoting. +% GFPP(T) is a matrix of order N for which Gaussian elimination +% with partial pivoting yields a growth factor 2^(N-1). +% T is an arbitrary nonsingular upper triangular matrix of order N-1. +% GFPP(T, C) sets all the multipliers to C (0 <= C <= 1) +% and gives growth factor (1+C)^(N-1) - but note that for T ~= EYE +% it is advisable to set C < 1, else rounding errors may cause +% computed growth factors smaller than expected. +% GFPP(N, C) (a special case) is the same as GFPP(EYE(N-1), C) and +% generates the well-known example of Wilkinson. + +% Reference: +% N. J. Higham and D. J. Higham, Large growth factors in +% Gaussian elimination with pivoting, SIAM J. Matrix Analysis and +% Appl., 10 (1989), pp. 155-164. +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 9.4. + +if ~isequal(T,triu(T)) | any(~diag(T)) + error('First argument must be a nonsingular upper triangular matrix.') +end + +if nargin == 1, c = 1; end + +if c < 0 | c > 1 + error('Second parameter must be a scalar between 0 and 1 inclusive.') +end + +m = length(T); +if m == 1 % Handle the special case T = scalar + n = T; + m = n-1; + T = eye(n-1); +else + n = m+1; +end + +A = zeros(n); +L = eye(n) - c*tril(ones(n), -1); +A(:,1:n-1) = L*[T; zeros(1,n-1)]; +theta = max(abs(A(:))); +A(:,n) = theta * ones(n,1); +A = A/theta;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/gj.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,36 @@ +function x = gj(A, b, piv) +%GJ Gauss-Jordan elimination to solve Ax = b. +% x = GJ(A, b, PIV) solves Ax = b by Gauss-Jordan elimination, +% where A is a square, nonsingular matrix. +% PIV determines the form of pivoting: +% PIV = 0: no pivoting, +% PIV = 1 (default): partial pivoting. + +% Reference: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 14.4. + +n = length(A); +if nargin < 3, piv = 1; end + +for k=1:n + if piv + % Partial pivoting (below the diagonal). + [colmax, i] = max( abs(A(k:n, k)) ); + i = k+i-1; + if i ~= k + A( [k, i], : ) = A( [i, k], : ); + b( [k, i] ) = b( [i, k] ); + end + end + + irange = [1:k-1 k+1:n]; + jrange = k:n; + mult = A(irange,k)/A(k,k); % Multipliers. + A(irange, jrange) = A(irange, jrange) - mult*A(k, jrange); + b(irange) = b(irange) - mult*b(k); + +end + +x = diag(diag(A))\b;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/gqr.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,55 @@ +function [U, Q, L, S] = gqr(A, B, partial) +%GQR Generalized QR factorization. +% [U, Q, L, S] = GQR(A, B, partial) factorizes +% the m-by-n A and p-by-n B, where m >= n >= p, as +% A = U*L*Q^T, B = S*Q^T, with U and Q orthogonal +% and L = [0; L1], S = [S1 0], with L1 and S1 lower triangular. +% If a nonzero third argument is present then only a partial reduction +% of A is performed: the first p columns of A are not reduced to +% triangular form (which is sufficient for solving the LSE problem). + +% Reference: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 20.9. + +[m n] = size(A); +[p n1] = size(B); +if nargin < 3, partial = 0; end + +if n ~= n1, error('A and B must have same number of columns!'), end + +if partial + limit = p+1; +else + limit = 1; +end + +[Q, S] = qr(B'); +S = S'; + +U = eye(m); +A = A*Q; + +% QL factorization of AQ. + +for i = n:-1:limit + + % Vector-reversal so that Householder eliminates leading + % rather than trailing elements. + temp = A(1:m-n+i,i); temp = temp(end:-1:1); + [v, beta] = gallery('house',temp); + v = v(end:-1:1); + + temp = A(1:m-n+i,1:i); + A(1:m-n+i,1:i) = temp - beta*v*(v'*temp); + + % Put zeros where they're supposed to be! + A(1:m-n+i-1,i) = zeros(m-n+i-1,1); + + temp = U(:,1:m-n+i); + U(:,1:m-n+i) = temp - beta*temp*v*v'; + +end + +L = A;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/gs_c.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,21 @@ +function [Q, R] = gs_c(A) +%GS_C Classical Gram-Schmidt QR factorization. +% [Q, R] = GS_C(A) uses the classical Gram-Schmidt method to compute the +% factorization A = Q*R for m-by-n A of full rank, +% where Q is m-by-n with orthonormal columns and R is n-by-n. + +% Reference: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec 19.8. + +[m, n] = size(A); +Q = zeros(m,n); +R = zeros(n); + +for j=1:n + R(1:j-1,j) = Q(:,1:j-1)'*A(:,j); + temp = A(:,j) - Q(:,1:j-1)*R(1:j-1,j); + R(j,j) = norm(temp); + Q(:,j) = temp/R(j,j); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/gs_m.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,21 @@ +function [Q, R] = gs_m(A) +%GS_M Modified Gram-Schmidt QR factorization. +% [Q, R] = GS_M(A) uses the modified Gram-Schmidt method to compute the +% factorization A = Q*R for m-by-n A of full rank, +% where Q is m-by-n with orthonormal columns and R is n-by-n. + +% Reference: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec 19.8. + +[m, n] = size(A); +Q = zeros(m,n); +R = zeros(n); + +for k=1:n + R(k,k) = norm(A(:,k)); + Q(:,k) = A(:,k)/R(k,k); + R(k,k+1:n) = Q(:,k)'*A(:,k+1:n); + A(:,k+1:n) = A(:,k+1:n) - Q(:,k)*R(k,k+1:n); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/ldlt_skew.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,83 @@ +function [L, D, P, rho] = ldlt_skew(A) +%LDLT_SKEW Block LDL^T factorization for a skew-symmetric matrix. +% Given a real, skew-symmetric A, +% [L, D, P, RHO] = LDLT_SKEW(A) computes a permutation P, +% a unit lower triangular L, and a block diagonal D +% with 1x1 and 2x2 diagonal blocks, such that P*A*P' = L*D*L'. +% A partial pivoting strategy of Bunch is used. +% RHO is the growth factor. + +% Reference: +% J. R. Bunch, A note on the stable decomposition of skew-symmetric +% matrices. Math. Comp., 38(158):475-479, 1982. +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; chap. 11. + +% This routine does not exploit skew-symmetry and is not designed to +% be efficient. + +if ~isreal(A) | ~isequal(triu(A,1)',-tril(A,-1)) + error('Must supply real, skew-symmetric matrix.') +end + +n = length(A); +k = 1; +D = zeros(n); +L = eye(n); +pp = 1:n; +if nargout >= 4 + maxA = norm(A(:), inf); + rho = maxA; +end + +while k < n + + if max( abs(A(k+1:n,k)) ) == 0 + + s = 1; + % Nothing to do. + + else + + s = 2; + + if k < n-1 + [colmaxima, rowindices] = max( abs(A(k+1:n, k:k+1)) ); + [biggest, colindex] = max(colmaxima); + row = rowindices(colindex)+k; col = colindex+k-1; + + % Permute largest element into (k+1,k) position. + % NB: k<->col permutation must be done before k+1<->row one. + A( [k, col], : ) = A( [col, k], : ); + A( :, [k, col] ) = A( :, [col, k] ); + A( [k+1, row], : ) = A( [row, k+1], : ); + A( :, [k+1, row] ) = A( :, [row, k+1] ); + L( [k, col], : ) = L( [col, k], : ); + L( :, [k, col] ) = L( :, [col, k] ); + L( [k+1, row], : ) = L( [row, k+1], : ); + L( :, [k+1, row] ) = L( :, [row, k+1] ); + pp( [k, col] ) = pp( [col, k] ); + pp( [k+1, row] ) = pp( [row, k+1] ); + end + + E = A(k:k+1,k:k+1); + D(k:k+1,k:k+1) = E; + C = A(k+2:n,k:k+1); + temp = C/E; + L(k+2:n,k:k+1) = temp; + A(k+2:n,k+2:n) = A(k+2:n,k+2:n) + temp*C'; % Note the plus sign. + % Restore skew-symmetry. + A(k+2:n,k+2:n) = 0.5 * (A(k+2:n,k+2:n) - A(k+2:n,k+2:n)'); + + if nargout >= 4, rho = max(rho, max(max(abs(A(k+2:n,k+2:n)))) ); end + + end + + k = k + s; + if k >= n-2, D(k:n,k:n) = A(k:n,k:n); break, end; + +end + +if nargout >= 3, P = eye(n); P = P(pp,:); end +if nargout >= 4, rho = rho/maxA; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/ldlt_symm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,161 @@ +function [L, D, P, rho, ncomp] = ldlt_symm(A, piv) +%LDLT_SYMM Block LDL^T factorization for a symmetric indefinite matrix. +% Given a Hermitian matrix A, +% [L, D, P, RHO, NCOMP] = LDLT_SYMM(A, PIV) computes a permutation P, +% a unit lower triangular L, and a real block diagonal D +% with 1x1 and 2x2 diagonal blocks, such that P*A*P' = L*D*L'. +% PIV controls the pivoting strategy: +% PIV = 'p': partial pivoting (Bunch and Kaufman), +% PIV = 'r': rook pivoting (Ashcraft, Grimes and Lewis). +% The default is partial pivoting. +% RHO is the growth factor. +% For PIV = 'r' only, NCOMP is the total number of comparisons. + +% References: +% J. R. Bunch and L. Kaufman, Some stable methods for calculating +% inertia and solving symmetric linear systems, Math. Comp., +% 31(137):163-179, 1977. +% C. Ashcraft, R. G. Grimes and J. G. Lewis, Accurate symmetric +% indefinite linear equation solvers. SIAM J. Matrix Anal. Appl., +% 20(2):513-561, 1998. +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; chap. 11. + +% This routine does not exploit symmetry and is not designed to be +% efficient. + +if ~isequal(triu(A,1)',tril(A,-1)), error('Must supply Hermitian matrix.'), end +if nargin < 2, piv = 'p'; end + +n = length(A); +k = 1; +D = eye(n); L = eye(n); if n == 1, D = A; end +pp = 1:n; +if nargout >= 4 + maxA = norm(A(:), inf); + rho = maxA; +end +ncomp = 0; + +alpha = (1 + sqrt(17))/8; + +while k < n + [lambda, r] = max( abs(A(k+1:n,k)) ); + r = r(1) + k; + + if lambda > 0 + swap = 0; + if abs(A(k,k)) >= alpha*lambda + s = 1; + else + if piv == 'p' + temp = A(k:n,r); temp(r-k+1) = 0; + sigma = norm(temp, inf); + if alpha*lambda^2 <= abs(A(k,k))*sigma + s = 1; + elseif abs(A(r,r)) >= alpha*sigma + swap = 1; + m1 = k; m2 = r; + s = 1; + else + swap = 1; + m1 = k+1; m2 = r; + s = 2; + end + if swap + A( [m1, m2], : ) = A( [m2, m1], : ); + L( [m1, m2], : ) = L( [m2, m1], : ); + A( :, [m1, m2] ) = A( :, [m2, m1] ); + L( :, [m1, m2] ) = L( :, [m2, m1] ); + pp( [m1, m2] ) = pp( [m2, m1] ); + end + elseif piv == 'r' + j = k; + pivot = 0; + lambda_j = lambda; + while ~pivot + [temp, r] = max( abs(A(k:n,j)) ); + ncomp = ncomp + n-k; + r = r(1) + k - 1; + temp = A(k:n,r); temp(r-k+1) = 0; + lambda_r = max( abs(temp) ); + ncomp = ncomp + n-k; + if alpha*lambda_r <= abs(A(r,r)) + pivot = 1; + s = 1; + A( [k, r], : ) = A( [r, k], : ); + L( [k, r], : ) = L( [r, k], : ); + A( :, [k, r] ) = A( :, [r, k] ); + L( :, [k, r] ) = L( :, [r, k] ); + pp( [k, r] ) = pp( [r, k] ); + elseif lambda_j == lambda_r + pivot = 1; + s = 2; + A( [k, j], : ) = A( [j, k], : ); + L( [k, j], : ) = L( [j, k], : ); + A( :, [k, j] ) = A( :, [j, k] ); + L( :, [k, j] ) = L( :, [j, k] ); + pp( [k, j] ) = pp( [j, k] ); + k1 = k+1; + A( [k1, r], : ) = A( [r, k1], : ); + L( [k1, r], : ) = L( [r, k1], : ); + A( :, [k1, r] ) = A( :, [r, k1] ); + L( :, [k1, r] ) = L( :, [r, k1] ); + pp( [k1, r] ) = pp( [r, k1] ); + else + j = r; + lambda_j = lambda_r; + end + end + end + end + + if s == 1 + + D(k,k) = A(k,k); + A(k+1:n,k) = A(k+1:n,k)/A(k,k); + L(k+1:n,k) = A(k+1:n,k); + i = k+1:n; + A(i,i) = A(i,i) - A(i,k) * A(k,i); + A(i,i) = 0.5 * (A(i,i) + A(i,i)'); + + elseif s == 2 + + E = A(k:k+1,k:k+1); + D(k:k+1,k:k+1) = E; + C = A(k+2:n,k:k+1); + temp = C/E; + L(k+2:n,k:k+1) = temp; + A(k+2:n,k+2:n) = A(k+2:n,k+2:n) - temp*C'; + A(k+2:n,k+2:n) = 0.5 * (A(k+2:n,k+2:n) + A(k+2:n,k+2:n)'); + + end + + % Ensure diagonal real (see LINPACK User's Guide, p. 5.17). + for i=k+s:n + A(i,i) = real(A(i,i)); + end + + if nargout >= 4 & k+s <= n + rho = max(rho, max(max(abs(A(k+s:n,k+s:n)))) ); + end + + else % Nothing to do. + + s = 1; + D(k,k) = A(k,k); + + end + + k = k + s; + + if k == n + D(n,n) = A(n,n); + break + end + +end + +if nargout >= 3, P = eye(n); P = P(pp,:); end +if nargout >= 4, rho = rho/maxA; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/ldlt_sytr.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,52 @@ +function [L, D] = ldlt_sytr(A) +%LDLT_SYTR Block LDL^T factorization for a symmetric tridiagonal matrix. +% [L, D] = LDLT_SYTR(A) factorizes A = L*D*L', where A is +% Hermitian tridiagonal, L is unit lower triangular, and D is +% block diagonal with 1x1 and 2x2 diagonal blocks. It uses +% Bunch's strategy for choosing the pivots. + +% References: +% J. R. Bunch, Partial pivoting strategies for symmetric +% matrices. SIAM J. Numer. Anal., 11(3):521-528, 1974. +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 11.1.4. + +n = length(A); +if norm( tril(A,-2), 1) | norm( triu(A,2), 1) | ~isequal(A,A') + error('Matrix must be Hermitian tridiagonal.') +end + +s = norm(A(:), inf); +a = (sqrt(5)-1)/2; +L = eye(n); +D = zeros(n); +k = 1; + +while k < n + + if s*abs(A(k,k)) >= a*abs(A(k+1,k))^2 + + % 1-by-1 pivot. + D(k,k) = A(k,k); + L(k+1,k) = A(k+1,k)/A(k,k); + A(k+1,k+1) = A(k+1,k+1) - abs(A(k+1,k))^2/A(k,k); + k = k+1; + + else + + % 2-by-2 pivot. + E = A(k:k+1,k:k+1); + D(k:k+1,k:k+1) = E; + if k+2 <= n + L(k+2:n,k:k+1) = A(k+2:n,k:k+1)/E; + A(k+2,k+2) = A(k+2,k+2) - abs(A(k+2,k+1))^2*A(k,k)/det(E); + end + k = k+2; + + end + +end +if k == n + D(k,k) = A(k,k); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/lse.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,61 @@ +function x = lse(A, b, B, d, method) +%LSE Solve the equality constrained least squares problem. +% [x, flopcount] = LSE(A, b, B, d, METHOD) finds the least squares +% solution of the system Ax = b subject to the constraint Bx = d. +% METHOD = 1 (default), 2 or 3 specifies the variant of the null-space +% method to be used. METHOD 3 is the most efficient. + +% Note: Should really apply Householder transformations in factored form. +% Not done here. +% +% Reference: +% A. J. Cox and N. J. Higham. Accuracy and stability of the null space +% method for solving the equality constrained least squares problem. +% BIT, 39(1):34-50, 1999. +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 20.9. + +b = b(:); d = d(:); +[m n] = size(A); +m2 = length(b); +[p n2] = size(B); +p2 = length(d); + +if m ~= m2 | p ~= p2 | n ~= n2 + error('Dimensions do not match!') +end + +if nargin < 5, method = 1; end + +partial = (method == 2 | method == 3); +[U, Q, L, S] = gqr(A, B, partial); +y1 = S(:,1:p)\d; + +L21 = L(m-n+p+1:m,1:p); +L22 = L(m-n+p+1:m,p+1:n); + +if method == 1 + + c = U'*b; + c3 = c(m-n+p+1:m); + y2 = L22 \ (c3 - L21*y1); + +elseif method == 2 + + W1 = A*Q(:,1:p); + g = U'*(b - W1*y1); + g2 = g(m-n+p+1:m); + y2 = L22\g2; + +elseif method == 3 + + W1y1 = A*(Q(:,1:p)*y1); + g = U'*(b - W1y1); + g2 = g(m-n+p+1:m); + y2 = L22\g2; + +end + +y = [y1; y2]; +x = Q*y;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/makejcf.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,39 @@ +function A = makejcf(n, e, m, X) +%MAKEJCF A matrix with specified Jordan canonical form. +% MAKEJCF(N, E, M) is a matrix having the Jordan canonical form +% whose i'th Jordan block is of dimension M(i) with eigenvalue E(i), +% and where N = SUM(M). +% Defaults: E = 1:N, M = ONES(SIZE(E)) with M(1) so that SUM(M) = N. +% The matrix is constructed by applying a random similarity +% transformation to the Jordan form. +% Alternatively, the matrix used in the similarity transformation +% can be specified as a fifth parameter. +% In particular, MAKEJCF(N, E, M, EYE(N)) returns the Jordan form +% itself. +% NB: The JCF is very sensitive to rounding errors. + +if nargin < 2, e = 1:n; end +if nargin < 3, m = ones(size(e)); m(1) = m(1) + n - sum(m); end + +if length(e) ~= length(m) + error('Parameters E and M must be of same dimension.') +end + +if sum(m) ~= n, error('Block dimensions must add up to N.'), end + +A = zeros(n); +j = 1; +for i=1:max(size(m)) + if m(i) > 1 + Jb = gallery('jordbloc',m(i),e(i)); + else + Jb = e(i); % JORDBLOC fails in n = 1 case. + end + A(j:j+m(i)-1,j:j+m(i)-1) = Jb; + j = j + m(i); +end + +if nargin < 4 + X = randn(n); +end +A = X\A*X;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/matrix.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,85 @@ +function A = matrix(k, n) +%MATRIX Test matrices accessed by number. +% MATRIX(K, N) is the N-by-N instance of matrix number K in +% a set of test matrices comprising those in MATLAB plus those +% in the Matrix Computation Toolbox, +% with all other parameters set to their default. +% N.B. - Only those matrices which are full and take an arbitrary +% dimension N are included. +% - Some of these matrices are random. +% MATRIX(K) is a string containing the name of the K'th matrix. +% MATRIX(0) is the number of matrices, i.e. the upper limit for K. +% Thus to set A to each N-by-N test matrix in turn use a loop like +% for k=1:matrix(0) +% A = matrix(k, N); +% Aname = matrix(k); % The name of the matrix +% end +% MATRIX(-1) returns the version number and date of the +% Matrix Computation Toolbox. +% MATRIX with no arguments lists the names and numbers of the M-files in the +% collection. + +% References: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 20.5. + +% Matrices from gallery: +matrices = [% +'cauchy '; 'chebspec'; 'chebvand'; 'chow '; +'circul '; 'clement '; 'condex '; +'cycol '; 'dramadah'; 'fiedler '; +'forsythe'; 'frank '; 'gearmat '; 'grcar '; +'invhess '; 'invol '; 'ipjfact '; 'jordbloc'; +'kahan '; 'kms '; 'krylov '; 'lehmer '; +'lesp '; 'lotkin '; 'minij '; 'moler '; +'orthog '; 'parter '; 'pei '; 'prolate '; +'randcolu'; 'randcorr'; 'rando '; 'randsvd '; +'redheff '; 'riemann '; 'ris '; 'smoke '; +'toeppd '; 'triw ';]; +n_gall = length(matrices); + +% Other MATLAB matrices: +matrices = [matrices; +'hilb '; 'invhilb '; 'magic '; 'pascal '; +'rand '; 'randn ';]; +n_MATLAB = length(matrices); + +% Matrices from Matrix Computation Toolbox: +matrices = [matrices; +'augment '; 'gfpp '; 'magic '; 'makejcf '; +'rschur '; 'vand ']; +n_mats = length(matrices); + +if nargin == 0 + + rows = ceil(n_mats/5); + temp = zeros(rows,5); + temp(1:n_mats) = 1:n_mats; + + for i = 1:rows + for j = 1:5 + if temp(i,j) == 0, continue, end + fprintf(['%2.0f: ' sprintf('%s',matrices(temp(i,j),:)) ' '], ... + temp(i,j)) + end + fprintf('\n') + end + fprintf('Matrices 1 to %1.0f are from MATLAB\.', n_MATLAB) + +elseif nargin == 1 + if k == 0 + A = length(matrices); + elseif k > 0 + A = deblank(matrices(k,:)); + else + % Version number and date of collection. + A = 'Version 1.2, September 5 2002'; + end +else + if k <= n_gall + A = eval( ['gallery(''' deblank(matrices(k,:)) ''',n)'] ); + else + A = eval( [deblank(matrices(k,:)) '(n)'] ); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/matsignt.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,38 @@ +function S = matsignt(T) +%MATSIGNT Matrix sign function of a triangular matrix. +% S = MATSIGN(T) computes the matrix sign function S of the +% upper triangular matrix T using a recurrence. + +% Called by SIGNM. + +if ~isequal(T,triu(T)), error('Matrix must be upper triangular.'), end + +n = length(T); + +S = diag( sign( diag(real(T)) ) ); +for p = 1:n-1 + for i = 1:n-p + + j = i+p; + d = T(j,j) - T(i,i); + + if S(i,i) ~= -S(j,j) % Solve via S^2 = I if we can. + + % Get S(i,j) from S^2 = I. + k = i+1:j-1; + S(i,j) = -S(i,k)*S(k,j) / (S(i,i)+S(j,j)); + + else + + % Get S(i,j) from S*T = T*S. + s = T(i,j)*(S(j,j)-S(i,i)); + if p > 1 + k = i+1:j-1; + s = s + T(i,k)*S(k,j) - S(i,k)*T(k,j); + end + S(i,j) = s/d; + + end + + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/mctdemo.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,233 @@ +clc +format compact +echo on +%MCTDEMO Demonstration of Matrix Computation Toolbox. +% N. J. Higham. + +% The Matrix Computation Toolbox contains test matrices, matrix +% factorizations, visualization functions, direct search optimization +% functions, and other miscellaneous functions. + +% The version of the toolbox is + +matrix(-1) +echo on + +% For this demonstration you will need to view both the command window +% and one figure window. +% This demonstration emphasises graphics and shows only +% some of the features of the toolbox. + +pause % Press any key to continue after pauses. + +% A list of test matrices available in MATLAB and in the Toolbox (all full, +% square, and of arbitrary dimension) is obtained by typing `matrix': + +matrix + +pause + +% The FV command plots the boundary of the field of values of a matrix +% (the set of all Rayleigh quotients) and plots the eigenvalues as +% crosses (`x'). Here are some examples: + +% Here is the field of values of the 10-by-10 Grcar matrix: + +clf +fv(gallery('grcar',10)); +title('fv(gallery(''grcar'',10))') + +pause + +% Next, we form a random orthogonal matrix and look at its field of values. +% The boundary is the convex hull of the eigenvalues since A is normal. + +A = gallery('randsvd',10, 1); +fv(A); +title('fv(gallery(''randsvd'',10, 1))') +pause + +% The PS command plots an approximation to a pseudospectrum of A, +% which is the set of complex numbers that are eigenvalues of some +% perturbed matrix A + E, with the norm of E at most epsilon +% (default: epsilon = 1E-3). +% The eigenvalues of A are plotted as crosses (`x'). +% Here are some interesting PS plots. + +% First, we use the Kahan matrix, a triangular matrix made up of sines and +% cosines. Here is an approximate pseudospectrum of the 10-by-10 matrix: + +ps(gallery('kahan',10),25); +title('ps(gallery(''kahan'',10),25)') +pause + +% Next, a different way of looking at pseudospectra, via norms of +% the resolvent. (The resolvent of A is INV(z*I-A), where z is a complex +% variable). PSCONT gives a color map with a superimposed contour +% plot. Here we specify a region of the complex plane in +% which the 8-by-8 Kahan matrix is interesting to look at. + +pscont(gallery('kahan',8), 0, 20, [0.2 1.2 -0.5 0.5]); +title('pscont(gallery(''kahan'',8))') +pause + +% The triw matrix is upper triangular, made up of 1s and -1s: + +gallery('triw',4) + +% Here is a combined surface and contour plot of the resolvent for N = 11. +% Notice how the repeated eigenvalue 1 `sucks in' the resolvent. + +pscont(gallery('triw',11), 2, 15, [-2 2 -2 2]); +title('pscont(gallery(''triw'',11))') +pause + +% The next PSCONT plot is for the companion matrix of the characteristic +% polynomial of the CHEBSPEC matrix: + +A = gallery('chebspec',8); C = compan(poly(A)); + +% The SHOW command shows the +/- pattern of the elements of a matrix, with +% blanks for zero elements: + +show(C) + +pscont(C, 2, 20, [-.1 .1 -.1 .1]); +title('pscont(gallery(''chebspec'',8))') +pause + +% The following matrix has a pseudospectrum in the form of a limacon. + +n = 25; A = gallery('triw',n,1,2) - eye(n); +sub(A, 6) % Leading principal 6-by-6 submatrix of A. +ps(A); +pause + +% We can get a visual representation of a matrix using the SEE +% command, which produces subplots with the following layout: +% /---------------------------------\ +% | MESH(A) SEMILOGY(SVD(A)) | +% | PS(A) FV(A) | +% \---------------------------------/ +% where PS is the 1e-3 pseudospectrum and FV is the field of values. +% RSCHUR is an upper quasi-triangular matrix: + +see(rschur(16,18)); + +pause + +% Matlab's MAGIC function produces magic squares: + +A = magic(5) + +% Using the toolbox routine PNORM we can estimate the matrix p-norm +% for any value of p. + +[pnorm(A,1) pnorm(A,1.5) pnorm(A,2) pnorm(A,pi) pnorm(A,inf)] + +% As this example suggests, the p-norm of a magic square is +% constant for all p! + +pause + +% GERSH plots Gershgorin disks. Here are some interesting examples. +clf +gersh(gallery('lesp',12)); +title('gersh(gallery(''lesp'',12))') +pause + +gersh(gallery('hanowa',10)); +title('gersh(gallery(''hanowa'',10))') +pause + +gersh(gallery('ipjfact',6,1)); +title('gersh(gallery(''ipjfact'',6,1))') +pause + +gersh(gallery('smoke',16,1)); +title('gersh(gallery(''smoke'',16,1))') +pause + +% GFPP generates matrices for which Gaussian elimination with partial +% pivoting produces a large growth factor. + +gfpp(6) +pause + +% Let's find the growth factor RHO for partial pivoting and complete pivoting +% for a bigger matrix: + +A = gfpp(20); + +[L, U, P, Q, rho] = gep(A,'p'); % Partial pivoting using Toolbox function GEP. +[rho, 2^19] + +[L, U, P, Q, rho] = gep(A,'c'); % Complete pivoting using Toolbox function GEP. +rho +% As expected, complete pivoting does not produce large growth here. +pause + +% Function MATRIX allows test matrices in the Toolbox and MATLAB to be +% accessed by number. The following piece of code steps through all the +% non-Hermitian matrices of arbitrary dimension, setting A to each +% 10-by-10 matrix in turn. It evaluates the 2-norm condition number and the +% ratio of the largest to smallest eigenvalue (in absolute values). + +% c = []; e = []; j = 1; +% for i=1:matrix(0) +% % Double on next line avoids bug in MATLAB 6.5 re. i = 35. +% A = double(matrix(i, 10)); +% if ~isequal(A,A') % If not Hermitian... +% c1 = cond(A); +% eg = eig(A); +% e1 = max(abs(eg)) / min(abs(eg)); +% % Filter out extremely ill-conditioned matrices. +% if c1 <= 1e10, c(j) = c1; e(j) = e1; j = j + 1; end +% end +% end +echo off + +c = []; e = []; j = 1; +for i=1:matrix(0) + % Double on next line avoids bug in MATLAB 6.5 re. i = 35. + A = double(matrix(i, 10)); + if ~isequal(A,A') % If not Hermitian... + c1 = cond(A); + eg = eig(A); + e1 = max(abs(eg)) / min(abs(eg)); + % Filter out extremely ill-conditioned matrices. + if c1 <= 1e10, c(j) = c1; e(j) = e1; j = j + 1; end + end +end +echo on + +% The following plots confirm that the condition number can be much +% larger than the extremal eigenvalue ratio. +echo off +j = max(size(c)); +subplot(2,1,1) +semilogy(1:j, c, 'x', 1:j, e, 'o'), hold on +semilogy(1:j, c, '-', 1:j, e, '--'), hold off +title('cond: x, eig\_ratio: o') +subplot(2,1,2) +semilogy(1:j, c./e) +title('cond/eig\_ratio') +echo on +pause + +% Finally, here are three interesting pseudospectra based on pentadiagonal +% Toeplitz matrices: + +clf +ps(full(gallery('toeppen',32,0,1/2,0,0,1))); % Propeller +pause + +ps(inv(full(gallery('toeppen',32,0,1,1,0,.25)))); % Man in the moon +pause + +ps(full(gallery('toeppen',32,0,1/2,1,1,1))); % Fish +pause + +echo off +clear A L U P Q V D +format
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/mdsmax.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,170 @@ +function [x, fmax, nf] = mdsmax(fun, x, stopit, savit, varargin) +%MDSMAX Multidirectional search method for direct search optimization. +% [x, fmax, nf] = MDSMAX(FUN, x0, STOPIT, SAVIT) attempts to +% maximize the function FUN, using the starting vector x0. +% The method of multidirectional search is used. +% Output arguments: +% x = vector yielding largest function value found, +% fmax = function value at x, +% nf = number of function evaluations. +% The iteration is terminated when either +% - the relative size of the simplex is <= STOPIT(1) +% (default 1e-3), +% - STOPIT(2) function evaluations have been performed +% (default inf, i.e., no limit), or +% - a function value equals or exceeds STOPIT(3) +% (default inf, i.e., no test on function values). +% The form of the initial simplex is determined by STOPIT(4): +% STOPIT(4) = 0: regular simplex (sides of equal length, the default), +% STOPIT(4) = 1: right-angled simplex. +% Progress of the iteration is not shown if STOPIT(5) = 0 (default 1). +% If a non-empty fourth parameter string SAVIT is present, then +% `SAVE SAVIT x fmax nf' is executed after each inner iteration. +% NB: x0 can be a matrix. In the output argument, in SAVIT saves, +% and in function calls, x has the same shape as x0. +% MDSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional +% arguments to be passed to fun, via feval(fun,x,P1,P2,...). + +% This implementation uses 2n^2 elements of storage (two simplices), where x0 +% is an n-vector. It is based on the algorithm statement in [2, sec.3], +% modified so as to halve the storage (with a slight loss in readability). + +% References: +% [1] V. J. Torczon, Multi-directional search: A direct search algorithm for +% parallel machines, Ph.D. Thesis, Rice University, Houston, Texas, 1989. +% [2] V. J. Torczon, On the convergence of the multidirectional search +% algorithm, SIAM J. Optimization, 1 (1991), pp. 123-145. +% [3] N. J. Higham, Optimization by direct search in matrix computations, +% SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993. +% [4] N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 20.5. + +x0 = x(:); % Work with column vector internally. +n = length(x0); + +mu = 2; % Expansion factor. +theta = 0.5; % Contraction factor. + +% Set up convergence parameters etc. +if nargin < 3 | isempty(stopit), stopit(1) = 1e-3; end +tol = stopit(1); % Tolerance for cgce test based on relative size of simplex. +if length(stopit) == 1, stopit(2) = inf; end % Max no. of f-evaluations. +if length(stopit) == 2, stopit(3) = inf; end % Default target for f-values. +if length(stopit) == 3, stopit(4) = 0; end % Default initial simplex. +if length(stopit) == 4, stopit(5) = 1; end % Default: show progress. +trace = stopit(5); +if nargin < 4, savit = []; end % File name for snapshots. + +V = [zeros(n,1) eye(n)]; T = V; +f = zeros(n+1,1); ft = f; +V(:,1) = x0; f(1) = feval(fun,x,varargin{:}); +fmax_old = f(1); + +if trace, fprintf('f(x0) = %9.4e\n', f(1)), end + +k = 0; m = 0; + +% Set up initial simplex. +scale = max(norm(x0,inf),1); +if stopit(4) == 0 + % Regular simplex - all edges have same length. + % Generated from construction given in reference [18, pp. 80-81] of [1]. + alpha = scale / (n*sqrt(2)) * [ sqrt(n+1)-1+n sqrt(n+1)-1 ]; + V(:,2:n+1) = (x0 + alpha(2)*ones(n,1)) * ones(1,n); + for j=2:n+1 + V(j-1,j) = x0(j-1) + alpha(1); + x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); + end +else + % Right-angled simplex based on co-ordinate axes. + alpha = scale*ones(n+1,1); + for j=2:n+1 + V(:,j) = x0 + alpha(j)*V(:,j); + x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); + end +end +nf = n+1; +size = 0; % Integer that keeps track of expansions/contractions. +flag_break = 0; % Flag which becomes true when ready to quit outer loop. + +while 1 %%%%%% Outer loop. +k = k+1; + +% Find a new best vertex x and function value fmax = f(x). +[fmax,j] = max(f); +V(:,[1 j]) = V(:,[j 1]); v1 = V(:,1); +if ~isempty(savit), x(:) = v1; eval(['save ' savit ' x fmax nf']), end +f([1 j]) = f([j 1]); +if trace + fprintf('Iter. %2.0f, inner = %2.0f, size = %2.0f, ', k, m, size) + fprintf('nf = %3.0f, f = %9.4e (%2.1f%%)\n', nf, fmax, ... + 100*(fmax-fmax_old)/(abs(fmax_old)+eps)) +end +fmax_old = fmax; + +% Stopping Test 1 - f reached target value? +if fmax >= stopit(3) + msg = ['Exceeded target...quitting\n']; + break % Quit. +end + +m = 0; +while 1 %%% Inner repeat loop. + m = m+1; + + % Stopping Test 2 - too many f-evals? + if nf >= stopit(2) + msg = ['Max no. of function evaluations exceeded...quitting\n']; + flag_break = 1; break % Quit. + end + + % Stopping Test 3 - converged? This is test (4.3) in [1]. + size_simplex = norm(V(:,2:n+1)- v1(:,ones(1,n)),1) / max(1, norm(v1,1)); + if size_simplex <= tol + msg = sprintf('Simplex size %9.4e <= %9.4e...quitting\n', ... + size_simplex, tol); + flag_break = 1; break % Quit. + end + + for j=2:n+1 % ---Rotation (reflection) step. + T(:,j) = 2*v1 - V(:,j); + x(:) = T(:,j); ft(j) = feval(fun,x,varargin{:}); + end + nf = nf + n; + + replaced = ( max(ft(2:n+1)) > fmax ); + + if replaced + for j=2:n+1 % ---Expansion step. + V(:,j) = (1-mu)*v1 + mu*T(:,j); + x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); + end + nf = nf + n; + % Accept expansion or rotation? + if max(ft(2:n+1)) > max(f(2:n+1)) + V(:,2:n+1) = T(:,2:n+1); f(2:n+1) = ft(2:n+1); % Accept rotation. + else + size = size + 1; % Accept expansion (f and V already set). + end + else + for j=2:n+1 % ---Contraction step. + V(:,j) = (1+theta)*v1 - theta*T(:,j); + x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); + end + nf = nf + n; + replaced = ( max(f(2:n+1)) > fmax ); + % Accept contraction (f and V already set). + size = size - 1; + end + + if replaced, break, end + if trace & rem(m,10) == 0, fprintf(' ...inner = %2.0f...\n',m), end + end %%% Of inner repeat loop. + +if flag_break, break, end +end %%%%%% Of outer loop. + +% Finished. +if trace, fprintf(msg), end +x(:) = v1;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/nmsmax.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,173 @@ +function [x, fmax, nf] = nmsmax(fun, x, stopit, savit, varargin) +%NMSMAX Nelder-Mead simplex method for direct search optimization. +% [x, fmax, nf] = NMSMAX(FUN, x0, STOPIT, SAVIT) attempts to +% maximize the function FUN, using the starting vector x0. +% The Nelder-Mead direct search method is used. +% Output arguments: +% x = vector yielding largest function value found, +% fmax = function value at x, +% nf = number of function evaluations. +% The iteration is terminated when either +% - the relative size of the simplex is <= STOPIT(1) +% (default 1e-3), +% - STOPIT(2) function evaluations have been performed +% (default inf, i.e., no limit), or +% - a function value equals or exceeds STOPIT(3) +% (default inf, i.e., no test on function values). +% The form of the initial simplex is determined by STOPIT(4): +% STOPIT(4) = 0: regular simplex (sides of equal length, the default) +% STOPIT(4) = 1: right-angled simplex. +% Progress of the iteration is not shown if STOPIT(5) = 0 (default 1). +% If a non-empty fourth parameter string SAVIT is present, then +% `SAVE SAVIT x fmax nf' is executed after each inner iteration. +% NB: x0 can be a matrix. In the output argument, in SAVIT saves, +% and in function calls, x has the same shape as x0. +% NMSMAX(fun, x0, STOPIT, SAVIT, P1, P2,...) allows additional +% arguments to be passed to fun, via feval(fun,x,P1,P2,...). + +% References: +% N. J. Higham, Optimization by direct search in matrix computations, +% SIAM J. Matrix Anal. Appl, 14(2): 317-333, 1993. +% C. T. Kelley, Iterative Methods for Optimization, Society for Industrial +% and Applied Mathematics, Philadelphia, PA, 1999. + +x0 = x(:); % Work with column vector internally. +n = length(x0); + +% Set up convergence parameters etc. +if nargin < 3 | isempty(stopit), stopit(1) = 1e-3; end +tol = stopit(1); % Tolerance for cgce test based on relative size of simplex. +if length(stopit) == 1, stopit(2) = inf; end % Max no. of f-evaluations. +if length(stopit) == 2, stopit(3) = inf; end % Default target for f-values. +if length(stopit) == 3, stopit(4) = 0; end % Default initial simplex. +if length(stopit) == 4, stopit(5) = 1; end % Default: show progress. +trace = stopit(5); +if nargin < 4, savit = []; end % File name for snapshots. + +V = [zeros(n,1) eye(n)]; +f = zeros(n+1,1); +V(:,1) = x0; f(1) = feval(fun,x,varargin{:}); +fmax_old = f(1); + +if trace, fprintf('f(x0) = %9.4e\n', f(1)), end + +k = 0; m = 0; + +% Set up initial simplex. +scale = max(norm(x0,inf),1); +if stopit(4) == 0 + % Regular simplex - all edges have same length. + % Generated from construction given in reference [18, pp. 80-81] of [1]. + alpha = scale / (n*sqrt(2)) * [ sqrt(n+1)-1+n sqrt(n+1)-1 ]; + V(:,2:n+1) = (x0 + alpha(2)*ones(n,1)) * ones(1,n); + for j=2:n+1 + V(j-1,j) = x0(j-1) + alpha(1); + x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); + end +else + % Right-angled simplex based on co-ordinate axes. + alpha = scale*ones(n+1,1); + for j=2:n+1 + V(:,j) = x0 + alpha(j)*V(:,j); + x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); + end +end +nf = n+1; +how = 'initial '; + +[temp,j] = sort(f); +j = j(n+1:-1:1); +f = f(j); V = V(:,j); + +alpha = 1; beta = 1/2; gamma = 2; + +while 1 %%%%%% Outer (and only) loop. +k = k+1; + + fmax = f(1); + if fmax > fmax_old + if ~isempty(savit) + x(:) = V(:,1); eval(['save ' savit ' x fmax nf']) + end + if trace + fprintf('Iter. %2.0f,', k) + fprintf([' how = ' how ' ']); + fprintf('nf = %3.0f, f = %9.4e (%2.1f%%)\n', nf, fmax, ... + 100*(fmax-fmax_old)/(abs(fmax_old)+eps)) + end + end + fmax_old = fmax; + + %%% Three stopping tests from MDSMAX.M + + % Stopping Test 1 - f reached target value? + if fmax >= stopit(3) + msg = ['Exceeded target...quitting\n']; + break % Quit. + end + + % Stopping Test 2 - too many f-evals? + if nf >= stopit(2) + msg = ['Max no. of function evaluations exceeded...quitting\n']; + break % Quit. + end + + % Stopping Test 3 - converged? This is test (4.3) in [1]. + v1 = V(:,1); + size_simplex = norm(V(:,2:n+1)-v1(:,ones(1,n)),1) / max(1, norm(v1,1)); + if size_simplex <= tol + msg = sprintf('Simplex size %9.4e <= %9.4e...quitting\n', ... + size_simplex, tol); + break % Quit. + end + + % One step of the Nelder-Mead simplex algorithm + % NJH: Altered function calls and changed CNT to NF. + % Changed each `fr < f(1)' type test to `>' for maximization + % and re-ordered function values after sort. + + vbar = (sum(V(:,1:n)')/n)'; % Mean value + vr = (1 + alpha)*vbar - alpha*V(:,n+1); x(:) = vr; fr = feval(fun,x,varargin{:}); + nf = nf + 1; + vk = vr; fk = fr; how = 'reflect, '; + if fr > f(n) + if fr > f(1) + ve = gamma*vr + (1-gamma)*vbar; x(:) = ve; fe = feval(fun,x,varargin{:}); + nf = nf + 1; + if fe > f(1) + vk = ve; fk = fe; + how = 'expand, '; + end + end + else + vt = V(:,n+1); ft = f(n+1); + if fr > ft + vt = vr; ft = fr; + end + vc = beta*vt + (1-beta)*vbar; x(:) = vc; fc = feval(fun,x,varargin{:}); + nf = nf + 1; + if fc > f(n) + vk = vc; fk = fc; + how = 'contract,'; + else + for j = 2:n + V(:,j) = (V(:,1) + V(:,j))/2; + x(:) = V(:,j); f(j) = feval(fun,x,varargin{:}); + end + nf = nf + n-1; + vk = (V(:,1) + V(:,n+1))/2; x(:) = vk; fk = feval(fun,x,varargin{:}); + nf = nf + 1; + how = 'shrink, '; + end + end + V(:,n+1) = vk; + f(n+1) = fk; + [temp,j] = sort(f); + j = j(n+1:-1:1); + f = f(j); V = V(:,j); + +end %%%%%% End of outer (and only) loop. + +% Finished. +if trace, fprintf(msg), end +x(:) = V(:,1);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/pnorm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,105 @@ +function [est, x, k] = pnorm(A, p, tol, prnt) +%PNORM Estimate of matrix p-norm (1 <= p <= inf). +% [EST, x, k] = PNORM(A, p, TOL) estimates the Holder p-norm of a +% matrix A, using the p-norm power method with a specially +% chosen starting vector. +% TOL is a relative convergence tolerance (default 1E-4). +% Returned are the norm estimate EST (which is a lower bound for the +% exact p-norm), the corresponding approximate maximizing vector x, +% and the number of power method iterations k. +% A nonzero fourth input argument causes trace output to the screen. +% If A is a vector, this routine simply returns NORM(A, p). +% +% See also NORM, NORMEST, NORMEST1. + +% Note: The estimate is exact for p = 1, but is not always exact for +% p = 2 or p = inf. Code could be added to treat p = 2 and p = inf +% separately. +% +% Calls DUAL. +% +% Reference: +% N. J. Higham, Estimating the matrix p-norm, Numer. Math., +% 62 (1992), pp. 539-555. +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; sec. 15.2. + +if nargin < 2, error('Must specify norm via second parameter.'), end +[m,n] = size(A); +if min(m,n) == 1, est = norm(A,p); return, end + +if nargin < 4, prnt = 0; end +if nargin < 3 | isempty(tol), tol = 1e-4; end + +% Stage I. Use Algorithm OSE to get starting vector x for power method. +% Form y = B*x, at each stage choosing x(k) = c and scaling previous +% x(k+1:n) by s, where norm([c s],p)=1. + +sm = 9; % Number of samples. +y = zeros(m,1); x = zeros(n,1); + +for k=1:n + + if k == 1 + c = 1; s = 0; + else + W = [A(:,k) y]; + + if p == 2 % Special case. Solve exactly for 2-norm. + [U,S,V] = svd(full(W)); + c = V(1,1); s = V(2,1); + + else + + fopt = 0; + for th=linspace(0,pi,sm) + c1 = cos(th); s1 = sin(th); + nrm = norm([c1 s1],p); + c1 = c1/nrm; s1 = s1/nrm; % [c1 s1] has unit p-norm. + f = norm( W*[c1 s1]', p ); + if f > fopt + fopt = f; + c = c1; s = s1; + end + end + + end + end + + x(k) = c; + y = x(k)*A(:,k) + s*y; + if k > 1, x(1:k-1) = s*x(1:k-1); end + +end + +est = norm(y,p); +if prnt, fprintf('Alg OSE: %9.4e\n', est), end + +% Stage II. Apply Algorithm PM (the power method). + +q = dual(p); +k = 1; + +while 1 + + y = A*x; + est_old = est; + est = norm(y,p); + + z = A' * dual(y,p); + + if prnt + fprintf('%2.0f: norm(y) = %9.4e, norm(z) = %9.4e', ... + k, norm(y,p), norm(z,q)) + fprintf(' rel_incr(est) = %9.4e\n', (est-est_old)/est) + end + + if ( norm(z,q) <= z'*x | abs(est-est_old)/est <= tol ) & k > 1 + return + end + + x = dual(z,q); + k = k + 1; + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/poldec.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,28 @@ +function [U, H] = poldec(A) +%POLDEC Polar decomposition. +% [U, H] = POLDEC(A) computes a matrix U of the same dimension +% (m-by-n) as A, and a Hermitian positive semi-definite matrix H, +% such that A = U*H. +% U has orthonormal columns if m >= n, and orthonormal rows if m <= n. +% U and H are computed via an SVD of A. +% U is a nearest unitary matrix to A in both the 2-norm and the +% Frobenius norm. + +% Reference: +% N. J. Higham, Computing the polar decomposition---with applications, +% SIAM J. Sci. Stat. Comput., 7(4):1160--1174, 1986. +% +% (The name `polar' is reserved for a graphics routine.) + +[m, n] = size(A); + +[P, S, Q] = svd(A, 0); % Economy size. +if m < n % Ditto for the m<n case. + S = S(:, 1:m); + Q = Q(:, 1:m); +end +U = P*Q'; +if nargout == 2 + H = Q*S*Q'; + H = (H + H')/2; % Force Hermitian by taking nearest Hermitian matrix. +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/ps.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,90 @@ +function y = ps(A, m, tol, rl, marksize) +%PS Dot plot of a pseudospectrum. +% PS(A, M, TOL, RL) plots an approximation to a pseudospectrum +% of the square matrix A, using M random perturbations of size TOL. +% M defaults to a SIZE(A)-dependent value and TOL to 1E-3. +% RL defines the type of perturbation: +% RL = 0 (default): absolute complex perturbations of 2-norm TOL. +% RL = 1: absolute real perturbations of 2-norm TOL. +% RL = -1: componentwise real perturbations of size TOL. +% The eigenvalues of A are plotted as crosses `x'. +% PS(A, M, TOL, RL, MARKSIZE) uses the specified marker size instead +% of a size that depends on the figure size, the matrix order, and M. +% If MARKSIZE < 0, the plot is suppressed and the plot data is returned +% as an output argument. +% PS(A, 0) plots just the eigenvalues of A. + +% For a given TOL, the pseudospectrum of A is the set of +% pseudo-eigenvalues of A, that is, the set +% { e : e is an eigenvalue of A+E, for some E with NORM(E) <= TOL }. +% +% References: +% L. N. Trefethen, Computation of pseudospectra, Acta Numerica, +% 8:247-295, 1999. +% L. N. Trefethen, Spectra and pseudospectra, in The Graduate +% Student's Guide to Numerical Analysis '98, M. Ainsworth, +% J. Levesley, and M. Marletta, eds., Springer-Verlag, Berlin, +% 1999, pp. 217-250. + +if diff(size(A)), error('Matrix must be square.'), end +n = length(A); + +if nargin < 5, marksize = 0; end +if nargin < 4, rl = 0; end +if nargin < 3, tol = 1e-3; end +if nargin < 2 | isempty(m), m = 5*max(1, round( 25*exp(-0.047*n) )); end + +if m == 0 + e = eig(A); + ax = cpltaxes(e); + plot(real(e), imag(e), 'x') + axis(ax), axis('square') + return +end + +x = zeros(m*n,1); +i = sqrt(-1); + +for j = 1:m + if rl == -1 % Componentwise. + dA = -ones(n) + 2*rand(n); % Uniform random numbers on [-1,1]. + dA = tol * A .* dA; + else + if rl == 0 % Complex absolute. + dA = randn(n) + i*randn(n); + else % Real absolute. + dA = randn(n); + end + dA = tol/norm(dA)*dA; + end + x((j-1)*n+1:j*n) = eig(A + dA); +end + +if marksize >= 0 + + ax = cpltaxes(x); + h = plot(real(x),imag(x),'.'); + axis(ax), axis('square') + + % Next block adapted from SPY.M. + if marksize == 0 + units = get(gca,'units'); + set(gca,'units','points'); + pos = get(gca,'position'); + nps = 2.4*sqrt(n*m); % Factor based on number of pseudo-ei'vals plotted. + myguess = round(3*min(pos(3:4))/nps); + marksize = max(1,myguess); + set(gca,'units',units); + end + + hold on + e = eig(A); + plot(real(e),imag(e),'x'); + set(h,'markersize',marksize); + hold off + +else + + y = x; + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/pscont.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,104 @@ +function [x, y, z, m] = pscont(A, k, npts, ax, levels) +%PSCONT Contours and colour pictures of pseudospectra. +% PSCONT(A, K, NPTS, AX, LEVELS) plots LOG10(1/NORM(R(z))), +% where R(z) = INV(z*I-A) is the resolvent of the square matrix A, +% over an NPTS-by-NPTS grid. +% NPTS defaults to a SIZE(A)-dependent value. +% The limits are AX(1) and AX(2) on the x-axis and +% AX(3) and AX(4) on the y-axis. +% If AX is omitted, suitable limits are guessed based on the +% eigenvalues of A. +% The eigenvalues of A are plotted as crosses `x'. +% K determines the type of plot: +% K = 0 (default) PCOLOR and CONTOUR +% K = 1 PCOLOR only +% K = 2 SURFC (SURF and CONTOUR) +% K = 3 SURF only +% K = 4 CONTOUR only +% The contours levels are specified by the vector LEVELS, which +% defaults to -10:-1 (recall we are plotting log10 of the data). +% Thus, by default, the contour lines trace out the boundaries of +% the epsilon pseudospectra for epsilon = 1e-10, ..., 1e-1. +% [X, Y, Z, NPTS] = PSCONT(A, ...) returns the plot data X, Y, Z +% and the value of NPTS used. +% +% After calling this function you may want to change the +% color map (e.g., type COLORMAP HOT - see HELP COLOR) and the +% shading (e.g., type SHADING INTERP - see HELP INTERP). +% For an explanation of the term `pseudospectra', and references, +% see PS.M. +% When A is real and the grid is symmetric about the x-axis, this +% routine exploits symmetry to halve the computational work. + +% Colour pseduospectral pictures of this type are referred to as +% `spectral portraits' by Godunov, Kostin, and colleagues. +% References: see PS. + +if diff(size(A)), error('Matrix must be square.'), end +n = length(A); +Areal = ~any(imag(A)); + +if nargin < 5, levels = -10:-1; end +e = eig(A); +if nargin < 4 | isempty(ax) + ax = cpltaxes(e); + if Areal, ax(3) = -ax(4); end % Make sure region symmetric about x-axis. +end +if nargin < 3 | isempty(npts) + npts = 3*round( min(max(5, sqrt(20^2*10^3/n^3) ), 30)); +end +if nargin < 2 | isempty(k), k = 0; end + +nptsx = npts; nptsy = npts; +Ysymmetry = (Areal & ax(3) == -ax(4)); + +x = linspace(ax(1), ax(2), npts); +y = linspace(ax(3), ax(4), npts); +if Ysymmetry % Exploit symmetry about x-axis. + nptsy = ceil(npts/2); + y1 = y; + y = y(1:nptsy); +end + +[xx, yy] = meshgrid(x,y); +z = xx + sqrt(-1)*yy; +I = eye(n); +Smin = zeros(nptsy, nptsx); + +for j=1:nptsx + for i=1:nptsy + Smin(i,j) = min( svd( z(i,j)*I-A ) ); + end +end + +z = log10( Smin + eps ); +if Ysymmetry + z = [z; z(nptsy-rem(npts,2):-1:1,:)]; + y = y1; +end + +if k == 0 | k == 1 + pcolor(x, y, z); hold on +elseif k == 2 + surfc(x, y, z); hold on +elseif k == 3 + surf(x, y, z); hold on +end + +if k == 0 + contour(x, y, z, levels,'-k'); hold on +elseif k == 4 + contour(x, y, z, levels); hold on +end + +if k ~= 2 & k ~= 3 + if k == 0 | k == 1 + s = 'w'; % White. + else + s = 'k'; % Black. + end + plot(real(e),imag(e),['x' s]); +end + +axis('square') +hold off
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/readme.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,16 @@ +echo on +% Welcome to the Matrix Computation Toolbox. +% The primary source for this toolbox is +% +% http://www.ma.man.ac.uk/~higham/mctoolbox +% +% The toolbox comprises the M-files in this directory and the accompanying +% document mctoolbox.pdf: +% +% The Matrix Computation Toolbox for MATLAB (version 1.0). +% Numerical Analysis Report No. 410, Manchester Centre for +% Computational Mathematics, Manchester, England, August 2002. +% +% For a descriptive list of M-files in the toolbox type +% help contents +echo off
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/rootm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,93 @@ +function [X, arg2] = rootm(A,p) +%ROOTM Pth root of a matrix. +% X = ROOTM(A,P) computes a Pth root X of a square matrix A. +% This function computes the Schur decomposition A = Q*T*Q' and then +% finds a Pth root U of T by a recursive formula, giving X = Q*U*Q'. +% +% X is the unique pth root for which every eigenvalue has nonnegative +% real part. If A has any eigenvalues with negative real parts then a +% complex result is produced. If A is singular then A may not have a +% pth root. A warning is printed if exact singularity is detected. +% +% With two output arguments, [X, RESNORM] = ROOTM(A) does not print any +% warning, and returns the residual, norm(A-X^2,'fro')/norm(A,'fro'). + +% Reference: +% M. I. Smith, A Schur Algorithm for Computing Matrix pth Roots, +% Numerical Analysis Report No. 392, Manchester Centre for +% Computational Mathematics, Manchester, UK, 2001; to appear in +% SIAM J. Matrix Anal. Appl. + +% Original function by Matthew Smith. + +n = length(A); + +[Q,T] = schur(A,'complex'); + +U = zeros(n); +R = zeros(n,(p-2)*n); + +for i = 1:n + U(i,i) = T(i,i)^(1/p); + for a = 1:p-2 + R(i,(a-1)*n+i) = U(i,i)^(a+1); + end +end + +warns = warning; +warning('off'); + +for c = 1:n-1 + for i = 1:n-c + sum1 = 0; + for d = 1:p-2 + sum2 = 0; + for k = i+1:i+c-1 + sum2 = sum2 + U(i,k)*R(k,(d-1)*n + i+c); + end + sum1 = sum1 + U(i,i)^(p-2-d)*sum2; + end + sum3 = 0; + for k = i+1:i+c-1 + sum3 = sum3 + U(i,k)*U(k,i+c); + end + sum1 = sum1 + U(i,i)^(p-2)*sum3; + sum4 = 0; + for j = 0:p-1 + sum4 = sum4 + U(i,i)^j*U(i+c,i+c)^(p-1-j); + end + + U(i,i+c) = (T(i,i+c) - sum1)/(sum4); + + for q = 1:p-2 + sum5 = 0; + for g = 0:q + sum5 = sum5 + U(i,i )^g*U(i+c,i+c)^(q-g); + end + sum6 = 0; + for h = 1:q-1 + sum7 = 0; + for w=i+1:i+c-1 + sum7 = sum7 + U(i,w)*R(w,(h-1)*n +i+c); + end + sum6 = sum6 + U(i,i)^(q-1-h)*sum7; + end + sum = sum6 + U(i,i)^(q-1)*sum3; + + R(i,(q-1)*n +i+c) = U(i,i+c)*sum5 + sum; + end + end +end + +X = Q*U*Q'; +warning(warns); + +nzeig = any(diag(T)==0); + +if nzeig & (nargout ~= 2) + warning('Matrix is singular and may not have a square root.') +end + +if nargout == 2 + arg2 = norm(X^p-A,'fro')/norm(A,'fro'); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/rschur.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,33 @@ +function A = rschur(n, mu, x, y) +%RSCHUR An upper quasi-triangular matrix. +% A = RSCHUR(N, MU, X, Y) is an N-by-N matrix in real Schur form. +% All the diagonal blocks are 2-by-2 (except for the last one, if N +% is odd) and the k'th has the form [x(k) y(k); -y(k) x(k)]. +% Thus the eigenvalues of A are x(k) +/- i*y(k). +% MU (default 1) controls the departure from normality. +% Defaults: X(k) = -k^2/10, Y(k) = -k, i.e., the eigenvalues +% lie on the parabola x = -y^2/10. + +% References: +% F. Chatelin, Eigenvalues of Matrices, John Wiley, Chichester, 1993; +% Section 4.2.7. +% F. Chatelin and V. Fraysse, Qualitative computing: Elements +% of a theory for finite precision computation, Lecture notes, +% CERFACS, Toulouse, France and THOMSON-CSF, Orsay, France, +% June 1993. + +m = floor(n/2)+1; +alpha = 10; beta = 1; + +if nargin < 4, y = -(1:m)/beta; end +if nargin < 3, x = -(1:m).^2/alpha; end +if nargin < 2, mu = 1; end + +A = diag( mu*ones(n-1,1), 1 ); +for i=1:2:2*(m-1) + j = (i+1)/2; + A(i:i+1,i:i+1) = [x(j) y(j); -y(j) x(j)]; +end +if 2*m ~= n, + A(n,n) = x(m); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/see.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,55 @@ +function see(A, k) +%SEE Pictures of a matrix. +% SEE(A) displays MESH(A), SEMILOGY(SVD(A),'o'), +% and (if A is square) PS(A) and FV(A) in four subplot windows. +% SEE(A, 1) plots MESH(PINV(A)) in the +% third window instead of the 1e-3-pseudospectrum. +% SEE(A, -1) plots only the eigenvalues in the third/fourth window, +% which is much quicker than PS or FV. +% If A is complex, only real parts are used for the mesh plots. +% If A is sparse, just SPY(A) is shown. + +if nargin < 2, k = 0; end +[m, n] = size(A); +square = (m == n); +clf + +if issparse(A) + + spy(A); + +else + + B = pinv(A); + s = svd(A); + zs = (s == zeros(size(s))); + if any( zs ) + s( zs ) = []; % Remove zero singular values for semilogy plot. + end + + subplot(2,2,1) + mesh(real(A)), axis('ij'), drawnow + subplot(2,2,2) + semilogy(s, 'o') + hold on, semilogy(s, '-'), hold off, drawnow + if any(zs), title('Zero(s) omitted'), end + + subplot(2,2,3) + if ~square + axis off + text(0,0,'Matrix not square.') + return + end + + if k == -1 + ps(A, 0); + elseif k == 1 + mesh(real(B)), axis('ij'), drawnow + else + ps(A); + end + + subplot(2,2,4) + fv(A); + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/seqcheb.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,17 @@ +function x = seqcheb(n, k) +%SEQCHEB Sequence of points related to Chebyshev polynomials. +% X = SEQCHEB(N, K) produces a row vector of length N. +% There are two choices: +% K = 1: zeros of T_N, (the default) +% K = 2: extrema of T_{N-1}, +% where T_k is the Chebsyhev polynomial of degree k. + +if nargin == 1, k = 1; end + +if k == 1 % Zeros of T_n + i = 1:n; j = .5*ones(1,n); + x = cos( (i-j) * (pi/n) ); +elseif k == 2 % Extrema of T_(n-1) + i = 0:n-1; + x = cos( i * (pi/(n-1)) ); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/seqm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,17 @@ +function y = seqm(a, b, n) +%SEQM Multiplicative sequence. +% Y = SEQM(A, B, N) produces a row vector comprising N +% logarithmically equally spaced numbers, starting at A ~= 0 +% and finishing at B ~= 0. +% If A*B < 0 and N > 2 then complex results are produced. +% If N is omitted then 10 points are generated. + +if nargin == 2, n = 10; end + +if n <= 1 + y = a; + return +end +p = [0:n-2]/(n-1); +r = (b/a).^p; +y = [a*r, b];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/show.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,9 @@ +function show(x) +%SHOW Display signs of matrix elements. +% SHOW(X) displays X in `FORMAT +' form, that is, +% with `+', `-' and blank representing positive, negative +% and zero elements respectively. + +format + +disp(x) +format
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/signm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,20 @@ +function [S, N] = signm(A) +%SIGNM Matrix sign decomposition. +% [S, N] = SIGNM(A) is the matrix sign decomposition A = S*N, +% computed via the Schur decomposition. +% S is the matrix sign function, sign(A). + +% Reference: +% N. J. Higham, The matrix sign decomposition and its relation to the +% polar decomposition, Linear Algebra and Appl., 212/213:3-20, 1994. + +[Q, T] = schur(A,'complex'); +S = Q * matsignt(T) * Q'; + +% Only problem with Schur method is possible nonzero imaginary part when +% A is real. Next line takes care of that. +if ~any(imag(A)), S = real(S); end + +if nargout == 2 + N = S*A; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/skewpart.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,8 @@ +function S = skewpart(A) +%SKEWPART Skew-symmetric (skew-Hermitian) part. +% SKEWPART(A) is the skew-symmetric (skew-Hermitian) part of A, +% (A - A')/2. +% It is the nearest skew-symmetric (skew-Hermitian) matrix to A in +% both the 2- and the Frobenius norms. + +S = (A - A')./2;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/sparsify.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,20 @@ +function A = sparsify(A, p) +%SPARSIFY Randomly set matrix elements to zero. +% S = SPARSIFY(A, P) is A with elements randomly set to zero +% (S = S' if A is square and A = A', i.e. symmetry is preserved). +% Each element has probability P of being zeroed. +% Thus on average 100*P percent of the elements of A will be zeroed. +% Default: P = 0.25. + +if nargin < 2, p = 0.25; end +if p<0 | p>1, error('Second parameter must be between 0 and 1 inclusive.'), end + +[m,n] = size(A); + +if ~isequal(A,A') + A = A .* (rand(m,n) > p); % Unsymmetric case +else + A = triu(A,1) .* (rand(m,n) > p); % Preserve symmetry + A = A + A'; + A = A + diag( diag(A) .* (rand(size(n,1)) > p) ); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/strassen.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,33 @@ +function C = strassen(A, B, nmin) +%STRASSEN Strassen's fast matrix multiplication algorithm. +% C = STRASSEN(A, B, NMIN), where A and B are matrices of dimension +% a power of 2, computes the product C = A*B. +% Strassen's algorithm is used recursively until dimension <= NMIN +% is reached, at which point standard multiplication is used. +% The default is NMIN = 8 (which minimizes the total number of +% operations). + +% Reference: +% V. Strassen, Gaussian elimination is not optimal, +% Numer. Math., 13 (1969), pp. 354-356. + +if nargin < 3, nmin = 8; end + +n = length(A); +if n ~= 2^( log2(n) ) + error('The matrix dimension must be a power of 2.') +end + +if n <= nmin + C = A*B; +else + m = n/2; i = 1:m; j = m+1:n; + P1 = strassen( A(i,i)+A(j,j), B(i,i)+B(j,j), nmin); + P2 = strassen( A(j,i)+A(j,j), B(i,i), nmin); + P3 = strassen( A(i,i), B(i,j)-B(j,j), nmin); + P4 = strassen( A(j,j), B(j,i)-B(i,i), nmin); + P5 = strassen( A(i,i)+A(i,j), B(j,j), nmin); + P6 = strassen( A(j,i)-A(i,i), B(i,i)+B(i,j), nmin); + P7 = strassen( A(i,j)-A(j,j), B(j,i)+B(j,j), nmin); + C = [ P1+P4-P5+P7 P3+P5; P2+P4 P1+P3-P2+P6 ]; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/strassenw.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,50 @@ +function C = strassenw(A, B, nmin) +%STRASSENW Strassen's fast matrix multiplication algorithm (Winograd variant). +% C = STRASSENW(A, B, NMIN), where A and B are matrices of dimension +% a power of 2, computes the product C = A*B. +% Winograd's variant of Strassen's algorithm is +% used recursively until dimension <= NMIN is reached, +% at which point standard multiplication is used. +% The default is NMIN = 8 (which minimizes the total number of +% operations). + +% Reference: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; chap. 23. + +if nargin < 3, nmin = 8; end + +n = length(A); +if n ~= 2^( log2(n) ) + error('The matrix dimension must be a power of 2.') +end + +if n <= nmin + C = A*B; +else + m = n/2; i = 1:m; j = m+1:n; + + S1 = A(j,i) + A(j,j); + S2 = S1 - A(i,i); + S3 = A(i,i) - A(j,i); + S4 = A(i,j) - S2; + S5 = B(i,j) - B(i,i); + S6 = B(j,j) - S5; + S7 = B(j,j) - B(i,j); + S8 = S6 - B(j,i); + + M1 = strassenw( S2, S6, nmin); + M2 = strassenw( A(i,i), B(i,i), nmin); + M3 = strassenw( A(i,j), B(j,i), nmin); + M4 = strassenw( S3, S7, nmin); + M5 = strassenw( S1, S5, nmin); + M6 = strassenw( S4, B(j,j), nmin); + M7 = strassenw( A(j,j), S8, nmin); + + T1 = M1 + M2; + T2 = T1 + M4; + + C = [ M2+M3 T1+M5+M6; T2-M7 T2+M5 ]; + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/sub.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,17 @@ +function S = sub(A, i, j) +%SUB Principal submatrix. +% SUB(A,i,j) is A(i:j,i:j). +% SUB(A,i) is the leading principal submatrix of order i, +% A(1:i,1:i), if i>0, and the trailing principal submatrix +% of order ABS(i) if i<0. + +if nargin == 2 + if i >= 0 + S = A(1:i, 1:i); + else + n = min(size(A)); + S = A(n+i+1:n, n+i+1:n); + end +else + S = A(i:j, i:j); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/symmpart.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,7 @@ +function S = symmpart(A) +%SYMMPART Symmetric (Hermitian) part. +% SYMMPART(A) is the symmetric (Hermitian) part of A, (A + A')/2. +% It is the nearest symmetric (Hermitian) matrix to A in both the +% 2- and the Frobenius norms. + +S = (A + A')./2;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/trap2tri.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,54 @@ +function [Q, T] = trap2tri(L) +%TRAP2TRI Unitary reduction of trapezoidal matrix to triangular form. +% [Q, T] = TRAP2TRI(L), where L is an m-by-n lower trapezoidal +% matrix with m >= n, produces a unitary Q such that Q*L = [T; 0], +% where T is n-by-n and lower triangular. +% Q is a product of Householder transformations. + +% Called by COD. +% +% Reference: +% G. H. Golub and C. F. Van Loan, Matrix Computations, third +% edition, Johns Hopkins University Press, Baltimore, Maryland, +% 1996; P5.2.5. + +[n, r] = size(L); + +if r > n | ~isequal(L,tril(L)) + error('Matrix must be lower trapezoidal and m-by-n with m >= n.') +end + +Q = eye(n); % To hold product of Householder transformations. + +if r ~= n + + % Reduce nxr L = r [L1] to lower triangular form: QL = [T]. + % n-r [L2] [0] + + for j=r:-1:1 + % x is the vector to be reduced, which we overwrite with the H.T. vector. + x = L(j:n,j); + x(2:r-j+1) = zeros(r-j,1); % These elts of column left unchanged. + [v,beta,s] = gallery('house',x); + + % Nothing to do if x is zero (or x=a*e_1, but we don't check for that). + if s ~= 0 + + % Implicitly apply H.T. to pivot column. + % L(r+1:n,j) = zeros(n-r,1); % We throw these elts away at the end. + L(j,j) = s; + + % Apply H.T. to rest of matrix. + if j > 1 + y = v'*L(j:n, 1:j-1); + L(j:n, 1:j-1) = L(j:n, 1:j-1) - beta*v*y; + end + + % Update H.T. product. + y = v'*Q(j:n,:); + Q(j:n,:) = Q(j:n,:) - beta*v*y; + end + end +end + +T = L(1:r,:); % Rows r+1:n have been zeroed out.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/treshape.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,51 @@ +function T = treshape(x,unit) +%TRESHAPE Reshape vector to or from (unit) triangular matrix. +% TRESHAPE(X) returns a square upper triangular matrix whose +% elements are taken columnwise from the matrix X. +% TRESHAPE(X,1) returns a UNIT upper triangular matrix, and +% the 1s should not be specified in X. +% An error results if X does not have a number of elements of the form +% N*(N+1)/2 (or N less than this in the unit triangular case). +% X = TRESHAPE(R,2) is the inverse operation to R = TRESHAPE(X). +% X = TRESHAPE(R,3) is the inverse operation to R = TRESHAPE(X,1). + +if nargin == 1, unit = 0; end + +[p,q] = size(x); + +if unit < 2 % Convert vector x to upper triangular R. + + m = p*q; + n = round( (-1 + sqrt(1+8*m))/2 ); + if n*(n+1)/2 ~= m + error('Matrix must have a ''triangular'' number of elements.') + end + + if unit == 1 + n = n+1; + end + + x = x(:); + T = unit*eye(n); + + i = 1; + for j = 1+unit:n + T(1:j-unit,j) = x(i:i+j-1-unit); + i = i+j-unit; + end + +elseif unit >= 2 % Convert upper triangular R to vector x. + + T = x; + if p ~= q, error('Must pass square matrix'), end + unit = unit - 2; + n = p*(p+1)/2 - unit*p; + x = zeros(n,1); + i = 1; + for j = 1+unit:p + x(i:i+j-1-unit) = T(1:j-unit,j); + i = i+j-unit; + end + T = x; + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/vand.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,29 @@ +function V = vand(m, p) +%VAND Vandermonde matrix. +% V = VAND(P), where P is a vector, produces the (primal) +% Vandermonde matrix based on the points P, i.e. V(i,j) = P(j)^(i-1). +% VAND(M,P) is a rectangular version of VAND(P) with M rows. +% Special case: If P is a scalar then P equally spaced points on [0,1] +% are used. + +% Reference: +% N. J. Higham, Accuracy and Stability of Numerical Algorithms, +% Second edition, Society for Industrial and Applied Mathematics, +% Philadelphia, PA, 2002; chap. 22. + +if nargin == 1, p = m; end +n = length(p); + +% Handle scalar p. +if n == 1 + n = p; + p = linspace(0,1,n); +end + +if nargin == 1, m = n; end + +p = p(:).'; % Ensure p is a row vector. +V = ones(m,n); +for i=2:m + V(i,:) = p.*V(i-1,:); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/matrixcomp/vecperm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,27 @@ +function P = vecperm(m, n) +%VECPERM Vec-permutation matrix. +% VECPERM(M, N) is the vec-permutation matrix, an MN-by-MN +% permutation matrix P with the property that if A is M-by-N then +% vec(A) = P*vec(A'). +% If N is omitted, it defaults to M. + +% P is formed by taking every n'th row from EYE(M*N), starting with +% the first and working down - see p. 277 of the reference. + +% Reference: +% H. V. Henderson and S. R. Searle The vec-permutation matrix, +% the vec operator and Kronecker products: A review Linear and +% Multilinear Algebra, 9 (1981), pp. 271-288. + +if nargin == 1, n = m; end + +P = zeros(m*n); +I = eye(m*n); + +k = 1; +for i=1:n + for j=i:n:m*n + P(k,:) = I(j,:); + k = k+1; + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/Contents.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,46 @@ +% Matrix Function Toolbox (MFT). +% Version 1.0 6-Mar-2008 +% Copyright (c) 2008 by N. J. Higham +% +% arnoldi - Arnoldi iteration +% ascent_seq - Ascent sequence for square (singular) matrix. +% cosm - Matrix cosine by double angle algorithm. +% cosm_pade - Evaluate Pade approximation to the matrix cosine. +% cosmsinm - Matrix cosine and sine by double angle algorithm. +% cosmsinm_pade - Evaluate Pade approximations to matrix cosine and sine. +% expm_cond - Relative condition number of matrix exponential. +% expm_frechet_pade - Frechet derivative of matrix exponential via Pade approx. +% expm_frechet_quad - Frechet derivative of matrix exponential via quadrature. +% fab_arnoldi - f(A)*b approximated by Arnoldi method. +% funm_condest1 - Estimate of 1-norm condition number of matrix function. +% funm_condest_fro - Estimate of Frobenius norm condition number of matrix function. +% funm_ev - Evaluate general matrix function via eigensystem. +% funm_simple - Simplified Schur-Parlett method for function of a matrix. +% logm_cond - Relative condition number of matrix logarithm. +% logm_frechet_pade - Frechet derivative of matrix logarithm via Pade approx. +% logm_iss - Matrix logarithm by inverse scaling and squaring method. +% logm_pade_pf - Evaluate Pade approximant to matrix log by partial fractions. +% mft_test - Test the Matrix Function Toolbox. +% mft_tolerance - Convergence tolerance for matrix iterations. +% polar_newton - Polar decomposition by scaled Newton iteration. +% polar_svd - Canonical polar decomposition via singular value decomposition. +% polyvalm_ps - Evaluate polynomial at matrix argument by Paterson-Stockmeyer alg. +% power_binary - Power of matrix by binary powering (repeated squaring). +% quasitriang_struct - Block structure of upper quasitriangular matrix. +% readme - Welcome to the Matrix Function Toolbox. +% riccati_xaxb - Solve Riccati equation XAX = B in positive definite matrices. +% rootpm_newton - Coupled Newton iteration for matrix pth root. +% rootpm_real - Pth root of real matrix via real Schur form. +% rootpm_schur_newton - Matrix pth root by Schur-Newton method. +% rootpm_sign - Matrix Pth root via matrix sign function. +% signm - Matrix sign decomposition. +% signm_newton - Matrix sign function by Newton iteration. +% sqrtm_db - Matrix square root by Denman-Beavers iteration. +% sqrtm_dbp - Matrix square root by product form of Denman-Beavers iteration. +% sqrtm_newton - Matrix square root by Newton iteration (unstable). +% sqrtm_newton_full - Matrix square root by full Newton method. +% sqrtm_pd - Square root of positive definite matrix via polar decomposition. +% sqrtm_pulay - Matrix square root by Pulay iteration. +% sqrtm_real - Square root of real matrix by real Schur method. +% sqrtm_triang_min_norm - Estimated min norm square root of triangular matrix. +% sylvsol - Solve Sylvester equation.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/arnoldi.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,28 @@ +function [Q,H] = arnoldi(A,q1,m) +%ARNOLDI Arnoldi iteration +% [Q,H] = ARNOLDI(A,q1,M) carries out M iterations of the +% Arnoldi iteration with N-by-N matrix A and starting vector q1 +% (which need not have unit 2-norm). For M < N it produces +% an N-by-(M+1) matrix Q with orthonormal columns and an +% (M+1)-by-M upper Hessenberg matrix H such that +% A*Q(:,1:M) = Q(:,1:M)*H(1:M,1:M) + H(M+1,M)*Q(:,M+1)*E_M', +% where E_M is the M'th column of the M-by-M identity matrix. + +n = length(A); +if nargin < 3, m = n; end +q1 = q1/norm(q1); +Q = zeros(n,m); Q(:,1) = q1; +H = zeros(min(m+1,m),n); + +for k=1:m + z = A*Q(:,k); + for i=1:k + H(i,k) = Q(:,i)'*z; + z = z - H(i,k)*Q(:,i); + end + if k < n + H(k+1,k) = norm(z); + if H(k+1,k) == 0, return, end + Q(:,k+1) = z/H(k+1,k); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/ascent_seq.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,25 @@ +function [d,a] = ascent_seq(A) +%ASCENT_SEQ Ascent sequence for square (singular) matrix. +% [d,a] = ASCENT_SEQ(A) returns symbolically computed +% a(i) = dim(null(A^(i-1))) and the ascent sequence d = DIFF(a). +% A has a square root if in the ascent sequence no two terms are +% the same odd integer. +% This function is intended for singular matrices of small +% dimension with exactly known entries. +% It requires the Symbolic Math Toolbox. + +if isempty(ver('symbolic')) + error('The Symbolic Math Toolbox is required.') +end + +n = length(A); +a = zeros(n,1); +A = sym(A); +X = sym(eye(n)); +for i = 2:n+1 + X = X*A; + N = null(X); + if isempty(N), break, end + a(i) = rank(N); +end +d = diff(a);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/cosm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,46 @@ +function [C,cost,s] = cosm(A) +%COSM Matrix cosine by double angle algorithm. +% [C,COST,s] = COSM(A) computes the cosine of the matrix A using the +% double angle algorithm with Pade approximation. The total number of +% matrix multiplications and linear systems solved is returned as +% COST and s specifies the amount of scaling (A is scaled to 2^(-s)*A). + +n = length(A); +% theta_m for m=2:2:20. +theta = [0.00613443965526 0.11110098037055 0.43324697422211 0.98367255274344 ... + 1.72463663220280 2.61357494421368 3.61521023400301 4.70271938553349 ... + 5.85623410320942 7.06109053959248]; +s = 0; + +B = A^2; +normA2 = sqrt(norm(B,inf)); +d = [2 4 6 8 12 16 20]; +for i = d(1:6) + if normA2 < theta(i/2) + m = i; + cost = (m<=8)*(m/2+1) + (m==12)*6; + C = cosm_pade(B,m); + s = m; + return + end +end + +if normA2 > theta(20/2) + s = ceil( log2( normA2/theta(20/2) ) ); + B = B/(4^s); + normA2 = normA2/(2^s); +end + +if normA2 > 2*theta(12/2) + m = 20; cost = 8; +elseif normA2 > theta(16/2) + B = B/4; m = 12; cost = 6; s = s+1; +else + m = 16; cost = 7; +end + +C = cosm_pade(B,m); +for i = 1:s + C = 2*(C^2) - eye(n); +end +cost = cost+s;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/cosm_pade.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,122 @@ +function C = cosm_pade(A,m,sq) +%COSM_PADE Evaluate Pade approximation to the matrix cosine. +% C = COSM_PADE(A,M,SQ) approximates the matrix cosine using the +% Mth order diagonal Pade approximation. +% If SQ = 1 (default) then C is an approximation to cos(sqrt(A)); +% otherwise C is an approximation to cos(A). + +if nargin < 3 + sq = 1; +end +if sq == 1 + A2 = A; +else + A2 = A^2; +end + +n = length(A2); +I = eye(n); +if m == 2 + X2 = A2; + P = I - (5/12)*X2; + Q = I + (1/12)*X2; +elseif m == 4 + X2 = A2; X4 = X2^2; + P = I - (115/252)*X2 + (313/15120)*X4; + Q = I + (11/252)*X2 + (13/15120)*X4; +elseif m == 6 + X2 = A2; X4 = X2^2; X6 = X4*X2; + P = I - (3665/7788)*X2 + (711/25960)*X4 - (2923/7850304)*X6; + Q = I + (229/7788)*X2 + (1/2360)*X4 + (127/39251520)*X6; +elseif m == 8 + X2 = A2; X4 = X2^2; X6 = X4*X2; X8 = X6*X2; + P = I - (260735/545628)*X2 + (4375409/141863280)*X4 ... + - (7696415/13108167072)*X6 + (80737373/23594700729600)*X8 ; + Q = I + (12079/545628)*X2 + (34709/141863280)*X4 ... + + (109247/65540835360)*X6 + (11321/1814976979200)*X8 ; +elseif m == 12 + X1 = A2; X2 = X1*X1; X3 = X2*X1; + p = [1,-220574348151635/454605030049116,20837207639809/606140040065488,... + -199961484798769/241849875986129712,38062401688454831/... + 4440363723105341512320,-116112688080827/2894459315802000393216,... + 151259208063389819/2133505961677654489839513600]; + q = [1,6728166872923/454605030049116,66817219029/606140040065488,... + 650617920073/1209249379930648560,8225608067111/4440363723105341512320,... + 2848116281867/651253346055450088473600,12170851069679/... + 2133505961677654489839513600]; + P = X3*((p(7)*eye(n))*X3+(p(4)*eye(n)+p(5)*X1+p(6)*X2)*eye(n))+... + (p(1)*eye(n)+p(2)*X1+p(3)*X2); + Q = X3*((q(7)*eye(n))*X3+(q(4)*eye(n)+q(5)*X1+q(6)*X2)*eye(n))+... + (q(1)*eye(n)+q(2)*X1+q(3)*X2); +elseif m == 16 + X1 = A2; X2 = X1*X1; X3 = X2*X1; X4 = X2*X2; + p = [1,-1126682407530029115789472765/2304577612359442026681336372,... + 145053661043845297596963732421/4009965045505429126425525287280,... + -1534672316720770887322573595/1603986018202171650570210114912,... + 718202654899849477670594159641/60630671488042088391553942343673600,... + -128936233968950140829066659951/1673406533069961639606888808685391360,... + 6524116556754642812271854422129/23929713422900451446378509964201096448000,... + -382586638331055978467487427009/... + 763836452458982410168402038057298998620160,... + 88555612088268453352055067469523/... + 233733954452448617511531023645533493577768960000]; + q = [1,25606398649691897551195421/2304577612359442026681336372,... + 22668270274336502918805611/364542276864129920584138662480,... + 1853378279158412863783499/8019930091010858252851050574560,... + 38226389122327179481602241/60630671488042088391553942343673600,... + 995615371594253927197913/760639333213618927094040367584268800,... + 225870994754204367988837/110275177064057379937228156517055744000,... + 42889724495628101076622829/19095911311474560254210050951432474965504000,... + 2603898999593850290644763/1931685573987178657120091104508541269237760000]; + P = X4*((p(9)*eye(n))*X4+(p(5)*eye(n)+p(6)*X1+p(7)*X2+p(8)*X3)*eye(n))+... + (p(1)*eye(n)+p(2)*X1+p(3)*X2+p(4)*X3); + Q = X4*((q(9)*eye(n))*X4+(q(5)*eye(n)+q(6)*X1+q(7)*X2+q(8)*X3)*eye(n))+... + (q(1)*eye(n)+q(2)*X1+q(3)*X2+q(4)*X3); +elseif m == 20 + X1 = A2; X2 = X1*X1; X3 = X2*X1; X4 = X2*X2; X5 = X4*X1; + p = [1,-18866133841442352341137832915472113127673/... + 38415527280635118612047973206722428679860,... + 917980006162069077942240197016800349995791/... + 24637158162647322736526766816577984260016880,... + -4028339250935885155796261896908967142863591/... + 3880352410616953331002965773611032520952658600,... + 3925400573997340625949450726927185904756763/... + 279385373564420639832213535699994341508591419200,... + -804035081520215224783821741744290679884325097/... + 7621632990837395054622785253895845636354373915776000,... + 19795406323827219175300218252334434555489703/... + 42100448901768467920773480450091337800814636868096000,... + -118523567829079039162888326742509818128969627/... + 92831489828399471765305524392451399850796274294151680000,... + 6151694105279089780298999575203793198700323571/... + 2954269332298984789459083008265373348851740633137083064320000,... + -438673281197605688527510681818034658057709668453/... + 232382825678638143538851469430154267620677918202562953839411200000,... + 31699084606166905465868332652040368902350407479/... + 42944346185412328925979751550692508656301279283833633869523189760000]; + q = [1,341629798875206964886153687889101212257/... + 38415527280635118612047973206722428679860,... + 981038224413663993784862242489461225499/... + 24637158162647322736526766816577984260016880,... + 461441299765418864926911910257258436499/... + 3880352410616953331002965773611032520952658600,... + 73764947345500690357380325430051300659/... + 279385373564420639832213535699994341508591419200,... + 3496016725011957790816142668159659762953/... + 7621632990837395054622785253895845636354373915776000,... + 562876526229442596170390468670872658343/... + 884109426937137826336243089451918093817107374230016000,... + 65309174262483666596220950666851746623/... + 92831489828399471765305524392451399850796274294151680000,... + 1768262649350763278383509302712194678051/... + 2954269332298984789459083008265373348851740633137083064320000,... + 83263779334467686055536878437959026858717/... + 232382825678638143538851469430154267620677918202562953839411200000,... + 24953265550459114615706087077367245444511/... + 214721730927061644629898757753462543281506396419168169347615948800000]; + P = X5*((p(11)*eye(n))*X5+(p(6)*eye(n)+p(7)*X1+p(8)*X2+p(9)*X3+p(10)*X4)*eye(n))... + +p(1)*eye(n)+p(2)*X1+p(3)*X2+p(4)*X3+p(5)*X4; + Q = X5*((q(11)*eye(n))*X5+(q(6)*eye(n)+q(7)*X1+q(8)*X2+q(9)*X3+q(10)*X4)*eye(n))... + +(q(1)*eye(n)+q(2)*X1+q(3)*X2+q(4)*X3+q(5)*X4); +end +C = Q\P;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/cosmsinm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,46 @@ +function [C,S,cost,s] = cosmsinm(A) +%COSMSINM Matrix cosine and sine by double angle algorithm. +% [C,S,COST,s] = COSMSINM(A) computes the cosine C and the sine S +% of the matrix A using the double angle algorithm with Pade +% approximation. The total number of matrix multiplications and +% linear systems solved is returned as COST and s specifies the +% amount of scaling (A is scaled to 2^(-s)*A). + +n = length(A); +theta = [0.00613443965526 0.11110098037055 0.43324697422211 0.98367255274344 ... + 1.72463663220280 2.61357494421368 3.61521023400301 4.70271938553349 ... + 5.85623410320942 7.06109053959248 0 9.58399601173102]; + +normA = norm(A,inf); +d = [2 4 6 8 10 12 14 16 20]; +for i = d(1:8) + if normA <= theta(i/2); + m = i; + cost = m/2+3; + [C,S] = cosmsinm_pade(A,m); + s = m; + return + end +end + +s = 0; +if normA > theta(20/2) + s = max( ceil( log2( normA/theta(20/2) ) ), 1 ); + A = A/(2^s); + normA = normA/(2^s); +end + +if normA > 2*theta(12/2) + m = 20; cost = 12; +elseif normA > theta(16/2) + A = A/2; m = 12; cost = 9; s = s+1; +else + m = 16; cost = 11; +end + +[C,S] = cosmsinm_pade(A,m); +for i = 1:s + S = 2*S*C; + C = 2*(C^2) - eye(n); +end +cost = cost + 2*s;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/cosmsinm_pade.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,231 @@ +function [C,S] = cosmsinm_pade(A,m) +%COSMSINM_PADE Evaluate Pade approximations to matrix cosine and sine. +% [C,S] = COSMSINM_PADE(A,M) approximates the matrix cosine +% C = cos(A) and the matrix sine S = sin(A) using the [M/M] and +% [M+1/M+1] Pade approximants, respectively, where M is even. + +n = length(A); +I = eye(n); +if m == 2 + X2 = A^2; + p = I - (5/12)*X2; + q = I + (1/12)*X2; + p2 = A*(-7/60*X2+I); + q2 = I+1/20*X2; +elseif m == 4 + X2 = A^2; X4 = X2^2; + p = I - (115/252)*X2 + (313/15120)*X4; + q = I + (11/252)*X2 + (13/15120)*X4; + p2 = A*(551/166320*X4-53/396*X2+I); + q2 = I+13/396*X2+5/11088*X4; +elseif m == 6 + X2 = A^2; X4 = X2^2; X6 = X4*X2; + p = I - (3665/7788)*X2 + (711/25960)*X4 - (2923/7850304)*X6; + q = I + (229/7788)*X2 + (1/2360)*X4 + (127/39251520)*X6; + p2 = A*(-479249/11511339840*X6+34911/7613320*X4-29593/207636*X2+I); + q2 = I+1671/69212*X2+97/351384*X4+2623/1644477120*X6; +elseif m == 8 + X2 = A^2; X4 = X2^2; X6 = X4*X2; X8 = X6*X2; + p = I - (260735/545628)*X2 + (4375409/141863280)*X4 ... + - (7696415/13108167072)*X6 + (80737373/23594700729600)*X8 ; + q = I + (12079/545628)*X2 + (34709/141863280)*X4 ... + + (109247/65540835360)*X6 + (11321/1814976979200)*X8 ; + p2 = A*(4585922449/15605159573203200*X8-269197963/3940696861920*X6... + + 38518909/7217393520*X4-53272705/360869676*X2+I); + q2 = I+2290747/120289892*X2+1281433/7217393520*X4+560401/... + 562956694560*X6+1029037/346781323848960*X8; +elseif m == 10 + X2 = A^2; X4 = X2^2; X6 = X4*X2; X8 = X6*X2; X10 = X8*X2; + p = I-213692663231/11226787947026555904*X10+2677576097699/... + 425257119205551360*X8-18894084119/25961973089472*X6+... + 53471234645/1622623318092*X4-5114526085/10605381164*X2; + q = I+2045322787/280669698675663897600*X10+1469628299/425257119205551360*... + X8+117715523/129809865447360*X6+256513745/1622623318092*X4+... + 188164497/10605381164*X2; + p2 = A*(-481959816488503/363275871831577908403200*X10+23704595077729/... + 42339845201815607040*X8-2933434889971/33603051747472704*X6+... + 3605886663403/617703157122660*X4-109061004303/722459832892*X2+I); + q2 = I+34046903537/2167379498676*X2+1679739379/13726736824948*X4+... + 101555058991/168015258737363520*X6+3924840709/2016183104848362240*X8+... + 37291724011/11008359752472057830400*X10; +elseif m == 12 + X2 = A^2; X4 = X2^2; X6 = X4*X2; X8 = X6*X2; X10 = X8*X2; X12 = X10*X2; + p = I-220574348151635/454605030049116*X2+20837207639809/606140040065488*X4-... + 199961484798769/241849875986129712*X6+38062401688454831/... + 4440363723105341512320*X8-116112688080827/2894459315802000393216*X10+... + 151259208063389819/2133505961677654489839513600*X12; + q = I+6728166872923/454605030049116*X2+66817219029/606140040065488*X4+... + 650617920073/1209249379930648560*X6+8225608067111/... + 4440363723105341512320*X8+2848116281867/651253346055450088473600*X10+... + 12170851069679/2133505961677654489839513600*X12; + p2 = A*(29255060349997508141/7036046883826051377386580480000*X12-... + 2614538885979566893/906830082707196962219136000*X10+366447091815846989/... + 467616734780258173795200*X8-404844663233524697/... + 3987959482775106258000*X6+235300071854130877/37980566502620059600*X4-... + 189866343920582047/1238496733781088900*X2+I); + q2 = I+5516592792088701/412832244593696300*X2+10149321432971387/... + 113941699507860178800*X4+307158833383841/797591896555021251600*X6+... + 234442582333727/202056613793938717072000*X8+5257566411989069/... + 2225855657554028907265152000*X10+547584669013343/... + 209866390569379868399285760000*X12; +elseif m == 14 + X2 = A^2; X4 = X2^2; X6 = X4*X2; X8 = X6*X2; X10 = X8*X2; X12 = X10*X2; ... + X14 = X12*X2; + p = I-9494011832127130075/19482625349840916996*X2+310367939726544641761/... + 8767181407428412648200*X4-290442017949426019417/... + 322632275793365585453760*X6+422064995289725174621/... + 40651666749964063767173760*X8-10143257867238927098753/... + 169923967014849786546786316800*X10+569672335133865317479319/... + 3379787703925362254415579841152000*X12-42948386132766526786783/... + 227121733703784343496726965325414400*X14; + q = I+247300842793328423/19482625349840916996*X2+711404045526343261/... + 8767181407428412648200*X4+184363345338626039/... + 537720459655609309089600*X6+42648708634036181/... + 40651666749964063767173760*X8+2008837845325770881/... + 849619835074248932733931584000*X10+1790291961275627717/... + 482826814846480322059368548736000*X12+1651330565298296101/... + 516185758417691689765288557557760000*X14; + p2 = A*(-91320110661332736729305509/... + 9540763134798175069489060319738357760000*X14+... + 261319443906679357131557951/25722645706563707295191093999294592000*X12... + -25455044236652128985986613/5785570334359808208545005397952000*X10+... + 145834078116038838318251/150274554139215797624545594752*X8-... + 264802008995134681176249/2352661207225153389183707200*X6+... + 1466350032750183270853069/226863759268139791099857480*X4-... + 390816793511920445322905/2520708436312664345553972*X2+I); + q2 = I+9767093068952315200919/840236145437554781851324*X2+... + 3067578723707839145789/45372751853627958219971496*X4+... + 7023288908523954335543/27223651112176774931982897600*X6+... + 2065373697229749176197/2922005219373640509366164342400*X8+... + 106107024079408237517/75137277069607898812272797376000*X10+... + 1279669772129105715749/659555018117018135774130615366528000*X12+... + 70807382924337520289663/48975917425297298690043842974656903168000*X14; +elseif m == 16 + X1 = A^2; X2 = X1*X1; X3 = X2*X1; X4 = X2*X2; + p = [1,-1126682407530029115789472765/2304577612359442026681336372,... + 145053661043845297596963732421/4009965045505429126425525287280,... + -1534672316720770887322573595/1603986018202171650570210114912,... + 718202654899849477670594159641/60630671488042088391553942343673600,... + -128936233968950140829066659951/1673406533069961639606888808685391360,... + 6524116556754642812271854422129/23929713422900451446378509964201096448000,... + -382586638331055978467487427009/... + 763836452458982410168402038057298998620160,... + 88555612088268453352055067469523/... + 233733954452448617511531023645533493577768960000]; + q = [1,25606398649691897551195421/2304577612359442026681336372,... + 22668270274336502918805611/364542276864129920584138662480,... + 1853378279158412863783499/8019930091010858252851050574560,... + 38226389122327179481602241/60630671488042088391553942343673600,... + 995615371594253927197913/760639333213618927094040367584268800,... + 225870994754204367988837/110275177064057379937228156517055744000,... + 42889724495628101076622829/19095911311474560254210050951432474965504000,... + 2603898999593850290644763/1931685573987178657120091104508541269237760000]; + p2 = fliplr([8061385294694990486176108590199657/... + 477924965253252594838839715519805972908219146240000,... + -4615365296919423054990558658047229/... + 177615978658790247602767966416950841324802560000,... + 99552990732049352416683777921586343/... + 5920532621959674920092265547231694710826752000,... + -2609410230536249996829038272149121/... + 450025282909674287024343687080548396992000,... + 307444630821656723644151024596157/... + 272742595702832901226874961866999028480,... + -1685658558102333722460472837754189/... + 13889669225607231080998261946930506080,... + 228262816046878642628376219569959/34211007944845396751227246174705680,... + -2876333034231579924039546769067/18393015024110428360874863534788,1]); + q2 = fliplr([1604612285445093/2658455991569831745807614120560689152,... + 2976480701836195/2596148429267413814265248164610048,... + 5974590438482987/5070602400912917605986812821504,520154651818787/... + 618970019642690137449562112,4332878172069797/9671406556917033397649408,... + 3408424760479185/18889465931478580854784,7823297585492293/... + 147573952589676412928,1482203634412391/144115188075855872,1]); + p = X4*((p(9)*eye(n))*X4+(p(5)*eye(n)+p(6)*X1+p(7)*X2+p(8)*X3)*eye(n))+... + (p(1)*eye(n)+p(2)*X1+p(3)*X2+p(4)*X3); + q = X4*((q(9)*eye(n))*X4+(q(5)*eye(n)+q(6)*X1+q(7)*X2+q(8)*X3)*eye(n))+... + (q(1)*eye(n)+q(2)*X1+q(3)*X2+q(4)*X3); + p2 = A*(X4*((p2(9)*eye(n))*X4+(p2(5)*eye(n)+p2(6)*X1+p2(7)*X2+p2(8)*X3)*eye(n))+... + (p2(1)*eye(n)+p2(2)*X1+p2(3)*X2+p2(4)*X3)); + q2 = X4*((q2(9)*eye(n))*X4+(q2(5)*eye(n)+q2(6)*X1+q2(7)*X2+q2(8)*X3)*eye(n))+... + (q2(1)*eye(n)+q2(2)*X1+q2(3)*X2+q2(4)*X3); +elseif m == 20 + X1 = A^2; X2 = X1*X1; X3 = X2*X1; X4 = X2*X2; X5 = X4*X1; + p = [1,-18866133841442352341137832915472113127673/... + 38415527280635118612047973206722428679860,... + 917980006162069077942240197016800349995791/... + 24637158162647322736526766816577984260016880,... + -4028339250935885155796261896908967142863591/... + 3880352410616953331002965773611032520952658600,... + 3925400573997340625949450726927185904756763/... + 279385373564420639832213535699994341508591419200,... + -804035081520215224783821741744290679884325097/... + 7621632990837395054622785253895845636354373915776000,... + 19795406323827219175300218252334434555489703/... + 42100448901768467920773480450091337800814636868096000,... + -118523567829079039162888326742509818128969627/... + 92831489828399471765305524392451399850796274294151680000,... + 6151694105279089780298999575203793198700323571/... + 2954269332298984789459083008265373348851740633137083064320000,... + -438673281197605688527510681818034658057709668453/... + 232382825678638143538851469430154267620677918202562953839411200000,... + 31699084606166905465868332652040368902350407479/... + 42944346185412328925979751550692508656301279283833633869523189760000]; + q = [1,341629798875206964886153687889101212257/... + 38415527280635118612047973206722428679860,... + 981038224413663993784862242489461225499/... + 24637158162647322736526766816577984260016880,... + 461441299765418864926911910257258436499/... + 3880352410616953331002965773611032520952658600,... + 73764947345500690357380325430051300659/... + 279385373564420639832213535699994341508591419200,... + 3496016725011957790816142668159659762953/... + 7621632990837395054622785253895845636354373915776000,... + 562876526229442596170390468670872658343/... + 884109426937137826336243089451918093817107374230016000,... + 65309174262483666596220950666851746623/... + 92831489828399471765305524392451399850796274294151680000,... + 1768262649350763278383509302712194678051/... + 2954269332298984789459083008265373348851740633137083064320000,... + 83263779334467686055536878437959026858717/... + 232382825678638143538851469430154267620677918202562953839411200000,... + 24953265550459114615706087077367245444511/... + 214721730927061644629898757753462543281506396419168169347615948800000]; + p2 = fliplr([504755453995739302254967861091843658659225385703072439/... + 19060545079818588288159374768330241280586535602550332513982470267073331200000,... + -8739283052502581091809652173945956169815253539783369/... + 114148670977473878836743171447659847170838038103667100934138640957440000,... + 7098206776968855944478646605493639889495273479386941/... + 73428969634632319719542390989722708706387042054990532764650587750400,... + -113535124075801033808719602712883055821671599003497/... + 1648104073175508145683904128728603495708248986725888205771264000,... + 159444168407032138165237007441577372335666527110947/... + 5304242994128072193005668460275965273543789842336191926620160,... + -250413931005302347677105318749990589129438025771967/... + 30554395127465853646346016476244039594146254852167004185600,... + 89477294557053179299513803506788857306761134861/... + 65142408168740093907440765129293961270139550682600640,... + -12973367660334266409984992764341733133099094183655/... + 96446954316495750146294243927538003769401056982850392,... + 2886168459322903018906421202468694698344454148801/... + 413757847775614543742146048595186631357361891818320,... + -755789275315695418647501273540203715920168810921/... + 4774129012795552427793992868405999592584944905596,1]); + q2 = fliplr([2290207460112795/44601490397061246283071436545296723011960832,... + 3841593531710827/21778071482940061661655974875633165533184,... + 3463624789758205/10633823966279326983230456482242756608,... + 8728650941242757/20769187434139310514121985316880384,4208410993374357/... + 10141204801825835211973625643008,1606756025554677/... + 4951760157141521099596496896,7804153743481773/38685626227668133590597632,... + 1845833986288687/18889465931478580854784,1293253388138245/... + 36893488147419103232,2408831652010703/288230376151711744,1]); + p = X5*((p(11)*eye(n))*X5+(p(6)*eye(n)+p(7)*X1+p(8)*X2+p(9)*X3+p(10)*X4)*eye(n))... + +p(1)*eye(n)+p(2)*X1+p(3)*X2+p(4)*X3+p(5)*X4; + q = X5*((q(11)*eye(n))*X5+(q(6)*eye(n)+q(7)*X1+q(8)*X2+q(9)*X3+q(10)*X4)*eye(n))... + +(q(1)*eye(n)+q(2)*X1+q(3)*X2+q(4)*X3+q(5)*X4); + p2 = A*(X5*((p2(11)*eye(n))*X5+(p2(6)*eye(n)+p2(7)*X1+p2(8)*X2+p2(9)*X3+p2(10)*X4)*eye(n))... + +p2(1)*eye(n)+p2(2)*X1+p2(3)*X2+p2(4)*X3+p2(5)*X4); + q2 = X5*((q2(11)*eye(n))*X5+(q2(6)*eye(n)+q2(7)*X1+q2(8)*X2+q2(9)*X3+q2(10)*X4)*eye(n))... + +(q2(1)*eye(n)+q2(2)*X1+q2(3)*X2+q2(4)*X3+q2(5)*X4); +end +C = q\p; +S = q2\p2;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/expm_cond.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,25 @@ +function [c,K] = expm_cond(A) +%EXPM_COND Relative condition number of matrix exponential. +% EXPM_COND(A) is the relative condition number in the Frobenius +% norm of the matrix exponential at the matrix A. +% [C,K] = EXPM_COND(A) returns the condition number C and the Kronecker +% matrix form K of the Frechet derivative. + +n = length(A); +N = n^2; +K = zeros(N); +E = zeros(n); + +if nargout < 2 && ~isequal(A,triu(A)) + % If returning K cannot use Schur form. + A = schur(A,'complex'); +end + +for j = 1:N + e = zeros(N,1); e(j) = 1; + E(:) = e; + X = expm_frechet_pade(A,E); + K(:,j) = X(:); +end + +c = norm(K) * norm(A,'fro') / norm(expm(A),'fro');
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/expm_frechet_pade.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,59 @@ +function L = expm_frechet_pade(A,E,k) +%EXPM_FRECHET_PADE Frechet derivative of matrix exponential via Pade approx. +% L = EXPM_FRECHET_PADE(A,E) evaluates the Frechet derivative of +% the matrix exponential at A in the direction E via scaling and +% squaring and a Pade approximant of the function tanh(x)/x. +% L = EXPM_FRECHET_PADE(A,E,k) uses either matrix exponentials +% (k = 0, the default) or repeated squaring (k = 1) in the final +% phase of the algorithm. + +if nargin < 3, k = 0; end + +real_data = isreal(A) && isreal(E); +% Form complex Schur form if A not already upper triangular. +use_Schur = false; +if ~isequal(A,triu(A)) + use_Schur = true; + [Q,T] = schur(A,'complex'); A = T; E = Q'*E*Q; +end + +Abound = 1; +if norm(A,1) <= Abound + s = 0; +else + s = ceil( log2(norm(A,1)/Abound) ); +end + +As = A/2^s; + +I = eye(size(A)); + +m = 8; +% Positive zeros of p8 and q8 in r8 = p8/q8 Pade approximant. +load tau_r8_zeros +% Zeros come in \pm pairs. +a = complex(0, [pzero; -pzero]); +b = complex(0, [qzero; -qzero]); + +G = 2^(-s)*E; +for i=1:m + rhs = (I + As/a(i)) * G + G * (I - As/a(i)); + AA = I + As/b(i); BB = I - As/b(i); + G = sylvsol(AA, BB, rhs); +end + +X = expm(As); +L = (G*X + X*G)/2; +for i=s:-1:1 + if i < s + if k == 0 + X = expm(2^(-i)*A); + else + X = X^2; + end + end + L = X*L + L*X; +end + +if use_Schur, L = Q*L*Q'; end +if real_data, L = real(L); end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/expm_frechet_quad.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,51 @@ +function R = expm_frechet_quad(A,E,theta,rule,k) +%EXPM_FRECHET_QUAD Frechet derivative of matrix exponential via quadrature. +% L = EXPM_FRECHET_QUAD(A,E,THETA,RULE) is an approximation to the +% Frechet derivative of the matrix exponential at A in the direction E +% intended to have norm of the correct order of magnitude. +% It is obtained from the repeated trapezium rule (RULE = 'T'), +% the repeated Simpson rule (RULE = 'S', default), +% or the repeated midpoint rule (RULE = 'M'). +% L = EXPM_FRECHET_QUAD(A,E,THETA,RULE,k) uses either matrix +% exponentials (k = 0, the default) or repeated squaring (k = 1) +% in the final phase of the algorithm. + +% A is scaled so that norm(A/2^s) <= THETA. Defalt: THETA = 1/2. + +if nargin < 3 || isempty(theta), theta = 1/2; end +if nargin < 4 || isempty(rule), rule = 'S'; end +if nargin < 5, k = 0; end + +s = ceil( log2(norm(A,1)/theta) ); +As = A/2^s; + +X = expm(As); + +switch upper(rule) + + case 'T' + R = 2^(-s) * (X*E + E*X)/2 ; + + case 'S' + Xmid = expm(As/2); + R = 2^(-s) * (X*E + 4*Xmid*E*Xmid + E*X)/6; + + case 'M' + Xmid = expm(As/2); + R = 2^(-s) * Xmid*E*Xmid; + + otherwise + error('Illegal value of RULE.') + +end + +for i = s:-1:1 + if i < s + if k == 0 + X = expm(2^(-i)*A); + else + X = X^2; + end + end + R = X*R + R*X; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/fab_arnoldi.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,14 @@ +function c = fab_arnoldi(A,b,fun,m) +%FAB_ARNOLDI f(A)*b approximated by Arnoldi method. +% C = FAB_ARNOLDI(A,B,FUN,M) approximates FUNM(A,FUN)*B +% for a square matrix A using M steps of the Arnoldi process +% with starting vector B/norm(B). +% FUN must be a function handle for which FUNM(A,FUN) is defined. +% For large matrices M is intended to be much less than LENGTH(A). + +q1 = b/norm(b); +[Q,H] = arnoldi(A,q1,m); +H = H(1:m,1:m); +Q = Q(:,1:m); +e = zeros(m,1); e(1) = 1; +c = norm(b)*Q*funm(H,fun)*e;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/funm_condest1.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,91 @@ +function [c,est] = funm_condest1(A,fun,fun_frechet,flag1,varargin) +%FUNM_CONDEST1 Estimate of 1-norm condition number of matrix function. +% C = FUNM_CONDEST1(A,FUN,FUN_FRECHET,FLAG) produces an estimate of +% the 1-norm relative condition number of function FUN at the matrix A. +% FUN and FUN_FRECHET are function handles: +% - FUN(A) evaluates the function at the matrix A. +% - If FLAG == 0 (default) +% FUN_FRECHET(B,E) evaluates the Frechet derivative at B +% in the direction E; +% if FLAG == 1 +% - FUN_FRECHET('notransp',E) evaluates the +% Frechet derivative at A in the direction E. +% - FUN_FRECHET('transp',E) evaluates the +% Frechet derivative at A' in the direction E. +% If FUN_FRECHET is empty then the Frechet derivative is approximated +% by finite differences. More reliable results are obtained when +% FUN_FRECHET is supplied. +% MATLAB'S NORMEST1 (block 1-norm power method) is used, with a random +% starting matrix, so the approximation can be different each time. +% C = FUNM_CONDEST1(A,FUN,FUN_FRECHET,FLAG,P1,P2,...) passes extra inputs +% P1,P2,... to FUN and FUN_FRECHET. +% [C,EST] = FUNM_CONDEST1(A,...) also returns an estimate EST of the +% 1-norm of the Frechet derivative. +% Note: this function makes an assumption on the adjoint of the +% Frechet derivative that, for f having a power series expansion, +% is equivalent to the series having real coefficients. + +if nargin < 3 || isempty(fun_frechet), fte_diff = 1; else fte_diff = 0; end +if nargin < 4 || isempty(flag1), flag1 = 0; end + +n = length(A); +funA = feval(fun,A,varargin{:}); +if fte_diff, d = sqrt( eps*norm(funA,1) ); end + +factor = norm(A,1)/norm(funA,1); + +[est,v,w,iter] = normest1(@afun); +c = est*factor; + + %%%%%%%%%%%%%%%%%%%%%%%%% Nested function. + function Z = afun(flag,X) + %AFUN Function to evaluate matrix products needed by NORMEST1. + + if isequal(flag,'dim') + Z = n^2; + elseif isequal(flag,'real') + Z = isreal(A); + else + + [p,q] = size(X); + if p ~= n^2, error('Dimension mismatch'), end + E = zeros(n); + Z = zeros(n^2,q); + for j=1:q + + E(:) = X(:,j); + + if isequal(flag,'notransp') + + if fte_diff + Y = (feval(fun,A+d*E/norm(E,1),varargin{:}) - funA)/d; + else + if flag1 + Y = feval(fun_frechet,'notransp',E,varargin{:}); + else + Y = feval(fun_frechet,A,E,varargin{:}); + end + end + + elseif isequal(flag,'transp') + + if fte_diff + Y = (feval(fun,A'+d*E/norm(E,1),varargin{:}) - funA')/d; + else + if flag1 + Y = feval(fun_frechet,'transp',E,varargin{:}); + else + Y = feval(fun_frechet,A',E,varargin{:}); + end + end + + end + + Z(:,j) = Y(:); + end + + end + + end + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/funm_condest_fro.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,66 @@ +function c = funm_condest_fro(A,fun,fun_frechet,its,flag,varargin) +%FUNM_CONDEST_FRO Estimate of Frobenius norm condition number of matrix function. +% C = FUNM_CONDEST_FRO(A,FUN,FUN_FRECHET,ITS,FLAG) produces an estimate of +% the Frobenius norm relative condition number of function FUN at +% the matrix A. FUN and FUN_FRECHET are function handles: +% - FUN(A) evaluates the function at the matrix A. +% - If FLAG == 0 (default) +% FUN_FRECHET(B,E) evaluates the Frechet derivative at B +% in the direction E; +% if FLAG == 1 +% - FUN_FRECHET('notransp',E) evaluates the +% Frechet derivative at A in the direction E. +% - FUN_FRECHET('transp',E) evaluates the +% Frechet derivative at A' in the direction E. +% If FUN_FRECHET is empty then the Frechet derivative is approximated +% by finite differences. More reliable results are obtained when +% FUN_FRECHET is supplied. +% The power method is used, with a random starting matrix, +% so the approximation can be different each time. +% ITS iterations (default 6) are used. +% C = FUNM_COND_EST_FRO(A,FUN,FUN_FRECHET,ITS,FLAG,P1,P2,...) +% passes extra inputs P1,P2,... to FUN and FUN_FRECHET. +% Note: this function makes an assumption on the adjoint of the +% Frechet derivative that, for f having a power series expansion, +% is equivalent to the series having real coefficients. + +if nargin < 5 || isempty(flag), flag = 0; end +if nargin < 4 || isempty(its), its = 6; end +if nargin < 3 || isempty(fun_frechet), fte_diff = 1; else fte_diff = 0; end + +funA = feval(fun,A,varargin{:}); +d = sqrt( eps*norm(funA,'fro') ); +Z = randn(size(A)); +Znorm = 1; + +factor = norm(A,'fro')/norm(funA,'fro'); + +for i=1:its + + Z = Z/norm(Z,'fro'); + if fte_diff + W = (feval(fun,A+d*Z,varargin{:}) - funA)/d; + else + if flag + W = feval(fun_frechet,'notransp',Z,varargin{:}); + else + W = feval(fun_frechet,A,Z,varargin{:}); + end + end + + W = W/norm(W,'fro'); + if fte_diff + Z = (feval(fun,A'+d*W,varargin{:}) - funA')/d; + else + if flag + Z = feval(fun_frechet,'transp',W,varargin{:}); + else + Z = feval(fun_frechet,A',W,varargin{:}); + end + end + + Znorm = norm(Z,'fro'); + +end + +c = Znorm*factor;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/funm_ev.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,9 @@ +function F = funm_ev(A,fun) +%FUNM_EV Evaluate general matrix function via eigensystem. +% F = FUNM_EV(A,FUN) evaluates the function FUN at the +% square matrix A using the eigensystem of A. +% This function is intended for diagonalizable matrices only +% and can be numerically unstable. + +[V,D] = eig(A); +F = V * diag(feval(fun,diag(D))) / V;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/funm_simple.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,30 @@ +function F = funm_simple(A,fun) +%FUNM_SIMPLE Simplified Schur-Parlett method for function of a matrix. +% F = FUNM_SIMPLE(A,FUN) evaluates the function FUN at the +% square matrix A by the Schur-Parlett method using the scalar +% Parlett recurrence (and hence without blocking or reordering). +% This function is intended for matrices with distinct eigenvalues +% only and can be numerically unstable. +% FUNM should in general be used in preference. + +n = length(A); + +[Q,T] = schur(A,'complex'); % Complex Schur form. +F = diag(feval(fun,diag(T))); % Diagonal of F. + +% Compute off-diagonal of F by scalar Parlett recurrence. +for j=2:n + for i = j-1:-1:1 + s = T(i,j)*(F(i,i)-F(j,j)); + if j-i >= 2 + k = i+1:j-1; + s = s + F(i,k)*T(k,j) - T(i,k)*F(k,j); + end + d = T(i,i) - T(j,j); + if d ~= 0 + F(i,j) = s/d; + end + end +end + +F = Q*F*Q';
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/logm_cond.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,22 @@ +function c = logm_cond(A) +%LOGM_COND Relative condition number of matrix logarithm. +% LOGM_COND(A) is the relative condition number in the Frobenius +% norm of the matrix logarithm at the matrix A. + +n = length(A); +N = n^2; +K = zeros(N); +E = zeros(n); + +if ~isequal(A,triu(A)) + A = schur(A,'complex'); +end + +for j = 1:N + e = zeros(N,1); e(j) = 1; + E(:) = e; + X = logm_frechet_pade(A,E); + K(:,j) = X(:); +end + +c = norm(K) * norm(A,'fro') / norm(logm(A),'fro');
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/logm_frechet_pade.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,47 @@ +function L = logm_frechet_pade(A,E) +%LOGM_FRECHET_PADE Frechet derivative of matrix logarithm via Pade approx. +% L = LOGM_FRECHET_PADE(A,E) evaluates the Frechet +% derivative of the matrix logarithm at A in the direction E via the +% inverse scaling and squaring and a Pade approximant of the function +% tanh(x)/x. A must have no eigenvalues on the negative real axis. + +real_data = isreal(A) && isreal(E); +% Form complex Schur form if A not already upper triangular. +use_Schur = false; +if ~isequal(A,triu(A)) + use_Schur = true; + [Q,T] = schur(A,'complex'); A = T; E = Q'*E*Q; +end + +I = eye(size(A)); +B = A; +for i = 0:inf + if norm(B-I,1) <= 1-1/exp(1), s = i; break, end + B = sqrtm(B); + Aroot{i+1} = B; +end + +% Positive zeros of p8 and q8 in r8 = p8/q8 Pade approximant. +load tau_r8_zeros +% Zeros come in \pm pairs. +a = complex(0, [pzero; -pzero]); +b = complex(0, [qzero; -qzero]); + +E = 2^s*E; +for i = 1:s + E = sylvsol(Aroot{i},Aroot{i},E); +end + +G = sylvsol(B,B,E); + +X = logm(B); + +for i=8:-1:1 + rhs = (I + X/b(i)) * G + G * (I - X/b(i)); + AA = I + X/a(i); BB = I - X/a(i); + G = sylvsol(AA, BB, rhs); +end + +L = 2*G; +if use_Schur, L = Q*L*Q'; end +if real_data, L = real(L); end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/logm_iss.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,44 @@ +function [X,k,m] = logm_iss(A) +%LOGM_ISS Matrix logarithm by inverse scaling and squaring method. +% X = LOGM_ISS(A) computes the logarithm of A, for a matrix with no +% nonpositive real eigenvalues, using the inverse scaling and squaring +% method with Pade approximation. +% Matrix square roots are computed by the product form of the +% Denman-Beavers teration. +% [X,K,M] = LOGM_ISS(A) returns the number K of square roots +% computed and the degree M of the Pade approximant. + +e = eig(A); +if any( imag(e) == 0 & real(e) <= 0 ) + error('A must not have any nonpositive real eigenvalues!') +end + +n = length(A); + +load log_pade_err_opt % mmax-by-3 matrix DATA. +% mvals = data(:,1); +xvals = data(:,2); + +X = A; +k = 0; p = 0; itk = 5; + +while 1 + + normdiff = norm(X-eye(n),1); + if normdiff <= xvals(16) + + p = p+1; + j1 = find(normdiff <= xvals(3:16)); + j1 = j1(1) + 2; + j2 = find(normdiff/2 <= xvals(3:16)); + j2 = j2(1) + 2; + if 2*(j1-j2)/3 < itk || p == 2, m = j1; break, end + + end + + [X,M,itk] = sqrtm_dbp(X,1); k = k+1; + +end + +X = 2^k*logm_pade_pf(X-eye(n),m); +if isreal(A), X = real(X); end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/logm_pade_pf.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,32 @@ +function S = logm_pade_pf(A,m) +%LOGM_PADE_PF Evaluate Pade approximant to matrix log by partial fractions. +% Y = LOGM_PADE_PF(A,M) evaluates the [M/M] Pade approximation to +% LOG(EYE(SIZE(A))+A) using a partial fraction expansion. + +[nodes,wts] = gauss_legendre(m); +% Convert from [-1,1] to [0,1]. +nodes = (nodes + 1)/2; +wts = wts/2; + +n = length(A); +S = zeros(n); + +for j=1:m + S = S + wts(j)*(A/(eye(n) + nodes(j)*A)); +end + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function [x,w] = gauss_legendre(n) +%GAUSS_LEGENDRE Nodes and weights for Gauss-Legendre quadrature. +% [X,W] = GAUSS_LEGENDRE(N) computes the nodes X and weights W +% for N-point Gauss-Legendre quadrature. + +% Reference: +% G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature +% rules, Math. Comp., 23(106):221-230, 1969. + +i = 1:n-1; +v = i./sqrt((2*i).^2-1); +[V,D] = eig( diag(v,-1)+diag(v,1) ); +x = diag(D); +w = 2*(V(1,:)'.^2);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/mft_test.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,223 @@ +function mft_test(n) +%MFT_TEST Test the Matrix Function Toolbox. +% MFT_TEST(N) tests most of the functions in the +% Matrix Function Toolbox on (mainly) random matrices of order N. +% The default is N = 8. +% For a run time of a few seconds choose a small +% value of N (such as the default). +% Each invocation uses different random matrices. +% This is not a thorough suite of tests. + +% For N much larger than 10 it may be necssary to adjust the fudge factors +% in this function and in MFT_TOLERANCE in order to achieve successful +% completion. + +if nargin < 1, n = 8; end + +fudge_factor = 200; +A = randn(n); +E = randn(n); +B = A^2; % No eigenvalues on nonpositive real axis. +m = 2*n; +c = randn(m,1); +Ap = A'*A; Bp = A'*A; % Positive definite. + +C = randn(n); +X = sylvsol(A,E,C); +assert_small( (A*X+X*E-C)/((norm(A,1)+norm(E,1))*norm(X,1) + norm(C,1))) +assert_eq(real(X),X) + +T = triu(randn(n)); U = triu(randn(n)); +X = sylvsol(T,U,C); +assert_eq(real(X),X) +assert_small( (T*X+X*U-C)/((norm(T,1)+norm(U,1))*norm(X,1) + norm(C,1))) + +if license('test','Symbolic_Toolbox') && ~isempty(ver('symbolic')) + % If Symbolic Math Toolbox licensed and installed... + for p = 2:6 + J = gallery('jordbloc',p,0); + [d,a] = ascent_seq(J); + assert_eq(d,ones(p,1)) + assert_eq(a,(0:p)') + end +end + +A1 = 2.5*A/sqrt(norm(A^2,inf)); +for k = 1:3 % Try different norms. + C1 = cosm(A1); + [C,S] = cosmsinm(A1); + C0 = funm(A1,@cos); + S0 = funm(A1,@sin); + assert_sim(C,C1) + assert_sim(C,C0) + assert_sim(S,S0) + assert_small( (C^2+S^2-eye(n)) / (norm(C,1)^1+norm(S,1)^2) ) + A1 = 2*A1; +end + +L = expm_frechet_pade(A,E); +R = expm_frechet_quad(A,E); +assert_sim(L,R,1e-2) + +% Check identity $L_{\exp}\bigl(\log(A), L_{\log}(A,E)\bigr) = E$. +L_log = logm_frechet_pade(B,E); +L_exp = expm_frechet_pade(logm(B),L_log); +assert_sim(L_exp,E,sqrt(eps)*norm(E,1)) + +m = n-1; +E = eye(m); +E = E(:,m); +[Q,H] = arnoldi(A,randn(n,1),m); +res = A*Q(:,1:m) - Q(:,1:m)*H(1:m,1:m) - H(m+1,m)*Q(:,m+1)*E'; +assert_small(res/(norm(H,1)*norm(Q,1))) + +b = randn(n,1); y = fab_arnoldi(A,b,@exp,n); +assert_sim(y,expm(A)*b) + +X = logm(B); +X1 = logm_iss(B); +tol = funm_condest1(B,@logm,@logm_frechet_pade)*eps*n; +assert_sim(X,X1,tol) + +c1 = expm_cond(A); +[c1a,K] = expm_cond(A); +assert_sim(c1,c1a) +st = randn('state'); +c2 = funm_condest_fro(A,@expm,@expm_frechet_pade); +randn('state',st); +c3 = funm_condest_fro(A,@expm,@fun_frechet_exp,[],1); +assert_sim(c2,c3,0.5) +fudge_factor1 = 2; +if c2 < c1/10 || c2 > c1*fudge_factor1, [c2 c1 c2/c1], error('Failure'), end +if c3 < c1/10 || c3 > c1*fudge_factor1, [c3 c1 c3/c1], error('Failure'), end + +c1 = funm_condest1(A,@expm); +st = rand('twister'); +c2 = funm_condest1(A,@expm,@expm_frechet_pade); +assert_sim(c1,c2,1) +rand('twister',st); +c3 = funm_condest1(A,@expm,@fun_frechet_exp,1); +assert_sim(c2,c3,0.5) + +c1 = funm_condest_fro(B,@logm); +c2 = funm_condest_fro(B,@logm,@logm_frechet_pade); +assert_sim(c1,c2,1) + +[U1,H1] = polar_newton(A); +assert_small((A-U1*H1)/norm(A,1)) +[U2,H2] = polar_svd(A); +assert_small((A-U2*H2)/norm(A,1)) +assert_sim(U1,U2) +assert_small((H1-H2)/norm(H1,1)) +assert_small(U1'*U1-eye(n)) +assert_small(U2'*U2-eye(n)) +assert_eq(H1,H1') +assert_eq(H2,H2') + +A2 = randn(n+2,n); +for i = 1:4 + [U,H] = polar_svd(A2); + assert_small((A2-U*H)/norm(A2,1)) + assert_small(U1'*U1-eye(n)) + assert_eq(H,H') + A2 = A2'; + if i == 2, A2(:,round(n/2)) = 0; end +end + +P1 = polyvalm_ps(c,A); +P2 = polyvalm(c,A); +assert_small((P1-P2)/norm(P1,1)) + +assert_sim(power_binary(A,m),A^m) + +X = riccati_xaxb(Ap,Bp); +assert_small( (X*Ap*X-Bp) / (norm(X,1)^2*norm(Ap,1)+norm(Bp,1)) ) +assert_eq(X,X') + +for p = [2 5 10 16] + X = rootpm_real(B,p); assert_small( (X^p-B)/(norm(X,1)^p + norm(B,1)) ); + X = rootpm_sign(B,p); assert_small( (X^p-B)/(norm(X,1)^p + norm(B,1)) ); + [X,Y] = rootpm_schur_newton(B,p); + assert_small( (X^p-B)/(norm(X,1)^p + norm(B,1)) ) + assert_small( (X*Y - eye(n))/(norm(X,1)*norm(Y,1)) ) +end + +[S,N] = signm(A); +assert_small( (S^2 - eye(n))/(norm(S,1)^2+1) ) +assert_small( (A-S*N)/(norm(A,1)+norm(S,1)*norm(N,1)) ) + +[X0,alpha,condest] = sqrtm(B); +tol = n*norm(X0,1)*condest*eps; + +[P,Q] = sqrtm_db(B); +assert_small( (P^2 - B)/(norm(P,1)^2+norm(B,1)) ) +assert_small( (Q^2 - inv(B))/(norm(Q,1)^2+norm(inv(B),1)) ) +assert_small(X0-P, tol) +[P,Q] = sqrtm_dbp(B); +assert_small( (P^2 - B)/(norm(P,1)^2+norm(B,1)) ) +assert_small( Q - eye(n) ) +assert_small(X0-P, tol) + +% Following usually succeeds but can fail: +% only local cgce conditions are known for full Newton. +% X = sqrtm_newton_full(B); +% assert_small( (X^2 - B)/(norm(X,1)^2+norm(B,1)) ) + +X = sqrtm_pd(Ap); +assert_small( (X^2 - Ap)/(norm(X,1)^2+norm(Ap,1)) ) +assert_eq(X,X') + +C = full(gallery('tridiag',n,1,4,1)); +D = diag(diag(C)); +X = sqrtm_pulay(C,D); +assert_small( (X^2 - C)/(norm(X,1)^2+norm(C,1)) ) + +X = sqrtm_real(B); +assert_small( (X^2 - B)/(norm(X,1)^2+norm(B,1)) ) +assert_small(X0-P, tol) + +T = schur(A,'complex'); +R = sqrtm_triang_min_norm(T); +assert_small( (R^2 - T)/(norm(R,1)^2+norm(T,1)) ) + +fprintf(['MFT_TEST: All tests of the Matrix Function Toolbox passed' ... + ' (n = %g).\n'], n) + + % Nested functions + + function L = fun_frechet_exp(flag,E) + % Frechet derivative of exponential. + + if strcmp(flag,'transp'), E = E'; end + L = expm_frechet_pade(A,E); + if strcmp(flag,'transp'), L = L'; end + + end + + % --------------------------------------------------------- + % Assertion functions. + + function assert_sim(a,b,tol) + if nargin < 3, tol = fudge_factor*eps(superiorfloat(a,b))*length(a); end + if norm(a-b,1)/max( norm(a,1), norm(b,1) ) > tol + fprintf('%9.2e, %9.2e\n', ... + norm(a-b,1)/max( norm(a,1), norm(b,1)), tol ) + error('Failure') + end + end + + function assert_small(a,tol) + if nargin < 2, tol = fudge_factor*eps(class(a))*length(a); end + if norm(a,1) > tol + fprintf('%9.2e, %9.2e\n',norm(a,1), tol), + error('Failure') + end + end + + function assert_eq(a,b) + if norm(a-b,1) + error('Failure') + end + end + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/mft_tolerance.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,8 @@ +function tol = mft_tolerance(A) +%MFT_TOLERANCE Convergence tolerance for matrix iterations. +% TOL = MFT_TOLERANCE(A) returns a convergence tolerance to use in +% the matrix iterations in the Matrix Function Toolbox applied to the +% matrix A. All functions in the toolbox call this function to set +% the convergence tolerance. + +tol = sqrt(length(A))*eps/2;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/polar_newton.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,43 @@ +function [U,H,k] = polar_newton(A) +%POLAR_NEWTON Polar decomposition by scaled Newton iteration. +% [U,H,k] = POLAR_NEWTON(A), where the matrix A is square and +% nonsingular, computes a unitary U and a Hermitian positive +% definite H such that A = U*H. k is the number of iterations. +% A Newton iteration with acceleration parameters is used. + +accel_tol = 1e-2; % Precise value not important. +need_accel = 1; + +tol = mft_tolerance(A); + +X = A; +reldiff = inf; +maxit = 16; + +for k = 1:maxit + + Xold = X; + Xinv = inv(X); + if need_accel + g = ( norm(Xinv,1)*norm(Xinv,inf) / (norm(X,1)*norm(X,inf)) )^(1/4); + else + g = 1; + end + X = 0.5*(g*X + Xinv'/g); + reldiff_old = reldiff; + diff_F = norm(X-Xold,'fro'); + reldiff = diff_F/norm(X,'fro'); + + if need_accel && (reldiff < accel_tol), need_accel = false; end + cged = (diff_F <= sqrt(tol)) || (reldiff > reldiff_old/2 && ~need_accel); + if cged, break, end + + if k == maxit, error('Not converged after %2.0f iterations', maxit), end + +end + +U = X; +if nargout >= 2 + H = U'*A; + H = (H + H')/2; % Force Hermitian by taking nearest Hermitian matrix. +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/polar_svd.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,18 @@ +function [U,H] = polar_svd(A) +%POLAR_SVD Canonical polar decomposition via singular value decomposition. +% [U,H] = POLAR_SVD(A) computes a matrix U of the same dimension +% (m-by-n) as A, and a Hermitian positive semi-definite matrix H, +% such that A = U*H. +% U is a partial isometry with range(U^*) = range(H). +% If A has full rank then U has orthonormal columns if m >= n +% and orthonormal rows if m <= n. +% U and H are computed via an SVD of A. + +[P,S,Q] = svd(A,'econ'); +U = P*Q'; +r = sum( diag(S) > norm(A,1)*eps/2 ); +U = P(:,1:r)*Q(:,1:r)'; +if nargout == 2 + H = Q*S*Q'; + H = (H + H')/2; % Force Hermitian by taking nearest Hermitian matrix. +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/polyvalm_ps.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,48 @@ +function [P,s,cost] = polyvalm_ps(c,A,s) +%POLYVALM_PS Evaluate polynomial at matrix argument by Paterson-Stockmeyer alg. +% [P,S,COST] = POLYVALM_PS(C,A,S) evaluates the polynomial whose +% coefficients are the vector C at the matrix A using the +% Paterson-Stockmeyer algorithm. If omitted, the integer parameter +% S is chosen automatically and its value is returned as an +% output argument. COST is the number of matrix multiplications used. + +m = length(c)-1; % Degree of poly. +c = c(end:-1:1); c = c(:); +n = length(A); + +if nargin < 3 + % Determine optimum parameter s. + s = ceil(sqrt(m)); +end +r = floor(m/s); +cost = s+r-(m==r*s)-1; + +% Apower{i+1} = A^i; +Apower = cell(s+1); +Apower{1} = eye(n); +for i=2:s+1 + Apower{i} = A*Apower{i-1}; +end + +B = cell(r+1); +for k=0:r-1 + temp = c(s*k+1)*eye(n); + for j=1:s-1 + temp = temp + c(s*k+j+1)*Apower{j+1}; + end + B{k+1} = temp; +end +B{r+1} = c(m+1)*Apower{m-s*r+1}; +for j=m-1:-1:s*r + if j == s*r + B{r+1} = B{r+1} + c(s*r+1)*eye(n); + else + B{r+1} = B{r+1} + c(j+1)*Apower{m-s*r-(m-j)+1}; + end +end + +As = Apower{s+1}; +P = zeros(n); +for k=r:-1:0 + P = P*As + B{k+1}; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/power_binary.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,22 @@ +function X = power_binary(A,m) +%POWER_BINARY Power of matrix by binary powering (repeated squaring). +% X = POWER_BINARY(A,m) computes A^m for a square matrix A and a +% positive integer m, by binary powering. + +s = double(dec2bin(m)) - 48; % Binary representation of s in double array. +k = length(s); + +P = A; +i = k; +while s(i) == 0 + P = P^2; + i = i-1; +end +X = P; +for j = i-1:-1:1 + P = P^2; + if s(j) == 1 + X = X*P; + end + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/quasitriang_struct.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,31 @@ +function [m, s, k] = quasitriang_struct(R) +%QUASITRIANG_STRUCT Block structure of upper quasitriangular matrix. +% [M,S,K] = QUASITRIANG_STRUCT(R), where R is an upper +% quasitriangular matrix, determines that R has M diagonal blocks, +% the i'th of which has order S(i) and starting position K(i). +% Any subdiagonal elements less than the tolerance EPS*NORM(R,'FRO') +% are treated as zero. + +n = length(R); +tol = eps*norm(R,'fro'); + +i = 1; j = 1; + +while i < n + k(j) = i; + if abs(R(i+1,i)) <= tol + s(j) = 1; + else + s(j) = 2; + end + i = i + s(j); + j = j+1; +end + +if i == n + k(j) = n; + s(j) = 1; + m = j; +else + m = j-1; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/readme.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,18 @@ +echo on +% Welcome to the Matrix Function Toolbox. +% The primary source for this toolbox is +% +% http://www.ma.man.ac.uk/~higham/mftoolbox +% +% The toolbox comprises the M-files in this directory. +% It accompanies the book +% +% Nicholas J. Higham, Functions of Matrices: Theory and Computation, +% SIAM, Philadelphia, PA, USA, 2008. ISBN 978-0-898716-46-7, +% +% which is the documentation for the toolbox. +% In particular, Appendix D of the book describes the toolbox. +% +% For a descriptive list of M-files in the toolbox type +% help mft +echo off
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/riccati_xaxb.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,15 @@ +function X = riccati_xaxb(A,B) +%RICCATI_XAXB Solve Riccati equation XAX = B in positive definite matrices. +% X = RICCATI_XAXB(A,B) is the Hermitian positive definite +% solution to XAX = B, where A and B are Hermitian positive +% definite matrices. + +R = chol(A); +S = chol(B); + +U = polar_newton(S*R'); +X = R\(U'*S); +X = (X + X')/2; + +% [U,H] = polar_newton(R*S'); % Variant derived in solution of Problem 6.21. +% X = R\(U*S);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/rootpm_newton.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,29 @@ +function [X,k] = rootpm_newton(A,p,c) +%ROOTPM_NEWTON Coupled Newton iteration for matrix pth root. +% [X,k] = ROOTPM_NEWTON(A,P,C) computes the principal +% Pth root of the matrix A. +% C (default 1) is a convergence parameter. +% k is the number of iterations. + +if nargin < 3, c = 1; end + +n = length(A); +M = A/c^p; +X = c*eye(n); +tol = mft_tolerance(A); +maxit = 20; + +relres = inf; + +for k=1:maxit + + X = ( ((p+1)*eye(n) - M)/p )\X; + M = power_binary( ((p+1)*eye(n) - M)/p, p) * M; + + relres_old = relres; + relres = norm(M-eye(n),inf); + + if relres <= tol || relres > relres_old/2, return, end + +end +error('Not converged after %2.0f iterations', maxit)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/rootpm_real.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,99 @@ +function X = rootpm_real(A,p) +%ROOTPM_REAL Pth root of real matrix via real Schur form. +% X = ROOTPM_REAL(A,P) is a Pth root of the nonsingular matrix A +% (A = X^P). When A has no eigenvalues on the negative real axis +% then X is the principal pth root, which is real when A is real. + +% Implementation is complicated by use of zero (and even "-1") +% subscripts in algorithm. Dealt with by increasing "k" by 1 +% each time and moving "-1" cases in front of loops. + +if norm(imag(A),1), error('A must be real.'), end +if p == 1, X = A; return; end + +n = length(A); +[Q,R] = schur(A,'real'); + +% m blocks: i'th has order s(i), starts at t(i). +[m,s,t] = quasitriang_struct(R); +U = zeros(n); B = cell(p); V = cell(p-1); +V{1} = eye(n); % U^k == V{k+1}. + +for j=1:m + rj = t(j):t(j)+s(j)-1; + if s(j) == 1 + % If A is real, X will be real unless R(p,p)<0 on the next line. + U(rj,rj) = R(rj,rj)^(1/p); + else + U(rj,rj) = root_block(R(rj,rj),p); + end + for k = 0:p-2, kk = k+1; V{kk}(rj,rj) = U(rj,rj)^(k+1); end + for i=j-1:-1:1 + ri = t(i):t(i)+s(i)-1; + for k = 0:p-2 + kk = k+1; + B{kk} = zeros(s(i),s(j)); + for ell = i+1:j-1 + rell = t(ell):t(ell)+s(ell)-1; + B{kk} = B{kk} + U(ri,rell)*V{kk}(rell,rj); + end + end + rhs = R(ri,rj) - B{p-2 +1}; + for k = 0:p-3 + rhs = rhs - V{p-3-k +1}(ri,ri)*B{k+1}; + end + coeff = kron( eye(s(j)), V{p-2 +1}(ri,ri) ) ... + + kron( V{p-2 +1}(rj,rj).', eye(s(i)) ); + for k = 1:p-2 + coeff = coeff + kron( V{k-1 +1}(rj,rj).', V{p-2-k +1}(ri,ri) ); + end + y = coeff\rhs(:); + rhs(:) = y; % `Un-vec' the solution. + U(ri,rj) = rhs; + for k = 0:p-2 + if k == 0 + S = U(ri,rj); + else + S = V{k-1 +1}(ri,ri)*U(ri,rj) + U(ri,rj)*V{k-1 +1}(rj,rj) ... + + B{k-1 +1}; + for ell = 1:k-1 + S = S + V{k-ell-1 +1}(ri,ri)*U(ri,rj)*V{ell-1 +1}(rj,rj); + end + for ell = 0:k-2 + S = S + V{k-2-ell +1}(ri,ri)*B{ell +1}; + end + end + V{k+1}(ri,rj) = S; + end + end +end + +X = Q*U*Q'; + +%%%%%%%%%%%%%%%%%%%%%%%%%%%% +function X = root_block(R,p) +%ROOT_BLOCK Pth root of a real 2x2 matrix with complex conjugate eigenvalues. + +if norm(imag(R)) ~= 0 || any(size(R) - [2,2]) + error('Matrix must be real, of dimension 2.') +end + +r11 = R(1,1); r12 = R(1,2); +r21 = R(2,1); r22 = R(2,2); + +theta = (r11 + r22) / 2; +musq = (-(r11 - r22)^2 - 4*r21*r12) / 4; +mu = sqrt(musq); + +if musq <= 0 + error('Matrix must have non-real complex conjugate eigenvalues.') +end + +r = sqrt(theta^2+musq); +phi = angle(complex(theta,mu)); +rootp = r^(1/p)*exp(i*phi/p); + +alpha = real(rootp); +beta = imag(rootp); + +X = alpha*eye(2) + (beta/mu)*(R - theta*eye(2));
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/rootpm_schur_newton.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,66 @@ +function [X,Y] = rootpm_schur_newton(A,p) +%ROOTPM_SCHUR_NEWTON Matrix pth root by Schur-Newton method. +% [X,Y] = ROOTPM_SCHUR_NEWTON(A,p) computes the principal pth root +% X of the real matrix A using the Schur-Newton algorithm. +% It also returns Y = INV(X). + +if norm(imag(A),1), error('A must be real.'), end +A = real(A); % Discard any zero imaginary part. +[Q,R] = schur(A,'real'); % Quasitriangular R. + +e = eig(R); +if any (e(find(e == real(e))) < 0 ) + error('A has a negative real eigenvalue: principal pth root not defined') +end + +f = factor(p); +k0 = length(find(f == 2)); % Number of factors 2. +q = p/2^(k0); +k1 = k0; +if q > 1 + + emax = max(abs(e)); emin = min(abs(e)); + if emax > emin % Avoid log(0). + k1 = max(k1, ceil( log2( log2(emax/emin) ) )); + end + + max_arg = norm(angle(e),inf); + if max_arg > pi/8 + k3 = 1; + if max_arg > pi/2 + k3 = 3; + elseif max_arg > pi/4 + k3 = 2; + end + k1 = max(k1,k3); + end + +end + +for i = 1:k1, R = sqrtm_real(R); end + + +if q ~= 1 + pw = 2^(-k1); emax = emax^pw; emin = emin^pw; + if ~any(imag(e)) + % Real eigenvalues. + if emax > emin + alpha = emax/emin; + c = ( (alpha^(1/q)*emax-emin)/( (alpha^(1/q)-1)*(q+1) ) )^(1/q); + else + c = emin^(1/q); + end + else + % Complex eigenvalues. + c = (( emax+emin)/2 )^(1/q); + end + + X = rootpm_newton(R,q,c); + for i = 1:k1-k0 + X = X*X; + end +else + X = R; +end +Y = Q*(X\Q'); % Return inverse pth root, too. +X = Q*X*Q';
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/rootpm_sign.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,47 @@ +function X = rootpm_sign(A,p) +%ROOTPM_SIGN Matrix Pth root via matrix sign function. +% X = ROOTPM_SIGN(A,P) computes the principal Pth root X +% of the matrix A using the matrix sign function and a block +% companion matrix approach. + +if p == 1, X = A; return, end + +n = length(A); +podd = rem(p,2); + +Y = A; + +if podd + p = 2*p; % Compensate by squaring at end. +else + while mod(p,4) == 0 + Y = sqrtm(Y); + p = round(p/2); + end +end + +if p == 2 + X = sqrtm(Y); + return +end + +% Form C, the block companion matrix. +C = zeros(p*n); +C(end-n+1:end, 1:n) = Y; +C = C + diag(ones(n*(p-1),1),n); + +S = signm(C); + +X = S(n+1:2*n,1:n); + +% Scale factor. +c = 0; +for l = 1:floor(p/4) + c = c+cos(2*pi*l/p); +end +c = c*4+2; +c = c/p; +X = X/c; +if podd + X = X*X; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/signm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,47 @@ +function [S,N] = signm(A) +%SIGNM Matrix sign decomposition. +% [S,N] = SIGNM(A) is the matrix sign decomposition A = S*N, +% computed via the Schur decomposition. +% S is the matrix sign function, sign(A). + +[Q, T] = schur(A,'complex'); +S = Q * matsignt(T) * Q'; + +if nargout == 2 + N = S*A; +end + +%%%%%%%%%%%%%%%%%%%%%%%% +function S = matsignt(T) +%MATSIGNT Matrix sign function of a triangular matrix. +% S = MATSIGN(T) computes the matrix sign function S of the +% upper triangular matrix T using a recurrence. + +n = length(T); +S = diag( sign( diag(real(T)) ) ); +for p = 1:n-1 + for i = 1:n-p + + j = i+p; + d = T(j,j) - T(i,i); + + if S(i,i) ~= -S(j,j) % Solve via S^2 = I if we can. + + % Get S(i,j) from S^2 = I. + k = i+1:j-1; + S(i,j) = -S(i,k)*S(k,j) / (S(i,i)+S(j,j)); + + else + + % Get S(i,j) from S*T = T*S. + s = T(i,j)*(S(j,j)-S(i,i)); + if p > 1 + k = i+1:j-1; + s = s + T(i,k)*S(k,j) - S(i,k)*T(k,j); + end + S(i,j) = s/d; + + end + + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/signm_newton.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,53 @@ +function [X,k] = signm_newton(A,scal) +%SIGNM_NEWTON Matrix sign function by Newton iteration. +% [S,k] = SIGNM_NEWTON(A,SCAL) computes the matrix sign +% function S of A using the scaled Newton iteration, with +% scale factors specified by SCAL: +% SCAL = 0: no scaling. +% SCAL = 1: determinantal scaling (default). +% SCAL = 2: spectral scaling. +% SCAL = 3: norm scaling. +% k is the number of iterations. + +if nargin < 2, scal = 1; end + +tol = mft_tolerance(A); +accel_tol = 1e-2; % Precise value not important. +need_accel = 1; +n = length(A); +maxit = 16; + +X = A; +reldiff = inf; + +for k = 1:maxit + + Xold = X; + Xinv = inv(X); + if need_accel && scal > 0 + % In practice should estimate spectral radius and 2-norms; + % here they are computed exactly. + switch scal + case 1 + g = abs(det(X))^(-1/n); + case 2 + s1 = max(abs(eig(Xinv))); + s2 = max(abs(eig(X))); + g = sqrt(s1/s2); + case 3 + g = sqrt( norm(Xinv) / (norm(X)) ); + end + X = g*X; Xinv = Xinv/g; + end + X = 0.5*(X + Xinv); + diff_F = norm(X-Xold,'fro'); + reldiff_old = reldiff; + reldiff = diff_F/norm(X,'fro'); + + if need_accel && (reldiff < accel_tol), need_accel = false; end + cged = (diff_F <= sqrt( tol*norm(X)/norm(Xinv) ) || ... + reldiff > reldiff_old/2 && ~need_accel); + if cged, return, end + +end +error('Not converged after %2.0f iterations', maxit)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sqrtm_db.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,43 @@ +function [P,Q,k] = sqrtm_db(A,scale) +%SQRTM_DB Matrix square root by Denman-Beavers iteration. +% [P,Q,k] = SQRTM_DB(A,SCAL) computes the principal square root +% P of the matrix A using the Denman-Beavers iteration. +% It also returns Q = INV(P). +% SCAL specifies the scaling: +% SCAL == 0, no scaling. +% SCAL == 1, determinantal scaling (default). +% k is the number of iterations. + +n = length(A); +if nargin < 2, scale = 1; end + +tol = mft_tolerance(A); +P = A; +Q = eye(n); +reldiff = inf; +maxit = 25; + +for k = 1:maxit + + if scale == 1 + g = (abs(det(P)*det(Q)))^(-1/(2*n)); + P = g*P; Q = g*Q; + end + + Pold = P; + + Poldinv = inv(Pold); + P = (P + inv(Q))/2; + Q = (Q + inv(Pold))/2; + + diff_F = norm(P-Pold,'fro'); + reldiff_old = reldiff; + reldiff = diff_F/norm(P,'fro'); + if reldiff < 1e-2, scale = 0; end % Switch to no scaling. + + cged = (diff_F <= sqrt( tol*norm(P)/norm(Poldinv) ) || ... + reldiff > reldiff_old/2 && ~scale); + if cged, return, end + +end +error('Not converged after %2.0f iterations', maxit)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sqrtm_dbp.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,39 @@ +function [X,M,k] = sqrtm_dbp(A,scale) +%SQRTM_DBP Matrix square root by product form of Denman-Beavers iteration. +% [X,M,k] = SQRTM_DBP(A,SCAL) computes the principal square root X +% of the matrix A using the product form of the Denman-Beavers +% iteration. The matrix M tends to EYE. +% SCAL specifies the scaling: +% SCAL == 0, no scaling. +% SCAL == 1, determinantal scaling (default). +% k is the number of iterations. + +n = length(A); +if nargin < 2, scale = 1; end + +tol = mft_tolerance(A); +X = A; +M = A; +maxit = 25; + +for k = 1:maxit + + if scale == 1 + g = (abs(det(M)))^(-1/(2*n)); + X = g*X; M = g^2*M; + end + + Xold = X; invM = inv(M); + + X = X*(eye(n) + invM)/2; + M = 0.5*(eye(n) + (M + invM)/2); + + Mres = norm(M - eye(n),'fro'); + + reldiff = norm(X - Xold,'fro')/norm(X,'fro'); + if reldiff < 1e-2, scale = 0; end % Switch to no scaling. + + if Mres <= tol, return; end + +end +error('Not converged after %2.0f iterations', maxit)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sqrtm_newton.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,36 @@ +function [X,k] = sqrtm_newton(A,scal,maxit) +%SQRTM_NEWTON Matrix square root by Newton iteration (unstable). +% [X,k] = SQRTM_NEWTON(A,SCAL,MAXIT) computes the principal +% square root X of the matrix A by an unstable Newton iteration. +% SCAL specifies whether scaling is used (SCAL = 1) or not +% (SCAL = 0, default). +% MAXIT (default 25) is the maximum number of iterations allowed. +% k is the number of iterations. + +n = length(A); +if nargin < 2, scal = 0; end +if nargin < 3, maxit = 25; end + +tol = mft_tolerance(A); +accel_tol = 1e-2; % Precise value not important. +need_accel = 1; + +X = A; + +for k = 1:maxit + + if need_accel && scal > 0 + mu = (abs(det(X))/sqrt(abs(det(A))))^(-1/n); + X = mu*X; + end + + Xold = X; + X = (X + X\A)/2; + + reldiff = norm(X-Xold,'fro')/norm(X,'fro'); + if need_accel && (reldiff < accel_tol), need_accel = false; end + + if reldiff <= tol, return, end + +end +error('Not converged after %2.0f iterations', maxit)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sqrtm_newton_full.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,26 @@ +function [X,k] = sqrtm_newton_full(A, X0) +%SQRTM_NEWTON_FULL Matrix square root by full Newton method. +% [X,K] = SQRTM_NEWTON_FULL(A, X0) applies Newton's method to +% compute a square root X of the matrix A, with starting matrix X0. +% Default: X0 = A. K is the number of iterations. + +if nargin < 2, X0 = A; end + +X = X0; +tol = mft_tolerance(A); +maxit = 50; + +for k = 1:maxit + + Xold = X; + R = A - X^2; + % Solve XE + EX = R. + E = sylvsol(X,X,R); + X = X + E; + + reldiff = norm(X - Xold,inf)/norm(X,inf); + + if reldiff <= tol; return; end + +end +error('Not converged after %2.0f iterations', maxit)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sqrtm_pd.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,10 @@ +function [X,k] = sqrtm_pd(A) +%SQRTM_PD Square root of positive definite matrix via polar decomposition. +% [X,K] = SQRT_PD(A) computes the Hermitian positive definite +% square root X of the Hermitian positive definite matrix A. +% It computes the Hermitian polar factor of the Cholesky factor of A. +% K is the number of Newton polar iterations used. + +R = chol(A); +[U,H,k] = polar_newton(R); +X = H;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sqrtm_pulay.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,35 @@ +function [X,k] = sqrtm_pulay(A,D) +%SQRTM_PULAY Matrix square root by Pulay iteration. +% [X,K] = SQRTM_PULAY(A,D) computes the principal square root of the +% matrix A using the Pulay iteration with diagonal matrix D +% (default: D = DIAG(DIAG(A))). D must have positive diagonal entries. +% K is the number of iterations. +% Note: this iteration is linearly converent and converges only when +% SQRTM(D) sufficiently well approximates SQRTM(A). + +if nargin < 2, D = diag(diag(A)); end + +if any(diag(D)<0) || ~isreal(D) + error('D must have positive, real diagonal.') +end + +n = length(A); +dhalf = sqrt(diag(D)); +Dhalf = diag(dhalf); +B = zeros(n); +maxit = 50; + +tol = mft_tolerance(A); + +for k = 1:maxit + + Bold = B; + B = (A - D - Bold^2) ./ (dhalf(:,ones(1,n)) + dhalf(:,ones(1,n))'); + X = Dhalf + B; + + reldiff = norm(B - Bold,inf)/norm(X,inf); + + if reldiff <= tol, return, end + +end +error('Not converged after %2.0f iterations', maxit)
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sqrtm_real.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,62 @@ +function X = sqrtm_real(A) +%SQRTM_REAL Square root of real matrix by real Schur method. +% X = SQRTM_REAL(A) is the principal square root of the real matrix A, +% computed in real arithmetic by the real Schur method. +% X is real unless A has a real negative eigenvalue +% (in this case a real primary square root does not exist). + +if norm(imag(A),1), error('A must be real.'), end +A = real(A); % Discard any zero imaginary part. +n = length(A); +[Q,R] = schur(A,'real'); % Quasitriangular R. + +% m blocks: i'th has order s(i), starts at t(i). +[m, s, k] = quasitriang_struct(R); +T = zeros(n); + +for j=1:m + p = k(j):k(j)+s(j)-1; + if s(j) == 1 + % If A is real, X will be real unless R(p,p)<0 on the next line. + T(p,p) = sqrt(R(p,p)); + else + T(p,p) = rsqrt2(R(p,p)); + end + for r=j-1:-1:1 + rind = k(r):k(r)+s(r)-1; + rj = k(r+1):k(j)-1; + if ~isempty(rj) + prod = T(rind,rj)*T(rj,p); % Gives [] when rj = []. + else + prod = zeros(s(r),s(j)); + end + B = R(rind,p) - prod; + % NB Unconjugated transpose on next line for complex case. + A = kron( eye(s(j)), T(rind,rind) ) + kron( T(p,p).', eye(s(r)) ); + y = A\B(:); + B(:) = y; % `Un-vec' the solution. + T(rind,p) = B; + end +end + +X = Q*T*Q'; + +%%%%%%%%%%%%%%%%%%%%%% +function X = rsqrt2(R) +%RSQRT2 Real square root of a real 2x2 matrix with complex conjugate +% eigenvalues. + +r11 = R(1,1); r12 = R(1,2); +r21 = R(2,1); r22 = R(2,2); + +theta = (r11 + r22) / 2; +musq = (-(r11 - r22)^2 - 4*r21*r12) / 4; +mu = sqrt(musq); + +if theta > 0 + alpha = sqrt( (theta + sqrt(theta^2+musq))/2 ); +else + alpha = mu / sqrt( 2*(-theta + sqrt(theta^2+musq)) ); +end + +X = (alpha-theta/(2*alpha)) * eye(2) + R/(2*alpha);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sqrtm_triang_min_norm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,25 @@ +function R = sqrtm_triang_min_norm(T) +%SQRTM_TRIANG_MIN_NORM Estimated min norm square root of triangular matrix. +% R = SQRTM_TRIANG_MIN_NORM(T) computes a primary square root of the +% upper triangular matrix T and attempts to minimize its 1-norm. + +if ~isequal(T,triu(T)), error('T must be upper triangular'), end + +n = length(T); +rp = zeros(n,1); +rm = zeros(n,1); + +R = zeros(n); +for j=1:n + rp(j) = sqrt(T(j,j)); + rm(j) = -sqrt(T(j,j)); + for i=j-1:-1:1 + rp(i) = (T(i,j) - R(i,i+1:j-1)*rp(i+1:j-1))/(R(i,i) + rp(j)); + rm(i) = (T(i,j) - R(i,i+1:j-1)*rm(i+1:j-1))/(R(i,i) + rm(j)); + end + if norm(rp(1:j),1) <= norm(rm(1:j),1) + R(1:j,j) = rp(1:j); + else + R(1:j,j) = rm(1:j); + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/mftoolbox/sylvsol.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,39 @@ +function X = sylvsol(A,B,C) +%SYLVSOL Solve Sylvester equation. +% X = SYLSOL(A, B, C) is the solution to the Sylvester equation +% AX + XB = C, where A is m-by-m, B is n-by-n and C is m-by-n. +% Schur decompositions are used to convert to a (quasi)-triangular +% system. + +% Reference: +% R. H. Bartels and G. W. Stewart. +% Algorithm 432: Solution of the matrix equation AX+XB=C. +% Comm. ACM, 15(9):820-826, 1972. + +[m,m] = size(A); +[n,n] = size(B); + +realdata = (isreal(A) && isreal(B) && isreal(C)); +if ~isequal(A,triu(A)) || ~isequal(B,triu(B)) + + [Q, T] = schur(A,'complex'); + [P, U] = schur(B,'complex'); + C = Q'*C*P; + schur_red = 1; + +else + + schur_red = 0; + T = A; U = B; + +end + +X = zeros(m,n); + +% Forward substitution. +for i = 1:n + X(:,i) = (T + U(i,i)*eye(m)) \ (C(:,i) - X(:,1:i-1)*U(1:i-1,i)); +end + +if schur_red, X = Q*X*P'; end +if realdata, X = real(X); end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/augment.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,37 @@ +function C = augment(A, alpha) +%AUGMENT Augmented system matrix. +% AUGMENT(A, ALPHA) is the square matrix +% [ALPHA*EYE(m) A; A' ZEROS(n)] of dimension m+n, where A is m-by-n. +% It is the symmetric and indefinite coefficient matrix of the +% augmented system associated with a least squares problem +% minimize NORM(A*x-b). ALPHA defaults to 1. +% Special case: if A is a scalar, n say, then AUGMENT(A) is the +% same as AUGMENT(RANDN(p,q)) where n = p+q and +% p = ROUND(n/2), that is, a random augmented matrix +% of dimension n is produced. +% The eigenvalues of AUGMENT(A) are given in terms of the singular +% values s(i) of A (where m>n) by +% 1/2 +/- SQRT( s(i)^2 + 1/4 ), i=1:n (2n eigenvalues), +% 1, (m-n eigenvalues). +% If m < n then the first expression provides 2m eigenvalues and the +% remaining n-m eigenvalues are zero. +% +% See also SPAUGMENT. + +% Reference: +% G.H. Golub and C.F. Van Loan, Matrix Computations, Second +% Edition, Johns Hopkins University Press, Baltimore, Maryland, +% 1989, sec. 5.6.4. + +[m, n] = size(A); +if nargin < 2, alpha = 1; end + +if max(m,n) == 1 + n = A; + p = round(n/2); + q = n - p; + A = randn(p,q); + m = p; n = q; +end + +C = [alpha*eye(m) A; A' zeros(n)];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/bandred.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,55 @@ +function A = bandred(A, kl, ku) +%BANDRED Band reduction by two-sided unitary transformations. +% B = BANDRED(A, KL, KU) is a matrix unitarily equivalent to A +% with lower bandwidth KL and upper bandwidth KU +% (i.e. B(i,j) = 0 if i > j+KL or j > i+KU). +% The reduction is performed using Householder transformations. +% If KU is omitted it defaults to KL. + +% Called by RANDSVD. +% This is a `standard' reduction. Cf. reduction to bidiagonal form +% prior to computing the SVD. This code is a little wasteful in that +% it computes certain elements which are immediately set to zero! +% +% Reference: +% G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, +% Johns Hopkins University Press, Baltimore, Maryland, 1989. +% Section 5.4.3. + +if nargin == 2, ku = kl; end + +if kl == 0 & ku == 0 + error('You''ve asked for a diagonal matrix. In that case use the SVD!') +end + +% Check for special case where order of left/right transformations matters. +% Easiest approach is to work on the transpose, flipping back at the end. +flip = 0; +if ku == 0 + A = A'; + temp = kl; kl = ku; ku = temp; flip = 1; +end + +[m, n] = size(A); + +for j=1 : min( min(m,n), max(m-kl-1,n-ku-1) ) + + if j+kl+1 <= m + [v, beta] = house(A(j+kl:m,j)); + temp = A(j+kl:m,j:n); + A(j+kl:m,j:n) = temp - beta*v*(v'*temp); + A(j+kl+1:m,j) = zeros(m-j-kl,1); + end + + if j+ku+1 <= n + [v, beta] = house(A(j,j+ku:n)'); + temp = A(j:m,j+ku:n); + A(j:m,j+ku:n) = temp - beta*(temp*v)*v'; + A(j,j+ku+1:n) = zeros(1,n-j-ku); + end + +end + +if flip + A = A'; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/cauchy.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,39 @@ +function C = cauchy(x, y) +%CAUCHY Cauchy matrix. +% C = CAUCHY(X, Y), where X, Y are N-vectors, is the N-by-N matrix +% with C(i,j) = 1/(X(i)+Y(j)). By default, Y = X. +% Special case: if X is a scalar CAUCHY(X) is the same as CAUCHY(1:X). +% Explicit formulas are known for DET(C) (which is nonzero if X and Y +% both have distinct elements) and the elements of INV(C). +% C is totally positive if 0 < X(1) < ... < X(N) and +% 0 < Y(1) < ... < Y(N). + +% References: +% N.J. Higham, Accuracy and Stability of Numerical Algorithms, +% Society for Industrial and Applied Mathematics, Philadelphia, PA, +% USA, 1996; sec. 26.1. +% D.E. Knuth, The Art of Computer Programming, Volume 1, +% Fundamental Algorithms, second edition, Addison-Wesley, Reading, +% Massachusetts, 1973, p. 36. +% E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, +% Linear Algebra and Appl., 149 (1991), pp. 1-18. +% O. Taussky and M. Marcus, Eigenvalues of finite matrices, in +% Survey of Numerical Analysis, J. Todd, ed., McGraw-Hill, New York, +% pp. 279-313, 1962. (States the totally positive property on p. 295.) + +n = max(size(x)); +% Handle scalar x. +if n == 1 + n = x; + x = 1:n; +end + +if nargin == 1, y = x; end + +x = x(:); y = y(:); % Ensure x and y are column vectors. +if any(size(x) ~= size(y)) + error('Parameter vectors must be of same dimension.') +end + +C = x*ones(1,n) + ones(n,1)*y.'; +C = ones(n) ./ C;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/chebspec.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,53 @@ +function C = chebspec(n, k) +%CHEBSPEC Chebyshev spectral differentiation matrix. +% C = CHEBSPEC(N, K) is a Chebyshev spectral differentiation +% matrix of order N. K = 0 (the default) or 1. +% For K = 0 (`no boundary conditions'), C is nilpotent, with +% C^N = 0 and it has the null vector ONES(N,1). +% C is similar to a Jordan block of size N with eigenvalue zero. +% For K = 1, C is nonsingular and well-conditioned, and its eigenvalues +% have negative real parts. +% For both K, the computed eigenvector matrix X from EIG is +% ill-conditioned (MESH(REAL(X)) is interesting). + +% References: +% C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral +% Methods in Fluid Dynamics, Springer-Verlag, Berlin, 1988; p. 69. +% L.N. Trefethen and M.R. Trummer, An instability phenomenon in +% spectral methods, SIAM J. Numer. Anal., 24 (1987), pp. 1008-1023. +% D. Funaro, Computing the inverse of the Chebyshev collocation +% derivative, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 1050-1057. + +if nargin == 1, k = 0; end + +% k = 1 case obtained from k = 0 case with one bigger n. +if k == 1, n = n + 1; end + +n = n-1; +C = zeros(n+1); + +one = ones(n+1,1); +x = cos( (0:n)' * (pi/n) ); +d = ones(n+1,1); d(1) = 2; d(n+1) = 2; + +% eye(size(C)) on next line avoids div by zero. +C = (d * (one./d)') ./ (x*one'-one*x' + eye(size(C))); + +% Now fix diagonal and signs. + +C(1,1) = (2*n^2+1)/6; +for i=2:n+1 + if rem(i,2) == 0 + C(:,i) = -C(:,i); + C(i,:) = -C(i,:); + end + if i < n+1 + C(i,i) = -x(i)/(2*(1-x(i)^2)); + else + C(n+1,n+1) = -C(1,1); + end +end + +if k == 1 + C = C(2:n+1,2:n+1); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/chebvand.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,34 @@ +function C = chebvand(m,p) +%CHEBVAND Vandermonde-like matrix for the Chebyshev polynomials. +% C = CHEBVAND(P), where P is a vector, produces the (primal) +% Chebyshev Vandermonde matrix based on the points P, +% i.e., C(i,j) = T_{i-1}(P(j)), where T_{i-1} is the Chebyshev +% polynomial of degree i-1. +% CHEBVAND(M,P) is a rectangular version of CHEBVAND(P) with M rows. +% Special case: If P is a scalar then P equally spaced points on +% [0,1] are used. + +% Reference: +% N.J. Higham, Stability analysis of algorithms for solving confluent +% Vandermonde-like systems, SIAM J. Matrix Anal. Appl., 11 (1990), +% pp. 23-41. + +if nargin == 1, p = m; end +n = max(size(p)); + +% Handle scalar p. +if n == 1 + n = p; + p = seqa(0,1,n); +end + +if nargin == 1, m = n; end + +p = p(:).'; % Ensure p is a row vector. +C = ones(m,n); +if m == 1, return, end +C(2,:) = p; +% Use Chebyshev polynomial recurrence. +for i=3:m + C(i,:) = 2.*p.*C(i-1,:) - C(i-2,:); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/cholp.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,69 @@ +function [R, P, I] = cholp(A, piv) +%CHOLP Cholesky factorization with pivoting of a pos. semidefinite matrix. +% [R, P] = CHOLP(A) returns R and a permutation matrix P such that +% R'*R = P'*A*P. Only the upper triangular part of A is used. +% If A is not positive definite, an error message is printed. +% +% [R, P, I] = CHOLP(A) never produces an error message. +% If A is positive definite then I = 0 and R is the Cholesky factor. +% If A is not positive definite then I is positive and +% R is (I-1)-by-N with P'*A*P - R'*R zeros in columns 1:I-1 and +% rows 1:I-1. +% [R, I] = CHOLP(A, 0) forces P = EYE(SIZE(A)), and therefore behaves +% like [R, I] = CHOL(A). + +% This routine is based on the LINPACK routine CCHDC. It works +% for both real and complex matrices. +% +% Reference: +% G.H. Golub and C.F. Van Loan, Matrix Computations, Second +% Edition, Johns Hopkins University Press, Baltimore, Maryland, +% 1989, sec. 4.2.9. + +if nargin == 1, piv = 1; end + +n = length(A); +pp = 1:n; +I = 0; + +for k = 1:n + + if piv + d = diag(A); + [big, m] = max( d(k:n) ); + m = m+k-1; + else + big = A(k,k); m = k; + end + if big <= 0, I = k; break, end + +% Symmetric row/column permutations. + if m ~= k + A(:, [k m]) = A(:, [m k]); + A([k m], :) = A([m k], :); + pp( [k m] ) = pp( [m k] ); + end + + A(k,k) = sqrt( A(k,k) ); + if k == n, break, end + A(k, k+1:n) = A(k, k+1:n) / A(k,k); + +% For simplicity update the whole of the remaining submatrix (rather +% than just the upper triangle). + + j = k+1:n; + A(j,j) = A(j,j) - A(k,j)'*A(k,j); + +end + +R = triu(A); +if I > 0 + if nargout < 3, error('Matrix must be positive definite.'), end + R = R(1:I-1,:); +end + +if piv == 0 + P = I; +else + P = eye(n); P = P(:,pp); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/chop.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,18 @@ +function c = chop(x, t) +%CHOP Round matrix elements. +% CHOP(X, t) is the matrix obtained by rounding the elements of X +% to t significant binary places. +% Default is t = 24, corresponding to IEEE single precision. + +if nargin < 2, t = 24; end +[m, n] = size(x); + +% Use the representation: +% x(i,j) = 2^e(i,j) * .d(1)d(2)...d(s) * sign(x(i,j)) + +% On the next line `+(x==0)' avoids passing a zero argument to LOG, which +% would cause a warning message to be generated. + +y = abs(x) + (x==0); +e = floor(log2(y) + 1); +c = pow2(round( pow2(x, t-e) ), e-t);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/chow.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,21 @@ +function A = chow(n, alpha, delta) +%CHOW Chow matrix - a singular Toeplitz lower Hessenberg matrix. +% A = CHOW(N, ALPHA, DELTA) is a Toeplitz lower Hessenberg matrix +% A = H(ALPHA) + DELTA*EYE, where H(i,j) = ALPHA^(i-j+1). +% H(ALPHA) has p = FLOOR(N/2) zero eigenvalues, the rest being +% 4*ALPHA*COS( k*PI/(N+2) )^2, k=1:N-p. +% Defaults: ALPHA = 1, DELTA = 0. + +% References: +% T.S. Chow, A class of Hessenberg matrices with known +% eigenvalues and inverses, SIAM Review, 11 (1969), pp. 391-395. +% G. Fairweather, On the eigenvalues and eigenvectors of a class of +% Hessenberg matrices, SIAM Review, 13 (1971), pp. 220-221. +% I. Singh, G. Poole and T. Boullion, A class of Hessenberg matrices +% with known pseudoinverse and Drazin inverse, Math. Comp., +% 29 (1975), pp. 615-619. + +if nargin < 3, delta = 0; end +if nargin < 2, alpha = 1; end + +A = toeplitz( alpha.^(1:n), [alpha 1 zeros(1,n-2)] ) + delta*eye(n);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/circul.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,25 @@ +function C = circul(v) +%CIRCUL Circulant matrix. +% C = CIRCUL(V) is the circulant matrix whose first row is V. +% (A circulant matrix has the property that each row is obtained +% from the previous one by cyclically permuting the entries one step +% forward; it is a special Toeplitz matrix in which the diagonals +% `wrap round'.) +% Special case: if V is a scalar then C = CIRCUL(1:V). +% The eigensystem of C (N-by-N) is known explicitly. If t is an Nth +% root of unity, then the inner product of V with W = [1 t t^2 ... t^N] +% is an eigenvalue of C, and W(N:-1:1) is an eigenvector of C. + +% Reference: +% P.J. Davis, Circulant Matrices, John Wiley, 1977. + +n = max(size(v)); + +if n == 1 + n = v; + v = 1:n; +end + +v = v(:).'; % Make sure v is a row vector. + +C = toeplitz( [ v(1) v(n:-1:2) ], v );
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/clement.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,38 @@ +function A = clement(n, k) +%CLEMENT Clement matrix - tridiagonal with zero diagonal entries. +% CLEMENT(N, K) is a tridiagonal matrix with zero diagonal entries +% and known eigenvalues. It is singular if N is odd. About 64 +% percent of the entries of the inverse are zero. The eigenvalues +% are plus and minus the numbers N-1, N-3, N-5, ..., (1 or 0). +% For K = 0 (the default) the matrix is unsymmetric, while for +% K = 1 it is symmetric. +% CLEMENT(N, 1) is diagonally similar to CLEMENT(N). + +% Similar properties hold for TRIDIAG(X,Y,Z) where Y = ZEROS(N,1). +% The eigenvalues still come in plus/minus pairs but they are not +% known explicitly. +% +% References: +% P.A. Clement, A class of triple-diagonal matrices for test +% purposes, SIAM Review, 1 (1959), pp. 50-52. +% A. Edelman and E. Kostlan, The road from Kac's matrix to Kac's +% random polynomials. In John~G. Lewis, editor, Proceedings of +% the Fifth SIAM Conference on Applied Linear Algebra Society +% for Industrial and Applied Mathematics, Philadelphia, 1994, +% pp. 503-507. +% O. Taussky and J. Todd, Another look at a matrix of Mark Kac, +% Linear Algebra and Appl., 150 (1991), pp. 341-360. + +if nargin == 1, k = 0; end + +n = n-1; + +x = n:-1:1; +z = 1:n; + +if k == 0 + A = diag(x, -1) + diag(z, 1); +else + y = sqrt(x.*z); + A = diag(y, -1) + diag(y, 1); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/cod.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,38 @@ +function [U, R, V] = cod(A, tol) +%COD Complete orthogonal decomposition. +% [U, R, V] = COD(A, TOL) computes a decomposition A = U*T*V, +% where U and V are unitary, T = [R 0; 0 0] has the same dimensions as +% A, and R is upper triangular and nonsingular of dimension rank(A). +% Rank decisions are made using TOL, which defaults to approximately +% MAX(SIZE(A))*NORM(A)*EPS. +% By itself, COD(A, TOL) returns R. + +% Reference: +% G.H. Golub and C.F. Van Loan, Matrix Computations, Second +% Edition, Johns Hopkins University Press, Baltimore, Maryland, +% 1989, sec. 5.4.2. + +[m, n] = size(A); + +% QR decomposition. +[U, R, P] = qr(A); % AP = UR +V = P'; % A = URV; +if nargin == 1, tol = max(m,n)*eps*abs(R(1,1)); end % |R(1,1)| approx NORM(A). + +% Determine r = effective rank. +r = sum(abs(diag(R)) > tol); +r = r(1); % Fix for case where R is vector. +R = R(1:r,:); % Throw away negligible rows (incl. all zero rows, m>n). + +if r ~= n + + % Reduce nxr R' = r [L] to lower triangular form: QR' = [Lbar]. + % n-r [M] [0] + + [Q, R] = trap2tri(R'); + V = Q*V; + R = R'; + +end + +if nargout <= 1, U = R; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/comp.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,41 @@ +function C = comp(A, k) +%COMP Comparison matrices. +% COMP(A) is DIAG(B) - TRIL(B,-1) - TRIU(B,1), where B = ABS(A). +% COMP(A, 1) is A with each diagonal element replaced by its +% absolute value, and each off-diagonal element replaced by minus +% the absolute value of the largest element in absolute value in +% its row. However, if A is triangular COMP(A, 1) is too. +% COMP(A, 0) is the same as COMP(A). +% COMP(A) is often denoted by M(A) in the literature. + +% Reference (e.g.): +% N.J. Higham, A survey of condition number estimation for +% triangular matrices, SIAM Review, 29 (1987), pp. 575-596. + +if nargin == 1, k = 0; end +[m, n] = size(A); +p = min(m, n); + +if k == 0 + +% This code uses less temporary storage than the `high level' definition above. + C = -abs(A); + for j=1:p + C(j,j) = abs(A(j,j)); + end + +elseif k == 1 + + C = A'; + for j=1:p + C(k,k) = 0; + end + mx = max(abs(C)); + C = -mx'*ones(1,n); + for j=1:p + C(j,j) = abs(A(j,j)); + end + if all( A == tril(A) ), C = tril(C); end + if all( A == triu(A) ), C = triu(C); end + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/compan.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,45 @@ +function A = compan(p) +%COMPAN Companion matrix. +% COMPAN(P) is a companion matrix. There are three cases. +% If P is a scalar then COMPAN(P) is the P-by-P matrix COMPAN(1:P+1). +% If P is an (n+1)-vector, COMPAN(P) is the n-by-n companion matrix +% whose first row is -P(2:n+1)/P(1). +% If P is a square matrix, COMPAN(P) is the companion matrix +% of the characteristic polynomial of P, computed as +% COMPAN(POLY(P)). + +% References: +% J.H. Wilkinson, The Algebraic Eigenvalue Problem, +% Oxford University Press, 1965, p. 12. +% G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, +% Johns Hopkins University Press, Baltimore, Maryland, 1989, +% sec 7.4.6. +% C. Kenney and A.J. Laub, Controllability and stability radii for +% companion form systems, Math. Control Signals Systems, 1 (1988), +% pp. 239-256. (Gives explicit formulas for the singular values of +% COMPAN(P).) + +[n,m] = size(p); + +if n == m & n > 1 + % Matrix argument. + A = compan(poly(p)); + return +end + +n = max(n,m); +% Handle scalar p. +if n == 1 + n = p+1; + p = 1:n; +end + +p = p(:)'; % Ensure p is a row vector. + +% Construct matrix of order n-1. +if n == 2 + A = 1; +else + A = diag(ones(1,n-2),-1); + A(1,:) = -p(2:n)/p(1); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/cond.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,38 @@ +function y = cond(A, p) +%COND Matrix condition number in 1, 2, Frobenius, or infinity norm. +% For p = 1, 2, 'fro', inf, COND(A,p) = NORM(A,p) * NORM(INV(A),p). +% If p is omitted then p = 2 is used. +% A may be a rectangular matrix if p = 2; in this case COND(A) +% is the ratio of the largest singular value of A to the smallest +% (and hence is infinite if A is rank deficient). + +% See also RCOND, NORM, CONDEST, NORMEST. +% Generalises and incorporates MATFUN/COND.M from Matlab 4. + +if length(A) == 0 % Handle null matrix. + y = NaN; + return +end +if issparse(A) + error('Matrix must be non-sparse.') +end + +if nargin == 1, p = 2; end + +[m, n] = size(A); +if m ~= n & p ~= 2 + error('A is rectangular. Use the 2 norm.') +end + +if p == 2 + s = svd(A); + if any(s == 0) % Handle singular matrix + disp('Condition is infinite') + y = Inf; + return + end + y = max(s)./min(s); +else +% We'll let NORM pick up any invalid p argument. + y = norm(A, p) * norm(inv(A), p); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/condex.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,63 @@ +function A = condex(n, k, theta) +%CONDEX `Counterexamples' to matrix condition number estimators. +% CONDEX(N, K, THETA) is a `counterexample' matrix to a condition +% estimator. It has order N and scalar parameter THETA (default 100). +% If N is not equal to the `natural' size of the matrix then +% the matrix is padded out with an identity matrix to order N. +% The matrix, its natural size, and the estimator to which it applies +% are specified by K (default K = 4) as follows: +% K = 1: 4-by-4, LINPACK (RCOND) +% K = 2: 3-by-3, LINPACK (RCOND) +% K = 3: arbitrary, LINPACK (RCOND) (independent of THETA) +% K = 4: N >= 4, SONEST (Higham 1988) +% (Note that in practice the K = 4 matrix is not usually a +% counterexample because of the rounding errors in forming it.) + +% References: +% A.K. Cline and R.K. Rew, A set of counter-examples to three +% condition number estimators, SIAM J. Sci. Stat. Comput., +% 4 (1983), pp. 602-611. +% N.J. Higham, FORTRAN codes for estimating the one-norm of a real or +% complex matrix, with applications to condition estimation +% (Algorithm 674), ACM Trans. Math. Soft., 14 (1988), pp. 381-396. + +if nargin < 3, theta = 100; end +if nargin < 2, k = 4; end + +if k == 1 % Cline and Rew (1983), Example B. + + A = [1 -1 -2*theta 0 + 0 1 theta -theta + 0 1 1+theta -(theta+1) + 0 0 0 theta]; + +elseif k == 2 % Cline and Rew (1983), Example C. + + A = [1 1-2/theta^2 -2 + 0 1/theta -1/theta + 0 0 1]; + +elseif k == 3 % Cline and Rew (1983), Example D. + + A = triw(n,-1)'; + A(n,n) = -1; + +elseif k == 4 % Higham (1988), p. 390. + + x = ones(n,3); % First col is e + x(2:n,2) = zeros(n-1,1); % Second col is e(1) + + % Third col is special vector b in SONEST + x(:, 3) = (-1).^[0:n-1]' .* ( 1 + [0:n-1]'/(n-1) ); + + Q = orth(x); % Q*Q' is now the orthogonal projector onto span(e(1),e,b)). + P = eye(n) - Q*Q'; + A = eye(n) + theta*P; + +end + +% Pad out with identity as necessary. +[m, m] = size(A); +if m < n + for i=n:-1:m+1, A(i,i) = 1; end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/contents.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,114 @@ +% Test Matrix Toolbox. +% Version 3.0, September 19 1995 +% Copyright (c) 1995 by N. J. Higham +% +% Demonstration +%TMTDEMO Demonstration of Test Matrix Toolbox. +% +% Test Matrices +%AUGMENT Augmented system matrix. +%CAUCHY Cauchy matrix. +%CHEBSPEC Chebyshev spectral differentiation matrix. +%CHEBVAND Vandermonde-like matrix for the Chebyshev polynomials. +%CHOW Chow matrix - a singular Toeplitz lower Hessenberg matrix. +%CIRCUL Circulant matrix. +%CLEMENT Clement matrix - tridiagonal with zero diagonal entries. +%COMPAN Companion matrix. +%CONDEX `Counterexamples' to matrix condition estimators. +%CYCOL Matrix whose columns repeat cyclically. +%DINGDONG Dingdong matrix - a symmetric Hankel matrix. +%DORR Dorr matrix - diag. dominant, ill-conditioned, tridiagonal (sparse). +%DRAMADAH A (0,1) matrix with large inverse. +%FIEDLER Fiedler matrix - symmetric. +%FORSYTHE Forsythe matrix - a perturbed Jordan block. +%FRANK Frank matrix - ill-conditioned eigenvalues. +%GALLERY Famous, and not so famous, test matrices. +%GEARM Gear matrix. +%GFPP Matrix giving maximal growth factor for Gaussian elim. with pivoting. +%GRCAR Grcar matrix - a Toeplitz matrix with sensitive eigenvalues. +%HADAMARD Hadamard matrix. +%HANOWA Hanowa matrix. +%HILB Hilbert matrix. +%INVHESS Inverse of an upper Hessenberg matrix. +%INVOL An involutory matrix. +%IPJFACT A Hankel matrix with factorial elements. +%JORDBLOC Jordan block. +%KAHAN Kahan matrix - upper trapezoidal. +%KMS Kac-Murdock-Szego Toeplitz matrix. +%KRYLOV Krylov matrix. +%LAUCHLI Lauchli matrix. +%LEHMER Lehmer matrix - symmetric positive definite. +%LESP A tridiagonal matrix with real, sensitive eigenvalues. +%LOTKIN Lotkin matrix. +%MAKEJCF A matrix with given Jordan canonical form. +%MINIJ Symmetric positive definite matrix MIN(i,j). +%MOLER Moler matrix - symmetric positive definite. +%NEUMANN Singular matrix from the discrete Neumann problem. +%OHESS Random, orthogonal upper Hessenberg matrix. +%ORTHOG Orthogonal and nearly orthogonal matrices. +%PARTER Parter matrix - a Toeplitz matrix with singular values near PI. +%PASCAL Pascal matrix. +%PDTOEP Symmetric positive definite Toeplitz matrix. +%PEI Pei matrix. +%PENTOEP Pentadiagonal Toeplitz matrix (sparse). +%POISSON Block tridiagonal matrix from Poisson's equation (sparse). +%PROLATE Prolate matrix - symmetric, ill-conditioned Toeplitz matrix. +%RANDO Random matrix with elements -1, 0 or 1. +%RANDSVD Random matrices with pre-assigned singular values. +%REDHEFF A matrix of 0s and 1s of Redheffer. +%RIEMANN A matrix associated with the Riemann hypothesis. +%RSCHUR An upper quasi-triangular matrix. +%SMOKE Smoke matrix - complex, with a `smoke ring' pseudospectrum. +%TRIDIAG Tridiagonal matrix (sparse). +%TRIW Upper triangular matrix discussed by Wilkinson and others. +%VAND Vandermonde matrix. +%WATHEN Wathen matrix - a finite element matrix (sparse, random entries). +%WILK Various specific matrices devised/discussed by Wilkinson. +% +% Visualization +%FV Field of values (or numerical range). +%GERSH Gershgorin disks. +%PS Approximation to the pseudospectrum. +%PSCONT Contours and colour pictures of pseudospectra. +%SEE Pictures of a matrix and its (pseudo-) inverse. +% +% Decompositions and factorizations. +%CGS Gram-Schmidt QR factorization. +%CHOLP Cholesky factorization with pivoting of a pos. semidefinite matrix. +%COD Complete orthogonal decomposition. +%DIAGPIV Diagonal pivoting factorization with partial pivoting. +%GE Gaussian elimination without pivoting. +%GECP Gaussian elimination with complete pivoting. +%GJ Gauss-Jordan elimination to solve Ax = b. +%MGS Modified Gram-Schmidt QR factorization. +%POLDEC Polar decomposition. +%SIGNM Matrix sign decomposition. +% +% Direct Search Optimization. +%ADSMAX Alternating directions direct search method. +%MDSMAX Multidirectional search method for direct search optimization. +%NMSMAX Nedler-Mead simplex method for direct search optimization. +% +% Miscellaneous +%BANDRED Band reduction by two-sided orthogonal transformations. +%CHOP Round matrix elements. +%COMP Comparison matrices. +%COND Matrix condition number in 1, 2, Frobenius, or infinity norm. +%CPLTAXES Determine suitable AXIS for plot of complex vector. +%DUAL Dual vector with respect to Holder p-norm. +%EIGSENS Eigenvalue condition numbers. +%HOUSE Householder matrix. +%MATRIX Test Matrix Collection information and access by number. +%MATSIGNT Matrix sign function of a triangular matrix. +%PNORM Estimate of matrix p-norm (1 <= p <= inf). +%QMULT Pre-multiply by random orthogonal matrix. +%RQ Rayleigh quotient. +%SEQA An additive sequence. +%SEQCHEB Sequence of points related to Chebyshev polynomials, T_N. +%SEQM A multiplicative sequence. +%SHOW Display signs of matrix elements. +%SKEWPART Skew-symmetric (Hermitian) part. +%SPARSIFY Randomly sets matrix elements to zero. +%SUB Principal submatrix. +%SYMMPART Symmetric (Hermitian) part. +%TRAP2TRI Trapezoidal matrix to triangular form.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/cpltaxes.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,37 @@ +function x = cpltaxes(z) +%CPLTAXES Determine suitable AXIS for plot of complex vector. +% X = CPLTAXES(Z), where Z is a complex vector, +% determines a 4-vector X such that AXIS(X) sets axes for a plot +% of Z that has axes of equal length and leaves a reasonable amount +% of space around the edge of the plot. + +% Called by FV, GERSH, PS and PSCONT. + +% Set x and y axis ranges so both have the same length. + +xmin = min(real(z)); xmax = max(real(z)); +ymin = min(imag(z)); ymax = max(imag(z)); + +% Fix for rare case of `trivial data'. +if xmin == xmax, xmin = xmin - 1/2; xmax = xmax + 1/2; end +if ymin == ymax, ymin = ymin - 1/2; ymax = ymax + 1/2; end + +if xmax-xmin >= ymax-ymin + ymid = (ymin + ymax)/2; + ymin = ymid - (xmax-xmin)/2; ymax = ymid + (xmax-xmin)/2; +else + xmid = (xmin + xmax)/2; + xmin = xmid - (ymax-ymin)/2; xmax = xmid + (ymax-ymin)/2; +end +axis('square') + +% Scale ranges by 1+2*alpha to give extra space around edges of plot. + +alpha = 0.1; +x(1) = xmin - alpha*(xmax-xmin); +x(2) = xmax + alpha*(xmax-xmin); +x(3) = ymin - alpha*(ymax-ymin); +x(4) = ymax + alpha*(ymax-ymin); + +if x(1) == x(2), x(2) = x(2) + 0.1; end +if x(3) == x(4), x(4) = x(3) + 0.1; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/cycol.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,22 @@ +function A = cycol(n, k) +%CYCOL Matrix whose columns repeat cyclically. +% A = CYCOL([M N], K) is an M-by-N matrix of the form A = B(1:M,1:N) +% where B = [C C C...] and C = RANDN(M, K). Thus A's columns repeat +% cyclically, and A has rank at most K. K need not divide N. +% K defaults to ROUND(N/4). +% CYCOL(N, K), where N is a scalar, is the same as CYCOL([N N], K). +% +% This type of matrix can lead to underflow problems for Gaussian +% elimination: see NA Digest Volume 89, Issue 3 (January 22, 1989). + +m = n(1); % Parameter n specifies dimension: m-by-n. +n = n(max(size(n))); + +if nargin < 2, k = max(round(n/4),1); end + +A = randn(m, k); +for i=2:ceil(n/k) + A = [A A(:,1:k)]; +end + +A = A(:, 1:n);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/diagpiv.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,108 @@ +function [L, D, P, rho] = diagpiv(A) +%DIAGPIV Diagonal pivoting factorization with partial pivoting. +% Given a Hermitian matrix A, +% [L, D, P, rho] = DIAGPIV(A) computes a permutation P, +% a unit lower triangular L, and a real block diagonal D +% with 1x1 and 2x2 diagonal blocks, such that +% P*A*P' = L*D*L'. +% The Bunch-Kaufman partial pivoting strategy is used. +% Rho is the growth factor. + +% Reference: +% J.R. Bunch and L. Kaufman, Some stable methods for calculating +% inertia and solving symmetric linear systems, Math. Comp., +% 31(137):163-179, 1977. + +% This routine does not exploit symmetry and is not designed to be +% efficient. + +if norm(triu(A,1)'-tril(A,-1),1), error('Must supply Hermitian matrix.'), end + +n = max(size(A)); +k = 1; +D = eye(n); +L = eye(n); +pp = 1:n; +normA = norm(A(:),inf); +rho = normA; + +alpha = (1 + sqrt(17))/8; + +while k < n + [lambda, r] = max( abs(A(k+1:n,k)) ); + r = r(1) + k; + + if lambda > 0 + swap = 0; + if abs(A(k,k)) >= alpha*lambda + s = 1; + else + temp = A(k:n,r); temp(r-k+1) = 0; + sigma = norm(temp, inf); + if alpha*lambda^2 <= abs(A(k,k))*sigma + s = 1; + elseif abs(A(r,r)) >= alpha*sigma + swap = 1; + m1 = k; m2 = r; + s = 1; + else + swap = 1; + m1 = k+1; m2 = r; + s = 2; + end + end + + if swap + A( [m1, m2], : ) = A( [m2, m1], : ); + L( [m1, m2], : ) = L( [m2, m1], : ); + A( :, [m1, m2] ) = A( :, [m2, m1] ); + L( :, [m1, m2] ) = L( :, [m2, m1] ); + pp( [m1, m2] ) = pp( [m2, m1] ); + end + + if s == 1 + + D(k,k) = A(k,k); + A(k+1:n,k) = A(k+1:n,k)/A(k,k); + L(k+1:n,k) = A(k+1:n,k); + i = k+1:n; + A(i,i) = A(i,i) - A(i,k) * A(k,i); + + elseif s == 2 + + E = A(k:k+1,k:k+1); + D(k:k+1,k:k+1) = E; + C = A(k+2:n,k:k+1); + temp = C/E; + L(k+2:n,k:k+1) = temp; + A(k+2:n,k+2:n) = A(k+2:n,k+2:n) - temp*C'; + + end + + % Make diagonal real (see LINPACK User's Guide, p. 5.17). + for i=k+s:n + A(i,i) = real(A(i,i)); + end + + if k+s <= n + rho = max(rho, max(max(abs(A(k+s:n,k+s:n)))) ); + end + + else % Nothing to do. + + s = 1; + D(k,k) = A(k,k); + + end + + k = k + s; + + if k == n + D(n,n) = A(n,n); + break + end + +end + +if nargout >= 3, P = eye(n); P = P(pp,:); end +rho = rho/normA;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/dingdong.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,15 @@ +function A = dingdong(n) +%DINGDONG Dingdong matrix - a symmetric Hankel matrix. +% A = DINGDONG(N) is the symmetric N-by-N Hankel matrix with +% A(i,j) = 0.5/(N-i-j+1.5). +% The eigenvalues of A cluster around PI/2 and -PI/2. + +% Invented by F.N. Ris. +% +% Reference: +% J.C. Nash, Compact Numerical Methods for Computers: Linear +% Algebra and Function Minimisation, second edition, Adam Hilger, +% Bristol, 1990 (Appendix 1). + +p= -2*(1:n) + (n+1.5); +A = cauchy(p);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/dorr.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,41 @@ +function [c, d, e] = dorr(n, theta) +%DORR Dorr matrix - diagonally dominant, ill conditioned, tridiagonal. +% [C, D, E] = DORR(N, THETA) returns the vectors defining a row diagonally +% dominant, tridiagonal M-matrix that is ill conditioned for small +% values of the parameter THETA >= 0. +% If only one output parameter is supplied then +% C = FULL(TRIDIAG(C,D,E)), i.e., the matrix iself is returned. +% The columns of INV(C) vary greatly in norm. THETA defaults to 0.01. +% The amount of diagonal dominance is given by (ignoring rounding errors): +% COMP(C)*ONES(N,1) = THETA*(N+1)^2 * [1 0 0 ... 0 1]'. + +% Reference: +% F.W. Dorr, An example of ill-conditioning in the numerical +% solution of singular perturbation problems, Math. Comp., 25 (1971), +% pp. 271-283. + +if nargin < 2, theta = 0.01; end + +c = zeros(n,1); e = c; d = c; +% All length n for convenience. Make c, e of length n-1 later. + +h = 1/(n+1); +m = floor( (n+1)/2 ); +term = theta/h^2; + +i = (1:m)'; + c(i) = -term*ones(m,1); + e(i) = c(i) - (0.5-i*h)/h; + d(i) = -(c(i) + e(i)); + +i = (m+1:n)'; + e(i) = -term*ones(n-m,1); + c(i) = e(i) + (0.5-i*h)/h; + d(i) = -(c(i) + e(i)); + +c = c(2:n); +e = e(1:n-1); + +if nargout <= 1 + c = tridiag(c, d, e); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/dramadah.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,55 @@ +function A = dramadah(n, k) +%DRAMADAH A (0,1) matrix whose inverse has large integer entries. +% An anti-Hadamard matrix A is a matrix with elements 0 or 1 for +% which MU(A) := NORM(INV(A),'FRO') is maximal. +% A = DRAMADAH(N, K) is an N-by-N (0,1) matrix for which MU(A) is +% relatively large, although not necessarily maximal. +% Available types (the default is K = 1): +% K = 1: A is Toeplitz, with ABS(DET(A)) = 1, and MU(A) > c(1.75)^N, +% where c is a constant. +% K = 2: A is upper triangular and Toeplitz. +% The inverses of both types have integer entries. +% +% Another interesting (0,1) matrix: +% K = 3: A has maximal determinant among (0,1) lower Hessenberg +% matrices: det(A) = the n'th Fibonacci number. A is Toeplitz. +% The eigenvalues have an interesting distribution in the complex +% plane. + +% References: +% R.L. Graham and N.J.A. Sloane, Anti-Hadamard matrices, +% Linear Algebra and Appl., 62 (1984), pp. 113-137. +% L. Ching, The maximum determinant of an nxn lower Hessenberg +% (0,1) matrix, Linear Algebra and Appl., 183 (1993), pp. 147-153. + +if nargin < 2, k = 1; end + +if k == 1 % Toeplitz + + c = ones(n,1); + for i=2:4:n + m = min(1,n-i); + c(i:i+m) = zeros(m+1,1); + end + r = zeros(n,1); + r(1:4) = [1 1 0 1]; + if n < 4, r = r(1:n); end + A = toeplitz(c,r); + +elseif k == 2 % Upper triangular and Toeplitz + + c = zeros(n,1); + c(1) = 1; + r = ones(n,1); + for i=3:2:n + r(i) = 0; + end + A = toeplitz(c,r); + +elseif k == 3 % Lower Hessenberg. + + c = ones(n,1); + for i=2:2:n, c(i)=0; end; + A = toeplitz(c, [1 1 zeros(1,n-2)]); + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/dual.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,47 @@ +function y = dual(x, p) +%DUAL Dual vector with respect to Holder p-norm. +% Y = DUAL(X, p), where 1 <= p <= inf, is a vector of unit q-norm +% that is dual to X with respect to the p-norm, that is, +% norm(Y, q) = 1 where 1/p + 1/q = 1 and there is +% equality in the Holder inequality: X'*Y = norm(X, p)*norm(Y, q). +% Special case: DUAL(X), where X >= 1 is a scalar, returns Y such +% that 1/X + 1/Y = 1. + +% Called by PNORM. + +if max(size(x)) == 1 & nargin == 1 + p = x; +end + +% The following test avoids a `division by zero message' when p = 1. +if p == 1 + q = inf; +else + q = 1/(1-1/p); +end + +if max(size(x)) == 1 & nargin == 1 + y = q; + return +end + +if norm(x,inf) == 0, y = x; return, end + +if p == 1 + + y = sign(x) + (x == 0); % y(i) = +1 or -1 (if x(i) real). + +elseif p == inf + + [xmax, k] = max(abs(x)); + f = find(abs(x)==xmax); k = f(1); + y = zeros(size(x)); + y(k) = sign(x(k)); % y is a multiple of unit vector e_k. + +else % 1 < p < inf. Dual is unique in this case. + + x = x/norm(x,inf); % This scaling helps to avoid under/over-flow. + y = abs(x).^(p-1) .* ( sign(x) + (x==0) ); + y = y / norm(y,q); % Normalize to unit q-norm. + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/eigsens.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,25 @@ +function [X, D, s] = eigsens(A) +%EIGSENS Eigenvalue condition numbers. +% EIGSENS(A) is a vector of condition numbers for the eigenvalues +% of A (reciprocals of the Wilkinson s(lambda) numbers). +% These condition numbers are the reciprocals of the cosines of the +% angles between the left and right eigenvectors. +% [V, D, s] = EIGSENS(A) is equivalent to +% [V, D] = EIG(A); s = EIGSENS(A); + +% Reference: +% G.H. Golub and C.F. Van Loan, Matrix Computations, Second +% Edition, Johns Hopkins University Press, Baltimore, Maryland, +% 1989, sec. 7.2.2. + +n = max(size(A)); +s = zeros(n,1); + +[X, D] = eig(A); +Y = inv(X); + +for i=1:n + s(i) = norm(Y(i,:)) * norm(X(:,i)) / abs( Y(i,:)*X(:,i) ); +end + +if nargout <= 1, X = s; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/fdemo.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,7 @@ +function f = fdemo(A) +%FDEMO Demonstration function for direct search maximizers. +% FDEMO(A) is the reciprocal of the underestimation ratio for RCOND +% applied to the square matrix A. +% Demonstration function for ADSMAX, MDSMAX and NMSMAX. + +f = norm(A,1)*norm(inv(A),1)*rcond(A); % f >= 1
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/fiedler.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,33 @@ +function A = fiedler(c) +%FIEDLER Fiedler matrix - symmetric. +% A = FIEDLER(C), where C is an n-vector, is the n-by-n symmetric +% matrix with elements ABS(C(i)-C(j)). +% Special case: if C is a scalar, then A = FIEDLER(1:C) +% (i.e. A(i,j) = ABS(i-j)). +% Properties: +% FIEDLER(N) has a dominant positive eigenvalue and all the other +% eigenvalues are negative (Szego, 1936). +% Explicit formulas for INV(A) and DET(A) are given by Todd (1977) +% and attributed to Fiedler. These indicate that INV(A) is +% tridiagonal except for nonzero (1,n) and (n,1) elements. +% [I think these formulas are valid only if the elements of +% C are in increasing or decreasing order---NJH.] + +% References: +% G. Szego, Solution to problem 3705, Amer. Math. Monthly, +% 43 (1936), pp. 246-259. +% J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, +% Birkhauser, Basel, and Academic Press, New York, 1977, p. 159. + +n = max(size(c)); + +% Handle scalar c. +if n == 1 + n = c; + c = 1:n; +end + +c = c(:).'; % Ensure c is a row vector. + +A = ones(n,1)*c; +A = abs(A - A.'); % NB. array transpose.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/forsythe.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,13 @@ +function A = forsythe(n, alpha, lambda) +%FORSYTHE Forsythe matrix - a perturbed Jordan block. +% FORSYTHE(N, ALPHA, LAMBDA) is the N-by-N matrix equal to +% JORDBLOC(N, LAMBDA) except it has an ALPHA in the (N,1) position. +% It has the characteristic polynomial +% DET(A-t*EYE) = (LAMBDA-t)^N - (-1)^N ALPHA. +% ALPHA defaults to SQRT(EPS) and LAMBDA to 0. + +if nargin < 2, alpha = sqrt(eps); end +if nargin < 3, lambda = 0; end + +A = jordbloc(n, lambda); +A(n,1) = alpha;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/frank.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,45 @@ +function F = frank(n, k) +%FRANK Frank matrix---ill conditioned eigenvalues. +% F = FRANK(N, K) is the Frank matrix of order N. It is upper +% Hessenberg with determinant 1. K = 0 is the default; if K = 1 the +% elements are reflected about the anti-diagonal (1,N)--(N,1). +% F has all positive eigenvalues and they occur in reciprocal pairs +% (so that 1 is an eigenvalue if N is odd). +% The eigenvalues of F may be obtained in terms of the zeros of the +% Hermite polynomials. +% The FLOOR(N/2) smallest eigenvalues of F are ill conditioned, +% the more so for bigger N. + +% DET(FRANK(N)') comes out far from 1 for large N---see Frank (1958) +% and Wilkinson (1960) for discussions. +% +% This version incorporates improvements suggested by W. Kahan. +% +% References: +% W.L. Frank, Computing eigenvalues of complex matrices by determinant +% evaluation and by methods of Danilewski and Wielandt, J. Soc. +% Indust. Appl. Math., 6 (1958), pp. 378-392 (see pp. 385, 388). +% G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the +% computation of the Jordan canonical form, SIAM Review, 18 (1976), +% pp. 578-619 (Section 13). +% H. Rutishauser, On test matrices, Programmation en Mathematiques +% Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, +% 1966, pp. 349-365. Section 9. +% J.H. Wilkinson, Error analysis of floating-point computation, +% Numer. Math., 2 (1960), pp. 319-340 (Section 8). +% J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University +% Press, 1965 (pp. 92-93). +% The next two references give details of the eigensystem, as does +% Rutishauser (see above). +% P.J. Eberlein, A note on the matrices denoted by B_n, SIAM J. Appl. +% Math., 20 (1971), pp. 87-92. +% J.M. Varah, A generalization of the Frank matrix, SIAM J. Sci. Stat. +% Comput., 7 (1986), pp. 835-839. + +if nargin == 1, k = 0; end + +p = n:-1:1; +F = triu( p( ones(n,1), :) - diag( ones(n-1,1), -1), -1 ); +if k ~= 0 + F = F(p,p)'; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/fv.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,95 @@ +function [f, e] = fv(B, nk, thmax, noplot) +%FV Field of values (or numerical range). +% FV(A, NK, THMAX) evaluates and plots the field of values of the +% NK largest leading principal submatrices of A, using THMAX +% equally spaced angles in the complex plane. +% The defaults are NK = 1 and THMAX = 16. +% (For a `publication quality' picture, set THMAX higher, say 32.) +% The eigenvalues of A are displayed as `x'. +% Alternative usage: [F, E] = FV(A, NK, THMAX, 1) suppresses the +% plot and returns the field of values plot data in F, with A's +% eigenvalues in E. Note that NORM(F,INF) approximates the +% numerical radius, +% max {abs(z): z is in the field of values of A}. + +% Theory: +% Field of values FV(A) = set of all Rayleigh quotients. FV(A) is a +% convex set containing the eigenvalues of A. When A is normal FV(A) is +% the convex hull of the eigenvalues of A (but not vice versa). +% z = x'Ax/(x'x), z' = x'A'x/(x'x) +% => REAL(z) = x'Hx/(x'x), H = (A+A')/2 +% so MIN(EIG(H)) <= REAL(z) <= MAX(EIG(H)) +% with equality for x = corresponding eigenvectors of H. For these x, +% RQ(A,x) is on the boundary of FV(A). +% +% Based on an original routine by A. Ruhe. +% +% References: +% R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge +% University Press, 1991, Section 1.5. +% A.S. Householder, The Theory of Matrices in Numerical Analysis, +% Blaisdell, New York, 1964, Section 3.3. +% C.R. Johnson, Numerical determination of the field of values of a +% general complex matrix, SIAM J. Numer. Anal., 15 (1978), +% pp. 595-602. + +if nargin < 2, nk = 1; end +if nargin < 3, thmax = 16; end +thmax = thmax - 1; % Because code below uses thmax + 1 angles. + +iu = sqrt(-1); +[n, p] = size(B); +if n ~= p, error('Matrix must be square.'), end +f = []; +z = zeros(2*thmax+1,1); +e = eig(B); + +% Filter out cases where B is Hermitian or skew-Hermitian, for efficiency. +if norm(skewpart(B),1) == 0 + + f = [min(e) max(e)]; + +elseif norm(symmpart(B),1) == 0 + + e = imag(e); + f = [min(e) max(e)]; + e = iu*e; f = iu*f; + +else + +for m = 1:nk + + ns = n+1-m; + A = B(1:ns, 1:ns); + + for i = 0:thmax + th = i/thmax*pi; + Ath = exp(iu*th)*A; % Rotate A through angle th. + H = 0.5*(Ath + Ath'); % Hermitian part of rotated A. + [X, D] = eig(H); + [lmbh, k] = sort(real(diag(D))); + z(1+i) = rq(A,X(:,k(1))); % RQ's of A corr. to eigenvalues of H + z(1+i+thmax) = rq(A,X(:,k(ns))); % with smallest/largest real part. + end + + f = [f; z]; + +end +% Next line ensures boundary is `joined up' (needed for orthogonal matrices). +f = [f; f(1,:)]; + +end +if thmax == 0; f = e; end + +if nargin < 4 + + ax = cpltaxes(f); + plot(real(f), imag(f),'k') % Plot the field of values + axis(ax); + axis('square'); + + hold on + plot(real(e), imag(e), 'xb') % Plot the eigenvalues too. + hold off + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/gallery.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,65 @@ +function [A, e] = gallery(n) +%GALLERY Famous, and not so famous, test matrices. +% A = GALLERY(N) is an N-by-N matrix with some special property. +% The following values of N are currently available: +% N = 3 is badly conditioned. +% N = 4 is the Wilson matrix. Symmetric pos def, integer inverse. +% N = 5 is an interesting eigenvalue problem: defective, nilpotent. +% N = 8 is the Rosser matrix, a classic symmetric eigenvalue problem. +% [A, e] = GALLERY(8) returns the exact eigenvalues in e. +% N = 21 is Wilkinson's tridiagonal W21+, another eigenvalue problem. + +% Original version supplied with MATLAB. Modified by N.J. Higham. +% +% References: +% J.R. Westlake, A Handbook of Numerical Matrix Inversion and Solution +% of Linear Equations, John Wiley, New York, 1968. +% J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University +% Press, 1965. + +if n == 3 + A = [ -149 -50 -154 + 537 180 546 + -27 -9 -25 ]; + +elseif n == 4 + A = [10 7 8 7 + 7 5 6 5 + 8 6 10 9 + 7 5 9 10]; + +elseif n == 5 +% disp('Try to find the EXACT eigenvalues and eigenvectors.') +% Matrix devised by Cleve Moler. Its Jordan form has just one block, with +% eigenvalue zero. Proof: A^k is nonzero for k<5, zero for k=5. +% TRACE(A)=0. No simple form for null vector. + A = [ -9 11 -21 63 -252 + 70 -69 141 -421 1684 + -575 575 -1149 3451 -13801 + 3891 -3891 7782 -23345 93365 + 1024 -1024 2048 -6144 24572 ]; + +elseif n == 8 + A = [ 611. 196. -192. 407. -8. -52. -49. 29. + 196. 899. 113. -192. -71. -43. -8. -44. + -192. 113. 899. 196. 61. 49. 8. 52. + 407. -192. 196. 611. 8. 44. 59. -23. + -8. -71. 61. 8. 411. -599. 208. 208. + -52. -43. 49. 44. -599. 411. 208. 208. + -49. -8. 8. 59. 208. 208. 99. -911. + 29. -44. 52. -23. 208. 208. -911. 99. ]; + + % Exact eigenvalues from Westlake (1968), p.150 (ei'vectors given too): + a = sqrt(10405); b = sqrt(26); + e = [-10*a, 0, 510-100*b, 1000, 1000, 510+100*b, ... + 1020, 10*a]'; + +elseif n == 21 + % W21+, Wilkinson (1965), p.308. + E = diag(ones(n-1,1),1); + m = (n-1)/2; + A = diag(abs(-m:m)) + E + E'; + +else + error('Sorry, that value of N is not available.') +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/gearm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,26 @@ +function A = gearm(n, i, j) +%GEARM Gear matrix. +% A = GEARM(N,I,J) is the N-by-N matrix with ones on the sub- and +% super-diagonals, SIGN(I) in the (1,ABS(I)) position, SIGN(J) +% in the (N,N+1-ABS(J)) position, and zeros everywhere else. +% Defaults: I = N, j = -N. +% All eigenvalues are of the form 2*COS(a) and the eigenvectors +% are of the form [SIN(w+a), SIN(w+2a), ..., SIN(w+Na)]. +% The values of a and w are given in the reference below. +% A can have double and triple eigenvalues and can be defective. +% GEARM(N) is singular. + +% (GEAR is a Simulink function, hence GEARM for Gear matrix.) +% Reference: +% C.W. Gear, A simple set of test matrices for eigenvalue programs, +% Math. Comp., 23 (1969), pp. 119-125. + +if nargin == 1, i = n; j = -n; end + +if ~(i~=0 & abs(i)<=n & j~=0 & abs(j)<=n) + error('Invalid I and J parameters') +end + +A = diag(ones(n-1,1),-1) + diag(ones(n-1,1),1); +A(1, abs(i)) = sign(i); +A(n, n+1-abs(j)) = sign(j);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/gersh.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,38 @@ +function [G, e] = gersh(A, noplot) +%GERSH Gershgorin disks. +% GERSH(A) draws the Gershgorin disks for the matrix A. +% The eigenvalues are plotted as crosses `x'. +% Alternative usage: [G, E] = GERSH(A, 1) suppresses the plot +% and returns the data in G, with A's eigenvalues in E. +% +% Try GERSH(LESP(N)) and GERSH(SMOKE(N,1)). + +if diff(size(A)), error('Matrix must be square.'), end + +n = max(size(A)); +m = 40; +G = zeros(m,n); + +d = diag(A); +r = sum( abs( A.'-diag(d) ) )'; +e = eig(A); + +radvec = exp(i * seqa(0,2*pi,m)'); + +for j=1:n + G(:,j) = d(j)*ones(size(radvec)) + r(j)*radvec; +end + +if nargin < 2 + + ax = cpltaxes(G(:)); + for j=1:n + plot(real(G(:,j)), imag(G(:,j)),'-c5') % Plot the disks. + hold on + end + plot(real(e), imag(e), 'xg') % Plot the eigenvalues too. + axis(ax); + axis('square'); + hold off + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/gfpp.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,40 @@ +function A = gfpp(T, c) +%GFPP Matrix giving maximal growth factor for Gaussian elim. with pivoting. +% GFPP(T) is a matrix of order N for which Gaussian elimination +% with partial pivoting yields a growth factor 2^(N-1). +% T is an arbitrary nonsingular upper triangular matrix of order N-1. +% GFPP(T, C) sets all the multipliers to C (0 <= C <= 1) +% and gives growth factor (1+C)^(N-1). +% GFPP(N, C) (a special case) is the same as GFPP(EYE(N-1), C) and +% generates the well-known example of Wilkinson. + +% Reference: +% N.J. Higham and D.J. Higham, Large growth factors in +% Gaussian elimination with pivoting, SIAM J. Matrix Analysis and +% Appl., 10 (1989), pp. 155-164. + +if norm(T-triu(T),1) | any(~diag(T)) + error('First argument must be a nonsingular upper triangular matrix.') +end + +if nargin == 1, c = 1; end + +if c < 0 | c > 1 + error('Second parameter must be a scalar between 0 and 1 inclusive.') +end + +[m, m] = size(T); +if m == 1 % Handle the special case T = scalar + n = T; + m = n-1; + T = eye(n-1); +else + n = m+1; +end + +d = 1+c; +L = eye(n) - c*tril(ones(n), -1); +U = [T (d.^[0:n-2])'; zeros(1,m) d^(n-1)]; +A = L*U; +theta = max(abs(A(:))); +A(:,n) = (theta/norm(A(:,n),inf)) * A(:,n);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/gj.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,31 @@ +function x = gj(A, b, piv) +%GJ Gauss-Jordan elimination to solve Ax = b. +% x = GJ(A, b, PIV) solves Ax = b by Gauss-Jordan elimination, +% where A is a square, nonsingular matrix. +% PIV determines the form of pivoting: +% PIV = 0: no pivoting, +% PIV = 1 (default): partial pivoting. + +n = max(size(A)); +if nargin < 3, piv = 1; end + +for k=1:n + if piv + % Partial pivoting (below the diagonal). + [colmax, i] = max( abs(A(k:n, k)) ); + i = k+i-1; + if i ~= k + A( [k, i], : ) = A( [i, k], : ); + b( [k, i] ) = b( [i, k] ); + end + end + + irange = [1:k-1 k+1:n]; + jrange = k:n; + mult = A(irange,k)/A(k,k); % Multipliers. + A(irange, jrange) = A(irange, jrange) - mult*A(k, jrange); + b(irange) = b(irange) - mult*b(k); + +end + +x = diag(diag(A))\b;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/grcar.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,18 @@ +function G = grcar(n, k) +%GRCAR Grcar matrix - a Toeplitz matrix with sensitive eigenvalues. +% GRCAR(N, K) is an N-by-N matrix with -1s on the +% subdiagonal, 1s on the diagonal, and K superdiagonals of 1s. +% The default is K = 3. The eigenvalues of this matrix form an +% interesting pattern in the complex plane (try PS(GRCAR(32))). + +% References: +% J.F. Grcar, Operator coefficient methods for linear equations, +% Report SAND89-8691, Sandia National Laboratories, Albuquerque, +% New Mexico, 1989 (Appendix 2). +% N.M. Nachtigal, L. Reichel and L.N. Trefethen, A hybrid GMRES +% algorithm for nonsymmetric linear systems, SIAM J. Matrix Anal. +% Appl., 13 (1992), pp. 796-825. + +if nargin == 1, k = 3; end + +G = tril(triu(ones(n)), k) - diag(ones(n-1,1), -1);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/hadamard.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,41 @@ +function H = hadamard(n) +%HADAMARD Hadamard matrix. +% HADAMARD(N) is a Hadamard matrix of order N, that is, +% a matrix H with elements 1 or -1 such that H*H' = N*EYE(N). +% An N-by-N Hadamard matrix with N>2 exists only if REM(N,4) = 0. +% This function handles only the cases where N, N/12 or N/20 +% is a power of 2. + +% Reference: +% S.W. Golomb and L.D. Baumert, The search for Hadamard matrices, +% Amer. Math. Monthly, 70 (1963) pp. 12-17. + +% History: +% NJH (11/14/91), revised by CBM, 6/24/92, +% comment lines revised by NJH, August 1993. + +[f,e] = log2([n n/12 n/20]); +k = find(f==1/2 & e>0); +if isempty(k) + error(['N, N/12 or N/20 must be a power of 2.']); +end +e = e(k)-1; + +if k == 1 % N = 1 * 2^e; + H = [1]; + +elseif k == 2 % N = 12 * 2^e; + H = [ones(1,12); ones(11,1) ... + toeplitz([-1 -1 1 -1 -1 -1 1 1 1 -1 1],[-1 1 -1 1 1 1 -1 -1 -1 1 -1])]; + +elseif k == 3 % N = 20 * 2^e; + H = [ones(1,20); ones(19,1) ... + hankel([-1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1 1], ... + [1 -1 -1 1 1 -1 -1 -1 -1 1 -1 1 -1 1 1 1 1 -1 -1])]; +end + +% Kronecker product construction. +for i = 1:e + H = [H H + H -H]; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/hanowa.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,22 @@ +function A = hanowa(n, d) +%HANOWA A matrix whose eigenvalues lie on a vertical line in the complex plane. +% HANOWA(N, d) is the N-by-N block 2x2 matrix (thus N = 2M must be even) +% [d*EYE(M) -DIAG(1:M) +% DIAG(1:M) d*EYE(M)] +% It has complex eigenvalues lambda(k) = d +/- k*i (1 <= k <= M). +% Parameter d defaults to -1. + +% Reference: +% E. Hairer, S.P. Norsett and G. Wanner, Solving Ordinary +% Differential Equations I: Nonstiff Problems, Springer-Verlag, +% Berlin, 1987. (pp. 86-87) + +if nargin == 1, d = -1; end + +m = n/2; +if round(m) ~= m + error('N must be even.') +end + +A = [ d*eye(m) -diag(1:m) + diag(1:m) d*eye(m)];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/hilb.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,27 @@ +function H = hilb(n) +%HILB Hilbert matrix. +% HILB(N) is the N-by-N matrix with elements 1/(i+j-1). +% It is a famous example of a badly conditioned matrix. +% COND(HILB(N)) grows like EXP(3.5*N). +% See INVHILB (standard MATLAB routine) for the exact inverse, which +% has integer entries. +% HILB(N) is symmetric positive definite, totally positive, and a +% Hankel matrix. + +% References: +% M.-D. Choi, Tricks or treats with the Hilbert matrix, Amer. Math. +% Monthly, 90 (1983), pp. 301-312. +% N.J. Higham, Accuracy and Stability of Numerical Algorithms, +% Society for Industrial and Applied Mathematics, Philadelphia, PA, +% USA, 1996; sec. 26.1. +% M. Newman and J. Todd, The evaluation of matrix inversion +% programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476. +% D.E. Knuth, The Art of Computer Programming, +% Volume 1, Fundamental Algorithms, second edition, Addison-Wesley, +% Reading, Massachusetts, 1973, p. 37. + +if n == 1 + H = 1; +else + H = cauchy( (1:n) - .5); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/house.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,36 @@ +function [v, beta] = house(x) +%HOUSE Householder matrix. +% If [v, beta] = HOUSE(x) then H = EYE - beta*v*v' is a Householder +% matrix such that Hx = -sign(x(1))*norm(x)*e_1. +% NB: If x = 0 then v = 0, beta = 1 is returned. +% x can be real or complex. +% sign(x) := exp(i*arg(x)) ( = x./abs(x) when x ~= 0). + +% Theory: (textbook references Golub & Van Loan 1989, 38-43; +% Stewart 1973, 231-234, 262; Wilkinson 1965, 48-50). +% Hx = y: (I - beta*v*v')x = -s*e_1. +% Must have |s| = norm(x), v = x+s*e_1, and +% x'y = x'Hx =(x'Hx)' real => arg(s) = arg(x(1)). +% So take s = sign(x(1))*norm(x) (which avoids cancellation). +% v'v = (x(1)+s)^2 + x(2)^2 + ... + x(n)^2 +% = 2*norm(x)*(norm(x) + |x(1)|). +% +% References: +% G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, +% Johns Hopkins University Press, Baltimore, Maryland, 1989. +% G.W. Stewart, Introduction to Matrix Computations, Academic Press, +% New York, 1973, +% J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University +% Press, 1965. + +[m, n] = size(x); +if n > 1, error('Argument must be a column vector.'), end + +s = norm(x) * (sign(x(1)) + (x(1)==0)); % Modification for sign(0)=1. +v = x; +if s == 0, beta = 1; return, end % Quit if x is the zero vector. +v(1) = v(1) + s; +beta = 1/(s'*v(1)); % NB the conjugated s. + +% beta = 1/(abs(s)*(abs(s)+abs(x(1)) would guarantee beta real. +% But beta as above can be non-real (due to rounding) only when x is complex.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/invhess.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,39 @@ +function A = invhess(x, y) +%INVHESS Inverse of an upper Hessenberg matrix. +% INVHESS(X, Y), where X is an N-vector and Y an N-1 vector, +% is the matrix whose lower triangle agrees with that of +% ONES(N,1)*X' and whose strict upper triangle agrees with +% that of [1 Y]*ONES(1,N). +% The matrix is nonsingular if X(1) ~= 0 and X(i+1) ~= Y(i) +% for all i, and its inverse is an upper Hessenberg matrix. +% If Y is omitted it defaults to -X(1:N-1). +% Special case: if X is a scalar INVHESS(X) is the same as +% INVHESS(1:X). + +% References: +% F.N. Valvi and V.S. Geroyannis, Analytic inverses and +% determinants for a class of matrices, IMA Journal of Numerical +% Analysis, 7 (1987), pp. 123-128. +% W.-L. Cao and W.J. Stewart, A note on inverses of Hessenberg-like +% matrices, Linear Algebra and Appl., 76 (1986), pp. 233-240. +% Y. Ikebe, On inverses of Hessenberg matrices, Linear Algebra and +% Appl., 24 (1979), pp. 93-97. +% P. Rozsa, On the inverse of band matrices, Integral Equations and +% Operator Theory, 10 (1987), pp. 82-95. + +n = max(size(x)); +% Handle scalar x. +if n == 1 + n = x; + x = 1:n; +end +x = x(:); + +if nargin < 2, y = -x; end +y = y(:); + +% On next line, z = x'; A = z(ones(n,1),:) would be more efficient. +A = ones(n,1)*x'; +for j=2:n + A(1:j-1,j) = y(1:j-1); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/invol.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,21 @@ +function A = invol(n) +%INVOL An involutory matrix. +% A = INVOL(N) is an N-by-N involutory (A*A = EYE(N)) and +% ill-conditioned matrix. +% It is a diagonally scaled version of HILB(N). +% NB: B = (EYE(N)-A)/2 and B = (EYE(N)+A)/2 are idempotent (B*B = B). + +% Reference: +% A.S. Householder and J.A. Carpenter, The singular values +% of involutory and of idempotent matrices, Numer. Math. 5 (1963), +% pp. 234-237. + +A = hilb(n); + +d = -n; +A(:, 1) = d*A(:, 1); + +for i = 1:n-1 + d = -(n+i)*(n-i)*d/(i*i); + A(i+1, :) = d*A(i+1, :); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/ipjfact.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,42 @@ +function [A, detA] = ipjfact(n, k) +%IPJFACT A Hankel matrix with factorial elements. +% A = IPJFACT(N, K) is the matrix with +% A(i,j) = (i+j)! (K = 0, default) +% A(i,j) = 1/(i+j)! (K = 1) +% Both are Hankel matrices. +% The determinant and inverse are known explicitly. +% If a second output argument is present, d = DET(A) is returned: +% [A, d] = IPJFACT(N, K); + +% Suggested by P. R. Graves-Morris. +% +% Reference: +% M.J.C. Gover, The explicit inverse of factorial Hankel matrices, +% Dept. of Mathematics, University of Bradford, 1993. + +if nargin == 1, k = 0; end + +c = cumprod(2:n+1); +d = cumprod(n+1:2*n) * c(n-1); + +A = hankel(c, d); + +if k == 1 + A = ones(n)./A; +end + +if nargout == 2 + d = 1; + if k == 0 + for i=1:n-1 + d = d*prod(1:i+1)*prod(1:n-i); + end + d = d*prod(1:n+1); + else + for i=0:n-1 + d = d*prod(1:i)/prod(1:n+1+i); + end + if rem(n*(n-1)/2,2), d = -d; end + end + detA = d; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/jordbloc.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,8 @@ +function J = jordbloc(n, lambda) +%JORDBLOC Jordan block. +% JORDBLOC(N, LAMBDA) is the N-by-N Jordan block with eigenvalue +% LAMBDA. LAMBDA = 1 is the default. + +if nargin == 1, lambda = 1; end + +J = lambda*eye(n) + diag(ones(n-1,1),1);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/kahan.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,43 @@ +function U = kahan(n, theta, pert) +%KAHAN Kahan matrix - upper trapezoidal. +% KAHAN(N, THETA) is an upper trapezoidal matrix +% that has some interesting properties regarding estimation of +% condition and rank. +% The matrix is N-by-N unless N is a 2-vector, in which case it +% is N(1)-by-N(2). +% The parameter THETA defaults to 1.2. +% The useful range of THETA is 0 < THETA < PI. +% +% To ensure that the QR factorization with column pivoting does not +% interchange columns in the presence of rounding errors, the diagonal +% is perturbed by PERT*EPS*diag( [N:-1:1] ). +% The default is PERT = 25, which ensures no interchanges for KAHAN(N) +% up to at least N = 90 in IEEE arithmetic. +% KAHAN(N, THETA, PERT) uses the given value of PERT. + +% The inverse of KAHAN(N, THETA) is known explicitly: see +% Higham (1987, p. 588), for example. +% The diagonal perturbation was suggested by Christian Bischof. +% +% References: +% W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, +% 9 (1966), pp. 757-801. +% N.J. Higham, A survey of condition number estimation for +% triangular matrices, SIAM Review, 29 (1987), pp. 575-596. + +if nargin < 3, pert = 25; end +if nargin < 2, theta = 1.2; end + +r = n(1); % Parameter n specifies dimension: r-by-n. +n = n(max(size(n))); + +s = sin(theta); +c = cos(theta); + +U = eye(n) - c*triu(ones(n), 1); +U = diag(s.^[0:n-1])*U + pert*eps*diag( [n:-1:1] ); +if r > n + U(r,n) = 0; % Extend to an r-by-n matrix. +else + U = U(1:r,:); % Reduce to an r-by-n matrix. +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/kms.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,27 @@ +function A = kms(n, rho) +%KMS Kac-Murdock-Szego Toeplitz matrix. +% A = KMS(N, RHO) is the N-by-N Kac-Murdock-Szego Toeplitz matrix with +% A(i,j) = RHO^(ABS((i-j))) (for real RHO). +% If RHO is complex, then the same formula holds except that elements +% below the diagonal are conjugated. +% RHO defaults to 0.5. +% Properties: +% A has an LDL' factorization with +% L = INV(TRIW(N,-RHO,1)'), +% D(i,i) = (1-ABS(RHO)^2)*EYE(N) except D(1,1) = 1. +% A is positive definite if and only if 0 < ABS(RHO) < 1. +% INV(A) is tridiagonal. + +% Reference: +% W.F. Trench, Numerical solution of the eigenvalue problem +% for Hermitian Toeplitz matrices, SIAM J. Matrix Analysis and Appl., +% 10 (1989), pp. 135-146 (and see the references therein). + +if nargin < 2, rho = 0.5; end + +A = (1:n)'*ones(1,n); +A = abs(A - A'); +A = rho .^ A; +if imag(rho) + A = conj(tril(A,-1)) + triu(A); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/krylov.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,28 @@ +function B = krylov(A, x, j) +%KRYLOV Krylov matrix. +% KRYLOV(A, x, j) is the Krylov matrix +% [x, Ax, A^2x, ..., A^(j-1)x], +% where A is an n-by-n matrix and x is an n-vector. +% Defaults: x = ONES(n,1), j = n. +% KRYLOV(n) is the same as KRYLOV(RANDN(n)). + +% Reference: +% G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, +% Johns Hopkins University Press, Baltimore, Maryland, 1989, p. 369. + +[n, n] = size(A); + +if n == 1 % Handle special case A = scalar. + n = A; + A = randn(n); +end + +if nargin < 3, j = n; end +if nargin < 2, x = ones(n,1); end + + +B = ones(n,j); +B(:,1) = x(:); +for i=2:j + B(:,i) = A*B(:,i-1); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/lauchli.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,14 @@ +function A = lauchli(n, mu) +%LAUCHLI Lauchli matrix - rectangular. +% LAUCHLI(N, MU) is the (N+1)-by-N matrix [ONES(1,N); MU*EYE(N))]. +% It is a well-known example in least squares and other problems +% that indicates the dangers of forming A'*A. +% MU defaults to SQRT(EPS). + +% Reference: +% P. Lauchli, Jordan-Elimination und Ausgleichung nach +% kleinsten Quadraten, Numer. Math, 3 (1961), pp. 226-240. + +if nargin < 2, mu = sqrt(eps); end +A = [ones(1,n); + mu*eye(n)];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/lehmer.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,19 @@ +function A = lehmer(n) +%LEHMER Lehmer matrix - symmetric positive definite. +% A = LEHMER(N) is the symmetric positive definite N-by-N matrix with +% A(i,j) = i/j for j >= i. +% A is totally nonnegative. INV(A) is tridiagonal, and explicit +% formulas are known for its entries. +% N <= COND(A) <= 4*N*N. + +% References: +% M. Newman and J. Todd, The evaluation of matrix inversion +% programs, J. Soc. Indust. Appl. Math., 6 (1958), pp. 466-476. +% Solutions to problem E710 (proposed by D.H. Lehmer): The inverse +% of a matrix, Amer. Math. Monthly, 53 (1946), pp. 534-535. +% J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, +% Birkhauser, Basel, and Academic Press, New York, 1977, p. 154. + +A = ones(n,1)*(1:n); +A = A./A'; +A = tril(A) + tril(A,-1)';
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/lesp.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,22 @@ +function T = lesp(n) +%LESP A tridiagonal matrix with real, sensitive eigenvalues. +% LESP(N) is an N-by-N matrix whose eigenvalues are real and smoothly +% distributed in the interval approximately [-2*N-3.5, -4.5]. +% The sensitivities of the eigenvalues increase exponentially as +% the eigenvalues grow more negative. +% The matrix is similar to the symmetric tridiagonal matrix with +% the same diagonal entries and with off-diagonal entries 1, +% via a similarity transformation with D = diag(1!,2!,...,N!). + +% References: +% H.W.J. Lenferink and M.N. Spijker, On the use of stability regions in +% the numerical analysis of initial value problems, +% Math. Comp., 57 (1991), pp. 221-237. +% L.N. Trefethen, Pseudospectra of matrices, in Numerical Analysis 1991, +% Proceedings of the 14th Dundee Conference, +% D.F. Griffiths and G.A. Watson, eds, Pitman Research Notes in +% Mathematics, volume 260, Longman Scientific and Technical, Essex, +% UK, 1992, pp. 234-266. + +x = 2:n; +T = full(tridiag( ones(size(x))./x, -(2*[x n+1]+1), x));
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/lotkin.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,12 @@ +function A = lotkin(n) +%LOTKIN Lotkin matrix. +% A = LOTKIN(N) is the Hilbert matrix with its first row altered to +% all ones. A is unsymmetric, ill-conditioned, and has many negative +% eigenvalues of small magnitude. +% The inverse has integer entries and is known explicitly. + +% Reference: +% M. Lotkin, A set of test matrices, MTAC, 9 (1955), pp. 153-161. + +A = hilb(n); +A(1,:) = ones(1,n);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/makejcf.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,39 @@ +function A = makejcf(n, e, m, X) +%MAKEJCF A matrix with given Jordan canonical form. +% MAKEJCF(N, E, M) is a matrix having the Jordan canonical form +% whose i'th Jordan block is of dimension M(i) with eigenvalue E(i), +% and where N = SUM(M). +% Defaults: E = 1:N, M = ONES(SIZE(E)) with M(1) so that SUM(M) = N. +% The matrix is constructed by applying a random similarity +% transformation to the Jordan form. +% Alternatively, the matrix used in the similarity transformation +% can be specified as a fifth parameter. +% In particular, MAKEJCF(N, E, M, EYE(N)) returns the Jordan form +% itself. +% NB: The JCF is very sensitive to rounding errors. + +if nargin < 2, e = 1:n; end +if nargin < 3, m = ones(size(e)); m(1) = m(1) + n - sum(m); end + +if any( size(e(:)) ~= size(m(:)) ) + error('Parameters E and M must be of same dimension.') +end + +if sum(m) ~= n, error('Block dimensions must add up to N.'), end + +A = zeros(n); +j = 1; +for i=1:max(size(m)) + if m(i) > 1 + Jb = jordbloc(m(i),e(i)); + else + Jb = e(i); % JORDBLOC fails in n = 1 case. + end + A(j:j+m(i)-1,j:j+m(i)-1) = Jb; + j = j + m(i); +end + +if nargin < 4 + X = randn(n); +end +A = X\A*X;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/matrix.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,99 @@ +function A = matrix(k, n) +%MATRIX Test Matrix Toolbox information and matrix access by number. +% MATRIX(K, N) is the N-by-N instance of the matrix number K in +% the Test Matrix Toolbox (including some of the matrices provided +% with MATLAB), with all other parameters set to their default. +% N.B. Only those matrices which take an arbitrary dimension N +% are included (thus GALLERY is omitted, for example). +% MATRIX(K) is a string containing the name of the K'th matrix. +% MATRIX(0) is the number of matrices, i.e. the upper limit for K. +% Thus to set A to each N-by-N test matrix in turn use a loop like +% for k=1:matrix(0) +% A = matrix(k, N); +% Aname = matrix(k); % The name of the matrix +% end +% MATRIX(-1) returns the version number and date of the toolbox. +% MATRIX with no arguments lists the names of the M-files in the +% collection. + +% References: +% N.J. Higham. The Test Matrix Toolbox for Matlab (version 3.0), +% Numerical Analysis Report No. 276, Manchester Centre for +% Computational Mathematics, Manchester, England, September 1995. +% N.J. Higham, Algorithm 694: A collection of test matrices in +% MATLAB, ACM Trans. Math. Soft., 17 (1991), pp. 289-305. +% +% Matrices omitted are: gallery, hadamard, hanowa, lauchli, +% neumann, wathen, wilk. +% Matrices provided with MATLAB that are included here: invhilb, +% magic. + +% Set up string array a few lines at a time to avoid `input buffer line +% overflow'. + +matrices = ''; + +matrices = [matrices +'augment '; 'cauchy '; 'chebspec'; 'chebvand'; +'chow '; 'circul '; 'clement '; 'compan '; +'condex '; 'cycol '; 'dingdong'; 'dorr '; +'dramadah'; 'fiedler '; 'forsythe'; 'frank ';]; + +matrices = [matrices +'gearm '; 'gfpp '; 'grcar '; 'hilb '; +'invhess '; 'invol '; 'ipjfact '; 'jordbloc'; +'kahan '; 'kms '; 'krylov '; 'lehmer '; +'lesp '; 'lotkin '; 'makejcf '; 'minij ';]; + +matrices = [matrices +'moler '; 'ohess '; 'orthog '; 'parter '; +'pascal '; 'pdtoep '; 'pei '; 'pentoep '; +'prolate '; 'rando '; 'randsvd '; +'redheff '; 'riemann '; 'rschur '; 'smoke '; +'tridiag '; 'triw '; 'vand ';]; + +if nargin == 0 + +fprintf('Test matrices: \n') +fprintf(' \n') +fprintf('augment cycol gfpp kahan moler poisson triw \n') +fprintf('cauchy dingdong grcar kms neumann prolate vand \n') +fprintf('chebspec dorr hadamard krylov ohess rando wathen \n') +fprintf('chebvand dramadah hanowa lauchli orthog randsvd wilk \n') +fprintf('chow fiedler hilb lehmer parter redheff \n') +fprintf('circul forsythe invhess lesp pascal riemann \n') +fprintf('clement frank invol lotkin pdtoep rschur \n') +fprintf('compan gallery ipjfact makejcf pei smoke \n') +fprintf('condex gearm jordbloc minij pentoep tridiag \n') +fprintf(' \n') +fprintf('Visualization: Decompositions: Miscellaneous: \n') +fprintf(' \n') +fprintf('fv cgs gj bandred matrix show \n') +fprintf('gersh cholp mgs chop matsignt skewpart\n') +fprintf('ps cod poldec comp pnorm sparsify\n') +fprintf('pscont diagpiv signm cond qmult sub \n') +fprintf('see ge cpltaxes rq symmpart\n') +fprintf(' gecp dual seqa trap2tri\n') +fprintf(' eigsens seqcheb \n') +fprintf(' house seqm \n') +fprintf(' eigsens seqcheb \n\n') +fprintf('Direct search optimization: adsmax, mdsmax, nmsmax \n') +fprintf('Demonstration: tmtdemo \n') + +elseif nargin == 1 + if k == 0 + [A, temp] = size(matrices); + elseif k > 0 + A = matrices(k,:); + else + % Version number and date of collection. +% A = 'Version 1.0, July 4 1989'; +% A = 'Version 1.1, November 15 1989'; +% A = 'Version 1.2, May 30 1990'; +% A = 'Version 1.3, November 14 1991'; +% A = 'Version 2.0, November 14 1993'; + A = 'Version 3.0, September 19 1995'; + end +else + A = eval( [matrices(k,:) '(n)'] ); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/matsignt.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,41 @@ +function S = matsignt(T) +%MATSIGNT Matrix sign function of a triangular matrix. +% S = MATSIGN(T) computes the matrix sign function S of the +% upper triangular matrix T using a recurrence. + +% Adapted from FUNM. Called by SIGNM. + +if norm(tril(T,-1),1), error('Matrix must be upper triangular.'), end + +n = max(size(T)); + +S = diag( sign( diag(real(T)) ) ); +tol = 0; +for p = 1:n-1 + for i = 1:n-p + + j = i+p; + d = T(j,j) - T(i,i); + + if S(i,i) ~= -S(j,j) % Solve via S^2 = I if we can. + + % Get S(i,j) from S^2 = I. + k = i+1:j-1; + RHS = 0; + if k, RHS = RHS - S(i,k)*S(k,j); end + S(i,j) = RHS / (S(i,i)+S(j,j)); + + else + + % Get S(i,j) from S*T = T*S. + s = T(i,j)*(S(j,j)-S(i,i)); + if p > 1 + k = i+1:j-1; + s = s + T(i,k)*S(k,j) - S(i,k)*T(k,j); + end + S(i,j) = s/d; + + end + + end +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/minij.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,25 @@ +function A = minij(n) +%MINIJ Symmetric positive definite matrix MIN(i,j). +% A = MINIJ(N) is the N-by-N symmetric positive definite matrix with +% A(i,j) = MIN(i,j). +% Properties, variations: +% A has eigenvalues .25*sec^2(r*PI/(2*N+1)), r=1:N, and the eigenvectors +% are also known explicitly. +% INV(A) is tridiagonal: it is minus the second difference matrix +% except its (N,N) element is 1. +% 2*A-ONES(N) (Givens' matrix) has tridiagonal inverse and +% eigenvalues .5*sec^2((2r-1)PI/4N), r=1:N. +% (N+1)*ONES(N)-A also has a tridiagonal inverse. +% FLIPUD(TRIW(N,1)) is a square root of A. + +% References: +% J. Fortiana and C. M. Cuadras, A family of matrices, the discretized +% Brownian bridge, and distance-based regression, Linear Algebra +% Appl., 264 (1997), 173-188. (For the eigensystem of A.) +% J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, +% Birkhauser, Basel, and Academic Press, New York, 1977, p. 158. +% D.E. Rutherford, Some continuant determinants arising in physics and +% chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. +% (For the eigenvalues of Givens' matrix.) + +A = min( ones(n,1)*(1:n), (1:n)'*ones(1,n) );
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/moler.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,17 @@ +function A = moler(n, alpha) +%MOLER Moler matrix - symmetric positive definite. +% A = MOLER(N, ALPHA) is the symmetric positive definite N-by-N matrix +% U'*U where U = TRIW(N, ALPHA). +% For ALPHA = -1 (the default) A(i,j) = MIN(i,j)-2, A(i,i) = i. +% A has one small eigenvalue. + +% Nash (1990) attributes the ALPHA = -1 matrix to Moler. +% +% Reference: +% J.C. Nash, Compact Numerical Methods for Computers: Linear +% Algebra and Function Minimisation, second edition, Adam Hilger, +% Bristol, 1990 (Appendix 1). + +if nargin == 1, alpha = -1; end + +A = triw(n, alpha)'*triw(n, alpha);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/neumann.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,24 @@ +function [A, T] = neumann(n) +%NEUMANN Singular matrix from the discrete Neumann problem (sparse). +% NEUMANN(N) is the singular, row diagonally dominant matrix resulting +% from discretizing the Neumann problem with the usual five point +% operator on a regular mesh. +% It has a one-dimensional null space with null vector ONES(N,1). +% The dimension N should be a perfect square, or else a 2-vector, +% in which case the dimension of the matrix is N(1)*N(2). + +% Reference: +% R.J. Plemmons, Regular splittings and the discrete Neumann +% problem, Numer. Math., 25 (1976), pp. 153-161. + +if max(size(n)) == 1 + m = sqrt(n); + if m^2 ~= n, error('N must be a perfect square.'), end + n(1) = m; n(2) = m; +end + +T = tridiag(n(1), -1, 2, -1); +T(1,2) = -2; +T(n(1),n(1)-1) = -2; + +A = kron(T, eye(n(2))) + kron(eye(n(2)), T);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/ohess.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,43 @@ +function H = ohess(x) +%OHESS Random, orthogonal upper Hessenberg matrix. +% H = OHESS(N) is an N-by-N real, random, orthogonal +% upper Hessenberg matrix. +% Alternatively, H = OHESS(X), where X is an arbitrary real +% N-vector (N > 1) constructs H non-randomly using the elements +% of X as parameters. +% In both cases H is constructed via a product of N-1 Givens rotations. + +% Note: See Gragg (1986) for how to represent an N-by-N (complex) +% unitary Hessenberg matrix with positive subdiagonal elements in terms +% of 2N-1 real parameters (the Schur parametrization). +% This M-file handles the real case only and is intended simply as a +% convenient way to generate random or non-random orthogonal Hessenberg +% matrices. +% +% Reference: +% W.B. Gragg, The QR algorithm for unitary Hessenberg matrices, +% J. Comp. Appl. Math., 16 (1986), pp. 1-8. + +if any(imag(x)), error('Parameter must be real.'), end + +n = max(size(x)); + +if n == 1 +% Handle scalar x. + n = x; + x = rand(n-1,1)*2*pi; + H = eye(n); + H(n,n) = sign(randn); +else + H = eye(n); + H(n,n) = sign(x(n)) + (x(n)==0); % Second term ensures H(n,n) nonzero. +end + +for i=n:-1:2 + % Apply Givens rotation through angle x(i-1). + theta = x(i-1); + c = cos(theta); + s = sin(theta); + H( [i-1 i], : ) = [ c*H(i-1,:)+s*H(i,:) + -s*H(i-1,:)+c*H(i,:) ]; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/orthog.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,80 @@ +function Q = orthog(n, k) +%ORTHOG Orthogonal and nearly orthogonal matrices. +% Q = ORTHOG(N, K) selects the K'th type of matrix of order N. +% K > 0 for exactly orthogonal matrices, K < 0 for diagonal scalings of +% orthogonal matrices. +% Available types: (K = 1 is the default) +% K = 1: Q(i,j) = SQRT(2/(n+1)) * SIN( i*j*PI/(n+1) ) +% Symmetric eigenvector matrix for second difference matrix. +% K = 2: Q(i,j) = 2/SQRT(2*n+1)) * SIN( 2*i*j*PI/(2*n+1) ) +% Symmetric. +% K = 3: Q(r,s) = EXP(2*PI*i*(r-1)*(s-1)/n) / SQRT(n) (i=SQRT(-1)) +% Unitary, the Fourier matrix. Q^4 is the identity. +% This is essentially the same matrix as FFT(EYE(N))/SQRT(N)! +% K = 4: Helmert matrix: a permutation of a lower Hessenberg matrix, +% whose first row is ONES(1:N)/SQRT(N). +% K = 5: Q(i,j) = SIN( 2*PI*(i-1)*(j-1)/n ) + COS( 2*PI*(i-1)*(j-1)/n ). +% Symmetric matrix arising in the Hartley transform. +% K = -1: Q(i,j) = COS( (i-1)*(j-1)*PI/(n-1) ) +% Chebyshev Vandermonde-like matrix, based on extrema of T(n-1). +% K = -2: Q(i,j) = COS( (i-1)*(j-1/2)*PI/n) ) +% Chebyshev Vandermonde-like matrix, based on zeros of T(n). + +% References: +% N.J. Higham and D.J. Higham, Large growth factors in Gaussian +% elimination with pivoting, SIAM J. Matrix Analysis and Appl., +% 10 (1989), pp. 155-164. +% P. Morton, On the eigenvectors of Schur's matrix, J. Number Theory, +% 12 (1980), pp. 122-127. (Re. ORTHOG(N, 3)) +% H.O. Lancaster, The Helmert Matrices, Amer. Math. Monthly, 72 (1965), +% pp. 4-12. +% D. Bini and P. Favati, On a matrix algebra related to the discrete +% Hartley transform, SIAM J. Matrix Anal. Appl., 14 (1993), +% pp. 500-507. + +if nargin == 1, k = 1; end + +if k == 1 + % E'vectors second difference matrix + m = (1:n)'*(1:n) * (pi/(n+1)); + Q = sin(m) * sqrt(2/(n+1)); + +elseif k == 2 + + m = (1:n)'*(1:n) * (2*pi/(2*n+1)); + Q = sin(m) * (2/sqrt(2*n+1)); + +elseif k == 3 % Vandermonde based on roots of unity + + m = 0:n-1; + Q = exp(m'*m*2*pi*sqrt(-1)/n) / sqrt(n); + +elseif k == 4 % Helmert matrix + + Q = tril(ones(n)); + Q(1,2:n) = ones(1,n-1); + for i=2:n + Q(i,i) = -(i-1); + end + Q = diag( sqrt( [n 1:n-1] .* [1:n] ) ) \ Q; + +elseif k == 5 % Hartley matrix + + m = (0:n-1)'*(0:n-1) * (2*pi/n); + Q = (cos(m) + sin(m))/sqrt(n); + +elseif k == -1 + % extrema of T(n-1) + m = (0:n-1)'*(0:n-1) * (pi/(n-1)); + Q = cos(m); + +elseif k == -2 + % zeros of T(n) + m = (0:n-1)'*(.5:n-.5) * (pi/n); + Q = cos(m); + +else + + error('Illegal value for second parameter.') + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/parter.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,18 @@ +function A = parter(n) +%PARTER Parter matrix - a Toeplitz matrix with singular values near PI. +% PARTER(N) is the matrix with (i,j) element 1/(i-j+0.5). +% It is a Cauchy matrix and a Toeplitz matrix. + +% At the Second SIAM Conference on Linear Algebra, Raleigh, N.C., +% 1985, Cleve Moler noted that most of the singular values of +% PARTER(N) are very close to PI. An explanation of the phenomenon +% was given by Parter; see also the paper by Tyrtyshnikov. +% +% References: +% The MathWorks Newsletter, Volume 1, Issue 1, March 1986, page 2. +% S.V. Parter, On the distribution of the singular values of Toeplitz +% matrices, Linear Algebra and Appl., 80 (1986), pp. 115-130. +% E.E. Tyrtyshnikov, Cauchy-Toeplitz matrices and some applications, +% Linear Algebra and Appl., 149 (1991), pp. 1-18. + +A = cauchy( (1:n)+0.5, -(1:n) );
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/pascal.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,58 @@ +function P = pascal(n, k) +%PASCAL Pascal matrix. +% P = PASCAL(N) is the Pascal matrix of order N: a symmetric positive +% definite matrix with integer entries taken from Pascal's +% triangle. +% The Pascal matrix is totally positive and its inverse has +% integer entries. Its eigenvalues occur in reciprocal pairs. +% COND(P) is approximately 16^N/(N*PI) for large N. +% PASCAL(N,1) is the lower triangular Cholesky factor (up to signs +% of columns) of the Pascal matrix. It is involutary (is its own +% inverse). +% PASCAL(N,2) is a transposed and permuted version of PASCAL(N,1) +% which is a cube root of the identity. + +% References: +% R. Brawer and M. Pirovino, The linear algebra of the Pascal matrix, +% Linear Algebra and Appl., 174 (1992), pp. 13-23 (this paper +% gives a factorization of L = PASCAL(N,1) and a formula for the +% elements of L^k). +% N.J. Higham, Accuracy and Stability of Numerical Algorithms, +% Society for Industrial and Applied Mathematics, Philadelphia, PA, +% USA, 1996; sec. 26.4. +% S. Karlin, Total Positivity, Volume 1, Stanford University Press, +% 1968. (Page 137: shows i+j-1 choose j is TP (i,j=0,1,...). +% PASCAL(N) is a submatrix of this matrix.) +% M. Newman and J. Todd, The evaluation of matrix inversion programs, +% J. Soc. Indust. Appl. Math., 6(4):466--476, 1958. +% H. Rutishauser, On test matrices, Programmation en Mathematiques +% Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, +% 1966, pp. 349-365. (Gives an integral formula for the +% elements of PASCAL(N).) +% J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, +% Birkhauser, Basel, and Academic Press, New York, 1977, p. 172. +% H.W. Turnbull, The Theory of Determinants, Matrices, and Invariants, +% Blackie, London and Glasgow, 1929. (PASCAL(N,2) on page 332.) + +if nargin == 1, k = 0; end + +P = diag( (-1).^[0:n-1] ); +P(:, 1) = ones(n,1); + +% Generate the Pascal Cholesky factor (up to signs). +for j=2:n-1 + for i=j+1:n + P(i,j) = P(i-1,j) - P(i-1,j-1); + end +end + +if k == 0 + + P = P*P'; + +elseif k == 2 + + P = rot90(P,3); + if n/2 == round(n/2), P = -P; end + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/pdtoep.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,29 @@ +function T = pdtoep(n, m, w, theta) +%PDTOEP Symmetric positive definite Toeplitz matrix. +% PDTOEP(N, M, W, THETA) is an N-by-N symmetric positive (semi-) +% definite (SPD) Toeplitz matrix, comprised of the sum of M rank 2 +% (or, for certain THETA, rank 1) SPD Toeplitz matrices. +% Specifically, +% T = W(1)*T(THETA(1)) + ... + W(M)*T(THETA(M)), +% where T(THETA(k)) has (i,j) element COS(2*PI*THETA(k)*(i-j)). +% Defaults: M = N, W = RAND(M,1), THETA = RAND(M,1). + +% Reference: +% G. Cybenko and C.F. Van Loan, Computing the minimum eigenvalue of +% a symmetric positive definite Toeplitz matrix, SIAM J. Sci. Stat. +% Comput., 7 (1986), pp. 123-131. + +if nargin < 2, m = n; end +if nargin < 3, w = rand(m,1); end +if nargin < 4, theta = rand(m,1); end + +if max(size(w)) ~= m | max(size(theta)) ~= m + error('Arguments W and THETA must be vectors of length M.') +end + +T = zeros(n); +E = 2*pi*( (1:n)'*ones(1,n) - ones(n,1)*(1:n) ); + +for i=1:m + T = T + w(i) * cos( theta(i)*E ); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/pei.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,14 @@ +function P = pei(n, alpha) +%PEI Pei matrix. +% PEI(N, ALPHA), where ALPHA is a scalar, is the symmetric matrix +% ALPHA*EYE(N) + ONES(N). +% If ALPHA is omitted then ALPHA = 1 is used. +% The matrix is singular for ALPHA = 0, -N. + +% Reference: +% M.L. Pei, A test matrix for inversion procedures, +% Comm. ACM, 5 (1962), p. 508. + +if nargin == 1, alpha = 1; end + +P = alpha*eye(n) + ones(n);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/pentoep.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,26 @@ +function P = pentoep(n, a, b, c, d, e) +%PENTOEP Pentadiagonal Toeplitz matrix (sparse). +% P = PENTOEP(N, A, B, C, D, E) is the N-by-N pentadiagonal +% Toeplitz matrix with diagonals composed of the numbers +% A =: P(3,1), B =: P(2,1), C =: P(1,1), D =: P(1,2), E =: P(1,3). +% Default: (A,B,C,D,E) = (1,-10,0,10,1) (a matrix of Rutishauser). +% This matrix has eigenvalues lying approximately on +% the line segment 2*cos(2*t) + 20*i*sin(t). +% +% Interesting plots are +% PS(FULL(PENTOEP(32,0,1,0,0,1/4))) - `triangle' +% PS(FULL(PENTOEP(32,0,1/2,0,0,1))) - `propeller' +% PS(FULL(PENTOEP(32,0,1/2,1,1,1))) - `fish' + +% References: +% R.M. Beam and R.F. Warming, The asymptotic spectra of +% banded Toeplitz and quasi-Toeplitz matrices, SIAM J. Sci. +% Comput. 14 (4), 1993, pp. 971-1006. +% H. Rutishauser, On test matrices, Programmation en Mathematiques +% Numeriques, Editions Centre Nat. Recherche Sci., Paris, 165, +% 1966, pp. 349-365. + +if nargin == 1, a = 1; b = -10; c = 0; d = 10; e = 1; end + +P = spdiags([ a*ones(n,1) b*ones(n,1) c*ones(n,1) d*ones(n,1) .... + e*ones(n,1) ], -2:2, n, n);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/pnorm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,102 @@ +function [est, x, k] = pnorm(A, p, tol, noprint) +%PNORM Estimate of matrix p-norm (1 <= p <= inf). +% [EST, x, k] = PNORM(A, p, TOL) estimates the Holder p-norm of a +% matrix A, using the p-norm power method with a specially +% chosen starting vector. +% TOL is a relative convergence tolerance (default 1E-4). +% Returned are the norm estimate EST (which is a lower bound for the +% exact p-norm), the corresponding approximate maximizing vector x, +% and the number of power method iterations k. +% A nonzero fourth argument causes trace output to the screen. +% If A is a vector, this routine simply returns NORM(A, p). +% +% See also NORM, NORMEST. + +% Note: The estimate is exact for p = 1, but is not always exact for +% p = 2 or p = inf. Code could be added to treat p = 2 and p = inf +% separately. +% +% Calls DUAL and SEQA. +% +% Reference: +% N.J. Higham, Estimating the matrix p-norm, +% Numer. Math., 62 (1992), pp. 539-555. + +if nargin < 2, error('Must specify norm via second parameter.'), end +[m,n] = size(A); +if min(m,n) == 1, est = norm(A,p); return, end + +if nargin < 4, noprint = 0; end +if nargin < 3, tol = 1e-4; end + +% Stage I. Use Algorithm OSE to get starting vector x for power method. +% Form y = B*x, at each stage choosing x(k) = c and scaling previous +% x(k+1:n) by s, where norm([c s],p)=1. + +sm = 9; % Number of samples. +y = zeros(m,1); x = zeros(n,1); + +for k=1:n + + if k == 1 + c = 1; s = 0; + else + W = [A(:,k) y]; + + if p == 2 % Special case. Solve exactly for 2-norm. + [U,S,V] = svd(full(W)); + c = V(1,1); s = V(2,1); + + else + + fopt = 0; + for th=seqa(0,pi,sm) + c1 = cos(th); s1 = sin(th); + nrm = norm([c1 s1],p); + c1 = c1/nrm; s1 = s1/nrm; % [c1 s1] has unit p-norm. + f = norm( W*[c1 s1]', p ); + if f > fopt + fopt = f; + c = c1; s = s1; + end + end + + end + end + + x(k) = c; + y = x(k)*A(:,k) + s*y; + if k > 1, x(1:k-1) = s*x(1:k-1); end + +end + +est = norm(y,p); +if noprint, fprintf('Alg OSE: %9.4e\n', est), end + +% Stage II. Apply Algorithm PM (the power method). + +q = dual(p); +k = 1; + +while 1 + + y = A*x; + est_old = est; + est = norm(y,p); + + z = A' * dual(y,p); + + if noprint + fprintf('%2.0f: norm(y) = %9.4e, norm(z) = %9.4e', ... + k, norm(y,p), norm(z,q)) + fprintf(' rel_incr(est) = %9.4e\n', (est-est_old)/est) + end + + if ( norm(z,q) <= z'*x | abs(est-est_old)/est <= tol ) & k > 1 + return + end + + x = dual(z,q); + k = k + 1; + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/poisson.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,14 @@ +function A = poisson(n) +%POISSON Block tridiagonal matrix from Poisson's equation (sparse). +% POISSON(N) is the block tridiagonal matrix of order N^2 +% resulting from discretizing Poisson's equation with the +% 5-point operator on an N-by-N mesh. + +% Reference: +% G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, +% Johns Hopkins University Press, Baltimore, Maryland, 1989 +% (Section 4.5.4). + +S = tridiag(n,-1,2,-1); +I = speye(n); +A = kron(I,S) + kron(S,I);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/poldec.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,28 @@ +function [U, H] = poldec(A) +%POLDEC Polar decomposition. +% [U, H] = POLDEC(A) computes a matrix U of the same dimension +% (m-by-n) as A, and a Hermitian positive semi-definite matrix H, +% such that A = U*H. +% U has orthonormal columns if m >= n, and orthonormal rows if m <= n. +% U and H are computed via an SVD of A. +% U is a nearest unitary matrix to A in both the 2-norm and the +% Frobenius norm. + +% Reference: +% N.J. Higham, Computing the polar decomposition---with applications, +% SIAM J. Sci. Stat. Comput., 7(4):1160--1174, 1986. +% +% (The name `polar' is reserved for a graphics routine.) + +[m, n] = size(A); + +[P, S, Q] = svd(A, 0); % Economy size. +if m < n % Ditto for the m<n case. + S = S(:, 1:m); + Q = Q(:, 1:m); +end +U = P*Q'; +if nargout == 2 + H = Q*S*Q'; + H = (H + H')/2; % Force Hermitian by taking nearest Hermitian matrix. +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/prolate.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,21 @@ +function A = prolate(n, w) +%PROLATE Prolate matrix - symmetric, ill-conditioned Toeplitz matrix. +% A = PROLATE(N, W) is the N-by-N prolate matrix with parameter W. +% It is a symmetric Toeplitz matrix. +% If 0 < W < 0.5 then +% - A is positive definite +% - the eigenvalues of A are distinct, lie in (0, 1), and +% tend to cluster around 0 and 1. +% W defaults to 0.25. + +% Reference: +% J.M. Varah. The Prolate matrix. Linear Algebra and Appl., +% 187:269--278, 1993. + +if nargin == 1, w = 0.25; end + +a = zeros(n,1); +a(1) = 2*w; +a(2:n) = sin( 2*pi*w*(1:n-1) ) ./ ( pi*(1:n-1) ); + +A = toeplitz(a);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/ps.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,94 @@ +function y = ps(A, m, tol, rl, marksize) +%PS Dot plot of a pseudospectrum. +% PS(A, M, TOL, RL) plots an approximation to a pseudospectrum +% of the square matrix A, using M random perturbations of size TOL. +% M defaults to a SIZE(A)-dependent value and TOL to 1E-3. +% RL defines the type of perturbation: +% RL = 0 (default): absolute complex perturbations of 2-norm TOL. +% RL = 1: absolute real perturbations of 2-norm TOL. +% RL = -1: componentwise real perturbations of size TOL. +% The eigenvalues of A are plotted as crosses `x'. +% PS(A, M, TOL, RL, MARKSIZE) uses the specified marker size instead +% of a size that depends on the figure size, the matrix order, and M. +% If MARKSIZE < 0, the plot is suppressed and the plot data is returned +% as an output argument. +% PS(A, 0) plots just the eigenvalues of A. + +% For a given TOL, the pseudospectrum of A is the set of +% pseudo-eigenvalues of A, that is, the set +% { e : e is an eigenvalue of A+E, for some E with NORM(E) <= TOL }. +% +% Reference: +% L.N. Trefethen, Pseudospectra of matrices, in D.F. Griffiths and +% G.A. Watson, eds, Numerical Analysis 1991, Proceedings of the 14th +% Dundee Conference, vol. 260, Pitman Research Notes in Mathematics, +% Longman Scientific and Technical, Essex, UK, 1992, pp. 234-266. + +if diff(size(A)), error('Matrix must be square.'), end +n = max(size(A)); + +if nargin < 5, marksize = 0; end +if nargin < 4, rl = 0; end +if nargin < 3, tol = 1e-3; end +if nargin < 2, m = max(1, round( 25*exp(-0.047*n) )); end + +if m == 0 + e = eig(A); + ax = cpltaxes(e); + plot(real(e), imag(e), 'xg') + axis(ax); + axis('square'); + return +end + +x = zeros(m*n,1); +i = sqrt(-1); + +for j = 1:m + if rl == -1 % Componentwise. + dA = -ones(n) + 2*rand(n); % Uniform random numbers on [-1,1]. + dA = tol * A .* dA; + else + if rl == 0 % Complex absolute. + dA = randn(n) + i*randn(n); + else % Real absolute. + dA = randn(n); + end + dA = tol/norm(dA)*dA; + end + e = eig(A + dA); + x((j-1)*n+1:j*n) = e; +end + +if marksize >= 0 + + ax = cpltaxes(x); + h = plot(real(x),imag(x),'.'); + axis(ax); + axis('square'); + + % Next block adapted from SPY.M. + if marksize == 0 + units = get(gca,'units'); + set(gca,'units','points'); + pos = get(gca,'position'); + nps = 2.4*sqrt(n*m); % Factor based on number of pseudo-ei'vals plotted. + myguess = round(3*min(pos(3:4))/nps); +% [nps myguess] + marksize = max(1,myguess); + set(gca,'units',units); + end + set(h,'markersize',marksize); +% set(h,'linemarkersize',marksize); + + hold on + e = eig(A); + plot(real(e),imag(e),'xw'); + set(h,'markersize',marksize); + hold off + +else + + y = x; + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/pscont.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,104 @@ +function [x, y, z, m] = pscont(A, k, npts, ax, levels) +%PSCONT Contours and colour pictures of pseudospectra. +% PSCONT(A, K, NPTS, AX, LEVELS) plots LOG10(1/NORM(R(z))), +% where R(z) = INV(z*I-A) is the resolvent of the square matrix A, +% over an NPTS-by-NPTS grid. +% NPTS defaults to a SIZE(A)-dependent value. +% The limits are AX(1) and AX(2) on the x-axis and +% AX(3) and AX(4) on the y-axis. +% If AX is omitted, suitable limits are guessed based on the +% eigenvalues of A. +% The eigenvalues of A are plotted as crosses `x'. +% K determines the type of plot: +% K = 0 (default) PCOLOR and CONTOUR +% K = 1 PCOLOR only +% K = 2 SURFC (SURF and CONTOUR) +% K = 3 SURF only +% K = 4 CONTOUR only +% The contours levels are specified by the vector LEVELS, which +% defaults to -10:-1 (recall we are plotting log10 of the data). +% Thus, by default, the contour lines trace out the boundaries of +% the epsilon pseudospectra for epsilon = 1e-10, ..., 1e-1. +% [X, Y, Z, NPTS] = PSCONT(A, ...) returns the plot data X, Y, Z +% and the value of NPTS used. +% +% After calling this function you may want to change the +% color map (e.g., type COLORMAP HOT - see HELP COLOR) and the +% shading (e.g., type SHADING INTERP - see HELP INTERP). +% For an explanation of the term `pseudospectra' see PS.M. +% When A is real and the grid is symmetric about the x-axis, this +% routine exploits symmetry to halve the computational work. + +% Colour pseduospectral pictures of this type are referred to as +% `spectral portraits' by Godunov, Kostin, and colleagues. +% References: +% V. I. Kostin, Linear algebra algorithms with guaranteed accuracy, +% Technical Report TR/PA/93/05, CERFACS, Toulouse, France, 1993. +% L.N. Trefethen, Pseudospectra of matrices, in D.F. Griffiths and +% G.A. Watson, eds, Numerical Analysis 1991, Proceedings of the 14th +% Dundee Conference, vol. 260, Pitman Research Notes in Mathematics, +% Longman Scientific and Technical, Essex, UK, 1992, pp. 234-266. + +Areal = ~norm(imag(A),1); + +if nargin < 5, levels = -10:-1; end +e = eig(A); +if nargin < 4 + ax = cpltaxes(e); + if Areal, ax(3) = -ax(4); end % Make sure region symmetric about x-axis. +end +n = max(size(A)); +if nargin < 3, npts = round( min(max(5, sqrt(20^2*10^3/n^3) ), 30)); end +if nargin < 2, k = 0; end + +nptsx = npts; nptsy = npts; +Ysymmetry = (Areal & ax(3) == -ax(4)); + +x = seqa(ax(1), ax(2), npts); +y = seqa(ax(3), ax(4), npts); +if Ysymmetry % Exploit symmetry about x-axis. + nptsy = ceil(npts/2); + y1 = y; + y = y(1:nptsy); +end + +[xx, yy] = meshgrid(x,y); +z = xx + sqrt(-1)*yy; +I = eye(n); +Smin = zeros(nptsy, nptsx); + +for j=1:nptsx + for i=1:nptsy + Smin(i,j) = min( svd( z(i,j)*I-A ) ); + end +end + +z = log10( Smin + eps ); +if Ysymmetry + z = [z; z(nptsy-rem(npts,2):-1:1,:)]; + y = y1; +end + +if k == 0 | k == 1 + pcolor(x, y, z); hold on +elseif k == 2 + surfc(x, y, z); hold on +elseif k == 3 + surf(x, y, z); hold on +end + +if k == 0 | k == 4 + contour(x, y, z, levels); hold on +end + +if k ~= 2 & k ~= 3 + if k == 0 | k == 1 + s = 'k'; % Black. + else + s = 'w'; % White. + end + plot(real(e),imag(e),['x' s]); +end + +axis('square'); +hold off
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/qmult.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,47 @@ +function B = qmult(A) +%QMULT Pre-multiply by random orthogonal matrix. +% QMULT(A) is Q*A where Q is a random real orthogonal matrix from +% the Haar distribution, of dimension the number of rows in A. +% Special case: if A is a scalar then QMULT(A) is the same as +% QMULT(EYE(A)). + +% Called by RANDSVD. +% +% Reference: +% G.W. Stewart, The efficient generation of random +% orthogonal matrices with an application to condition estimators, +% SIAM J. Numer. Anal., 17 (1980), 403-409. + +[n, m] = size(A); + +% Handle scalar A. +if max(n,m) == 1 + n = A; + A = eye(n); +end + +d = zeros(n); + +for k = n-1:-1:1 + + % Generate random Householder transformation. + x = randn(n-k+1,1); + s = norm(x); + sgn = sign(x(1)) + (x(1)==0); % Modification for sign(1)=1. + s = sgn*s; + d(k) = -sgn; + x(1) = x(1) + s; + beta = s*x(1); + + % Apply the transformation to A. + y = x'*A(k:n,:); + A(k:n,:) = A(k:n,:) - x*(y/beta); + +end + +% Tidy up signs. +for i=1:n-1 + A(i,:) = d(i)*A(i,:); +end +A(n,:) = A(n,:)*sign(randn); +B = A;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/rando.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,21 @@ +function A = rando(n, k) +%RANDO Random matrix with elements -1, 0 or 1. +% A = RANDO(N, K) is a random N-by-N matrix with elements from +% one of the following discrete distributions (default K = 1): +% K = 1: A(i,j) = 0 or 1 with equal probability, +% K = 2: A(i,j) = -1 or 1 with equal probability, +% K = 3: A(i,j) = -1, 0 or 1 with equal probability. +% N may be a 2-vector, in which case the matrix is N(1)-by-N(2). + +if nargin < 2, k = 1; end + +m = n(1); % Parameter n specifies dimension: m-by-n. +n = n(max(size(n))); + +if k == 1 % {0, 1} + A = floor( rand(m,n) + .5 ); +elseif k == 2 % {-1, 1} + A = 2*floor( rand(m,n) + .5 ) - 1; +elseif k == 3 % {-1, 0, 1} + A = round( 3*rand(m,n) - 1.5 ); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/randsvd.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,104 @@ +function A = randsvd(n, kappa, mode, kl, ku) +%RANDSVD Random matrix with pre-assigned singular values. +% RANDSVD(N, KAPPA, MODE, KL, KU) is a (banded) random matrix of order N +% with COND(A) = KAPPA and singular values from the distribution MODE. +% N may be a 2-vector, in which case the matrix is N(1)-by-N(2). +% Available types: +% MODE = 1: one large singular value, +% MODE = 2: one small singular value, +% MODE = 3: geometrically distributed singular values, +% MODE = 4: arithmetically distributed singular values, +% MODE = 5: random singular values with unif. dist. logarithm. +% If omitted, MODE defaults to 3, and KAPPA defaults to SQRT(1/EPS). +% If MODE < 0 then the effect is as for ABS(MODE) except that in the +% original matrix of singular values the order of the diagonal entries +% is reversed: small to large instead of large to small. +% KL and KU are the lower and upper bandwidths respectively; if they +% are omitted a full matrix is produced. +% If only KL is present, KU defaults to KL. +% Special case: if KAPPA < 0 then a random full symmetric positive +% definite matrix is produced with COND(A) = -KAPPA and +% eigenvalues distributed according to MODE. +% KL and KU, if present, are ignored. + +% Reference: +% N.J. Higham, Accuracy and Stability of Numerical Algorithms, +% Society for Industrial and Applied Mathematics, Philadelphia, PA, +% USA, 1996; sec. 26.3. + +% This routine is similar to the more comprehensive Fortran routine xLATMS +% in the following reference: +% J.W. Demmel and A. McKenney, A test matrix generation suite, +% LAPACK Working Note #9, Courant Institute of Mathematical Sciences, +% New York, 1989. + +if nargin < 2, kappa = sqrt(1/eps); end +if nargin < 3, mode = 3; end +if nargin < 4, kl = n-1; end % Full matrix. +if nargin < 5, ku = kl; end % Same upper and lower bandwidths. + +if abs(kappa) < 1, error('Condition number must be at least 1!'), end +posdef = 0; if kappa < 0, posdef = 1; kappa = -kappa; end % Special case. + +p = min(n); +m = n(1); % Parameter n specifies dimension: m-by-n. +n = n(max(size(n))); + +if p == 1 % Handle case where A is a vector. + A = randn(m, n); + A = A/norm(A); + return +end + +j = abs(mode); + +% Set up vector sigma of singular values. +if j == 3 + factor = kappa^(-1/(p-1)); + sigma = factor.^[0:p-1]; + +elseif j == 4 + sigma = ones(p,1) - (0:p-1)'/(p-1)*(1-1/kappa); + +elseif j == 5 % In this case cond(A) <= kappa. + sigma = exp( -rand(p,1)*log(kappa) ); + +elseif j == 2 + sigma = ones(p,1); + sigma(p) = 1/kappa; + +elseif j == 1 + sigma = ones(p,1)./kappa; + sigma(1) = 1; +end + +% Convert to diagonal matrix of singular values. +if mode < 0 + sigma = sigma(p:-1:1); +end +sigma = diag(sigma); + +if posdef % Handle special case. + Q = qmult(p); + A = Q'*sigma*Q; + A = (A + A')/2; % Ensure matrix is symmetric. + return +end + +if m ~= n + sigma(m, n) = 0; % Expand to m-by-n diagonal matrix. +end + +if kl == 0 & ku == 0 % Diagonal matrix requested - nothing more to do. + A = sigma; + return +end + +% A = U*sigma*V, where U, V are random orthogonal matrices from the +% Haar distribution. +A = qmult(sigma'); +A = qmult(A'); + +if kl < n-1 | ku < n-1 % Bandwidth reduction. + A = bandred(A, kl, ku); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/redheff.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,28 @@ +function A = redheff(n) +%REDHEFF A (0,1) matrix of Redheffer associated with the Riemann hypothesis. +% A = REDHEFF(N) is an N-by-N matrix of 0s and 1s defined by +% A(i,j) = 1 if j = 1 or if i divides j, +% A(i,j) = 0 otherwise. +% It has N - FLOOR(LOG2(N)) - 1 eigenvalues equal to 1, +% a real eigenvalue (the spectral radius) approximately SQRT(N), +% a negative eigenvalue approximately -SQRT(N), +% and the remaining eigenvalues are provably ``small''. +% Barrett and Jarvis (1992) conjecture that +% ``the small eigenvalues all lie inside the unit circle +% ABS(Z) = 1'', +% and a proof of this conjecture, together with a proof that some +% eigenvalue tends to zero as N tends to infinity, would yield +% a new proof of the prime number theorem. +% The Riemann hypothesis is true if and only if +% DET(A) = O( N^(1/2+epsilon) ) for every epsilon > 0 +% (`!' denotes factorial). +% See also RIEMANN. + +% Reference: +% W.W. Barrett and T.J. Jarvis, +% Spectral Properties of a Matrix of Redheffer, +% Linear Algebra and Appl., 162 (1992), pp. 673-683. + +i = (1:n)'*ones(1,n); +A = ~rem(i',i); +A(:,1) = ones(n,1);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/riemann.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,23 @@ +function A = riemann(n) +%RIEMANN A matrix associated with the Riemann hypothesis. +% A = RIEMANN(N) is an N-by-N matrix for which the +% Riemann hypothesis is true if and only if +% DET(A) = O( N! N^(-1/2+epsilon) ) for every epsilon > 0 +% (`!' denotes factorial). +% A = B(2:N+1, 2:N+1), where +% B(i,j) = i-1 if i divides j and -1 otherwise. +% Properties include, with M = N+1: +% Each eigenvalue E(i) satisfies ABS(E(i)) <= M - 1/M. +% i <= E(i) <= i+1 with at most M-SQRT(M) exceptions. +% All integers in the interval (M/3, M/2] are eigenvalues. +% +% See also REDHEFF. + +% Reference: +% F. Roesler, Riemann's hypothesis as an eigenvalue problem, +% Linear Algebra and Appl., 81 (1986), pp. 153-198. + +n = n+1; +i = (2:n)'*ones(1,n-1); +j = i'; +A = i .* (~rem(j,i)) - ones(n-1);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/rq.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,7 @@ +function z = rq(A,x) +%RQ Rayleigh quotient. +% RQ(A,x) is the Rayleigh quotient of A and x, x'*A*x/(x'*x). + +% Called by FV. + +z = x'*A*x/(x'*x);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/rschur.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,33 @@ +function A = rschur(n, mu, x, y) +%RSCHUR An upper quasi-triangular matrix. +% A = RSCHUR(N, MU, X, Y) is an N-by-N matrix in real Schur form. +% All the diagonal blocks are 2-by-2 (except for the last one, if N +% is odd) and the k'th has the form [x(k) y(k); -y(k) x(k)]. +% Thus the eigenvalues of A are x(k) +/- i*y(k). +% MU (default 1) controls the departure from normality. +% Defaults: X(k) = -k^2/10, Y(k) = -k, i.e., the eigenvalues +% lie on the parabola x = -y^2/10. + +% References: +% F. Chatelin, Eigenvalues of Matrices, John Wiley, Chichester, 1993; +% Section 4.2.7. +% F. Chatelin and V. Fraysse, Qualitative computing: Elements +% of a theory for finite precision computation, Lecture notes, +% CERFACS, Toulouse, France and THOMSON-CSF, Orsay, France, +% June 1993. + +m = floor(n/2)+1; +alpha = 10; beta = 1; + +if nargin < 4, y = -(1:m)/beta; end +if nargin < 3, x = -(1:m).^2/alpha; end +if nargin < 2, mu = 1; end + +A = diag( mu*ones(n-1,1), 1 ); +for i=1:2:2*(m-1) + j = (i+1)/2; + A(i:i+1,i:i+1) = [x(j) y(j); -y(j) x(j)]; +end +if 2*m ~= n, + A(n,n) = x(m); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/see.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,64 @@ +function see(A, k) +%SEE Pictures of a matrix and its (pseudo-) inverse. +% SEE(A) displays MESH(A), MESH(PINV(A)), SEMILOGY(SVD(A),'o'), +% and (if A is square) FV(A) in four subplot windows. +% SEE(A, 1) plots an approximation to the pseudospectrum in the +% third window instead of the singular values. +% SEE(A, -1) plots only the eigenvalues in the fourth window, +% which is much quicker than plotting the field of values. +% If A is complex, only real parts are used for the mesh plots. +% If A is sparse, just SPY(A) is shown. + +if nargin < 2, k = 0; end +[m, n] = size(A); +square = (m == n); +clf + +if issparse(A) + + spy(A); + +else + + B = pinv(A); + s = svd(A); + zs = (s == zeros(size(s))); + if any( zs ) + s( zs ) = []; % Remove zero singular values for semilogy plot. + end + + subplot(2,2,1) + mesh(real(A)), axis('ij'), drawnow + subplot(2,2,2) + mesh(real(B)), axis('ij'), drawnow + + if k <= 0 + subplot(2,2,3) + semilogy(s, 'og') + hold on, semilogy(s, '-'), hold off, drawnow + if any(zs), subplot(2,2,3), title('Zero(s) omitted'), subplot(2,2,4), end + elseif k == 1 + subplot(2,2,3) + ps(A); drawnow + end + + if square + if k == -1 + subplot(2,2,4) + ps(A, 0); + else + subplot(2,2,4) + fv(A); + end + else + if k == 0 + subplot(2,2,4) + axis off + else + clf + end + text(0,0,'Matrix not square.') + end + subplot; + +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/seqa.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,13 @@ +function y = seqa(a, b, n) +%SEQA Additive sequence. +% Y = SEQA(A, B, N) produces a row vector comprising N equally +% spaced numbers starting at A and finishing at B. +% If N is omitted then 10 points are generated. + +if nargin == 2, n = 10; end + +if n <= 1 + y = a; + return +end +y = [a+(0:n-2)*(b-a)/(n-1), b];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/seqcheb.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,17 @@ +function x = seqcheb(n, k) +%SEQCHEB Sequence of points related to Chebyshev polynomials. +% X = SEQCHEB(N, K) produces a row vector of length N. +% There are two choices: +% K = 1: zeros of T_N, (the default) +% K = 2: extrema of T_{N-1}, +% where T_k is the Chebsyhev polynomial of degree k. + +if nargin == 1, k = 1; end + +if k == 1 % Zeros of T_n + i = 1:n; j = .5*ones(1,n); + x = cos( (i-j) * (pi/n) ); +elseif k == 2 % Extrema of T_(n-1) + i = 0:n-1; + x = cos( i * (pi/(n-1)) ); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/seqm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,17 @@ +function y = seqm(a, b, n) +%SEQM Multiplicative sequence. +% Y = SEQM(A, B, N) produces a row vector comprising N +% logarithmically equally spaced numbers, starting at A ~= 0 +% and finishing at B ~= 0. +% If A*B < 0 and N > 2 then complex results are produced. +% If N is omitted then 10 points are generated. + +if nargin == 2, n = 10; end + +if n <= 1 + y = a; + return +end +p = [0:n-2]/(n-1); +r = (b/a).^p; +y = [a*r, b];
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/show.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,9 @@ +function show(x) +%SHOW Display signs of matrix elements. +% SHOW(X) displays X in `FORMAT +' form, that is, +% with `+', `-' and blank representing positive, negative +% and zero elements respectively. + +format + +disp(x) +format
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/signm.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,21 @@ +function [S, N] = signm(A) +%SIGNM Matrix sign decomposition. +% [S, N] = SIGNM(A) is the matrix sign decomposition A = S*N, +% computed via the Schur decomposition. +% S is the matrix sign function, sign(A). + +% Reference: +% N.J. Higham, The matrix sign decomposition and its relation to the +% polar decomposition, Linear Algebra and Appl., 212/213:3-20, 1994. + +[Q, T] = schur(A); +[Q, T] = rsf2csf(Q, T); +S = Q * matsignt(T) * Q'; + +% Only problem with Schur method is possible nonzero imaginary part when +% A is real. Next line takes care of that. +if ~norm(imag(A),1), S = real(S); end + +if nargout == 2 + N = S*A; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/skewpart.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,8 @@ +function S = skewpart(A) +%SKEWPART Skew-symmetric (skew-Hermitian) part. +% SKEWPART(A) is the skew-symmetric (skew-Hermitian) part of A, +% (A - A')/2. +% It is the nearest skew-symmetric (skew-Hermitian) matrix to A in +% both the 2- and the Frobenius norms. + +S = (A - A')./2;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/smoke.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,22 @@ +function A = smoke(n, k) +%SMOKE Smoke matrix - complex, with a `smoke ring' pseudospectrum. +% SMOKE(N) is an N-by-N matrix with 1s on the +% superdiagonal, 1 in the (N,1) position, and powers of +% roots of unity along the diagonal. +% SMOKE(N, 1) is the same except for a zero (N,1) element. +% The eigenvalues of SMOKE(N, 1) are the N'th roots of unity; +% those of SMOKE(N) are the N'th roots of unity times 2^(1/N). +% +% Try PS(SMOKE(32)). For SMOKE(N, 1) the pseudospectrum looks +% like a sausage folded back on itself. +% GERSH(SMOKE(N, 1)) is interesting. + +% Reference: +% L. Reichel and L.N. Trefethen, Eigenvalues and pseudo-eigenvalues of +% Toeplitz matrices, Linear Algebra and Appl., 162-164:153-185, 1992. + +if nargin < 2, k = 0; end + +w = exp(2*pi*i/n); +A = diag( [w.^(1:n-1) 1] ) + diag(ones(n-1,1),1); +if k == 0, A(n,1) = 1; end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/sparsify.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,24 @@ +function A = sparsify(A, p) +%SPARSIFY Randomly sets matrix elements to zero. +% S = SPARSIFY(A, P) is A with elements randomly set to zero +% (S = S' if A is square and A = A', i.e. symmetry is preserved). +% Each element has probability P of being zeroed. +% Thus on average 100*P percent of the elements of A will be zeroed. +% Default: P = 0.25. + +if nargin < 2, p = 0.25; end +if p<0 | p>1, error('Second parameter must be between 0 and 1 inclusive.'), end + +% Is A square and symmetric? +symm = 0; +if min(size(A)) == max(size(A)) + if norm(A-A',1) == 0, symm = 1; end +end + +if ~symm + A = A .* (rand(size(A)) > p); % Unsymmetric case +else + A = triu(A,1) .* (rand(size(A)) > p); % Preserve symmetry + A = A + A'; + A = A + diag( diag(A) .* (rand(size(diag(A))) > p) ); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/sub.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,17 @@ +function S = sub(A, i, j) +%SUB Principal submatrix. +% SUB(A,i,j) is A(i:j,i:j). +% SUB(A,i) is the leading principal submatrix of order i, +% A(1:i,1:i), if i>0, and the trailing principal submatrix +% of order ABS(i) if i<0. + +if nargin == 2 + if i >= 0 + S = A(1:i, 1:i); + else + n = min(size(A)); + S = A(n+i+1:n, n+i+1:n); + end +else + S = A(i:j, i:j); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/symmpart.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,7 @@ +function S = symmpart(A) +%SYMMPART Symmetric (Hermitian) part. +% SYMMPART(A) is the symmetric (Hermitian) part of A, (A + A')/2. +% It is the nearest symmetric (Hermitian) matrix to A in both the +% 2- and the Frobenius norms. + +S = (A + A')./2;
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/tmtdemo.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,263 @@ +clc +format compact +echo on + +%TMTDEMO Demonstration of Test Matrix Toolbox. +% N. J. Higham. + +% The Test Matrix Toolbox contains test matrices, visualization routines, +% and other miscellaneous routines. + +% The version of the toolbox is + +matrix(-1) +echo on + +% For this demonstration you will need to view both the command window +% and one figure window. +% This demonstration emphasises graphics and shows only +% some of the features of the toolbox. + +pause % Press any key to continue after pauses. + +% A list of the available M-files is obtained by typing `matrix': + +matrix + +pause + +% The FV command plots the boundary of the field of values of a matrix +% (the set of all Rayleigh quotients) and plots the eigenvalues as +% crosses (`x'). Here are some examples: + +% The Grcar matrix is a Toeplitz matrix of the following form: + +grcar(5) + +% Here is the field of values of the 10-by-10 Grcar matrix: + +fv(grcar(10)); +title('fv(grcar(10))') + +pause + +% Next, we form a random orthogonal matrix and look at its field of values. +% The boundary is the convex hull of the eigenvalues since A is normal. + +A = randsvd(10, 1); +fv(A); +title('randsvd(10, 1)') +pause + +% The RANDSVD commands generates random matrices with pre-assigned +% condition number, and various singular value distributions: + +A = randsvd(6, 1e6, 3); % Exponential distribution. + +format short e +svd(A)' +cond(A) +pause + +% The PS command plots an approximation to a pseudospectrum of A, +% which is the set of complex numbers that are eigenvalues of some +% perturbed matrix A + E, with the norm of E at most epsilon +% (default: epsilon = 1E-3). +% The eigenvalues of A are plotted as crosses (`x'). +% Here are some interesting PS plots. + +% First, we use the KAHAN matrix, a triangular matrix made up of sines and +% cosines. Here is an approximate pseudospectrum of the 10-by-10 matrix: + +ps(kahan(10),25); +title('ps(kahan(10))') +pause + +% Next, a different way of looking at pseudospectra, via norms of +% the resolvent. (The resolvent of A is INV(z*I-A), where z is a complex +% variable). PSCONT gives a color map with a superimposed contour +% plot. Here we specify a region of the complex plane in +% which the 8-by-8 Kahan matrix is interesting to look at. + +pscont(kahan(8), 0, 20, [0.2 1.2 -0.5 0.5]); +title('pscont(kahan(8))') +pause + +% The TRIW matrix is upper triangular, made up of 1s and -1s: + +triw(4) + +% Here is a combined surface and contour plot of its resolvent. +% Notice how the repeated eigenvalue 1 `sucks in' the resolvent. + +pscont(triw(11), 2, 15, [-2 2 -2 2]); +title('pscont(triw(11))') +pause + +% The next PSCONT plot is for the companion matrix of the characteristic +% polynomial of the CHEBSPEC matrix: + +A = chebspec(8); C = compan(A); + +% The SHOW command shows the +/- pattern of the elements of a matrix, with +% blanks for zero elements: + +show(C) + +pscont(C, 2, 20, [-.1 .1 -.1 .1]); +title('pscont(compan(chebspec(8)))') +pause + +% The following matrix has a pseudospectrum in the form of a limacon. + +n = 25; A = triw(n,1,2) - eye(n); +sub(A, 6) % Leading principal 6-by-6 submatrix of A. +ps(A); +pause + +% Here is the 8-by-8 Frank matrix. +A = frank(8) + +% We can get a visual representation of the matrix using the SEE +% command, which produces subplots with the following layout: +% /---------------------------------\ +% | MESH(A) MESH(INV(A)) | +% | SEMILOGY(SVD(A)) FV(A) | +% \---------------------------------/ +% where FV is the field of values. + +see(A) + +pause + +% The Frank matrix is well-known for having ill-conditioned eigenvalues. +% Here are the eigenvalues (in column 1) together with the corresponding +% eigenvalue condition numbers (in column 2): + +format short e +[V, D, c] = eigsens(A); +[diag(D) c 1./diag(D)] + +% In the last column are shown the reciprocals of the eigenvalues. +% Notice that if LAMBDA is an eigenvalue, so is 1/LAMBDA! + +pause + +% Matlab's MAGIC function produces magic squares: + +A = magic(5) + +% Using the toolbox routine PNORM we can estimate the matrix p-norm +% for any value of p. + +[pnorm(A,1) pnorm(A,1.5) pnorm(A,2) pnorm(A,pi) pnorm(A,inf)] + +% As this example suggests, the p-norm of a magic square is +% constant for all p! + +pause + +% GERSH plots Gershgorin disks. Here are some interesting examples. + +gersh(lesp(12)); +title('gersh(lesp(12))') +pause + +gersh(hanowa(10)); +title('gersh(hanowa(10))') +pause + +gersh(ipjfact(6,1)); +title('gersh(ipjfact(6,1))') +pause + +gersh(smoke(16,1)); +title('gersh(smoke(16,1))') +pause + +% A Hadamard matrix has elements 1 or -1 and mutually orthogonal rows: + +show(hadamard(16)) + +% A CONTOUR plot of this matrix is interesting: + +contour(hadamard(16)) +pause + +% There are a few sparse matrices in the toolbox. +% WATHEN is a finite element matrix with random entries. + +spy(wathen(6,6)); % SPY plot of sparsity pattern. + +pause + +% GFPP generates matrices for which Gaussian elimination with partial +% pivoting produces a large growth factor. + +gfpp(6) +pause + +% Let's find the growth factor for partial pivoting and complete pivoting +% for a bigger matrix: + +A = gfpp(20); +[L, U] = lu(A); % Partial pivoting. +max(max(abs(U))) / max(max(abs(A))) + +[L, U, P, Q, rho] = gecp(A); % Complete pivoting using toolbox routine GECP. +rho +% As expected, complete pivoting does not produce large growth here. +pause + +% The toolbox function MATRIX allows the test matrices to be accessed +% by number. The following piece of code steps through all the +% square matrices of arbitrary dimension, setting A to each 10-by-10 +% matrix in turn. It evaluate the 2-norm condition number and the +% ratio of the largest to smallest eigenvalue (in absolute values). +c = []; e = []; j = 1; +for i=1:matrix(0) + A = full(matrix(i, 10)); + if norm(skewpart(A),1) % If not Hermitian... + c1 = cond(A); + eg = eig(A); + e1 = max(abs(eg)) / min(abs(eg)); + % Filter out extremely ill-conditioned matrices. + if c1 <= 1e10, c(j) = c1; e(j) = e1; j = j + 1; end + end +end + +% The following plots confirm that the condition number can be much +% larger than the extremal eigenvalue ratio. +echo off +j = max(size(c)); +subplot(2,1,1) +semilogy(1:j, c, 'x', 1:j, e, 'o'), hold on +semilogy(1:j, c, '-', 1:j, e, '--'), hold off +title('cond: x, eig_ratio: o') +subplot(2,1,2) +semilogy(1:j, c./e) +title('cond/eig_ratio') +echo on +pause + +% Finally, here are three interesting pseudospectra based on pentadiagonal +% Toeplitz matrices: + +A = full(pentoep(6,0,1/2,0,0,1)) + +subplot(1,1,1) +ps(full(pentoep(32,0,1/2,0,0,1))); % Propeller +title('ps(full(pentoep(32,0,1/2,0,0,1))') +pause + +ps(inv(full(pentoep(32,0,1,1,0,.25)))); % Man in the moon +title('ps(inv(full(pentoep(32,0,1,1,0,.25))') +pause + +ps(full(pentoep(32,0,1/2,1,1,1))); % Fish +title('ps(full(pentoep(32,0,1/2,1,1,1)))') +pause + +echo off +clear A L U P Q V D +format
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/trap2tri.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,56 @@ +function [Q, T] = trap2tri(L) +%TRAP2TRI Unitary reduction of trapezoidal matrix to triangular form. +% [Q, T] = TRAP2TRI(L), where L is an m-by-n lower trapezoidal +% matrix with m >= n, produces a unitary Q such that QL = [T; 0], +% where T is n-by-n and lower triangular. +% Q is a product of Householder transformations. + +% Called by RANDSVD. +% +% Reference: +% G.H. Golub and C.F. Van Loan, Matrix Computations, second edition, +% Johns Hopkins University Press, Baltimore, Maryland, 1989. +% P5.2.5, p. 220. + +[n, r] = size(L); + +if r > n | norm(L-tril(L),1) + error('Matrix must be lower trapezoidal and m-by-n with m >= n.') +end + +Q = eye(n); % To hold product of H.T.s + +if r ~= n + + % Reduce nxr L = r [L1] to lower triangular form: QL = [T]. + % n-r [L2] [0] + + for j=r:-1:1 + % x is the vector to be reduced, which we overwrite with the H.T. vector. + x = L(j:n,j); + x(2:r-j+1) = zeros(r-j,1); % These elts of column left unchanged. + s = norm(x)*(sign(x(1)) + (x(1)==0)); % Modification for sign(1)=1. + + % Nothing to do if x is zero (or x=a*e_1, but we don't check for that). + if s ~= 0 + x(1) = x(1) + s; + beta = s'*x(1); + + % Implicitly apply H.T. to pivot column. + % L(r+1:n,j) = zeros(n-r,1); % We throw these elts away at the end. + L(j,j) = -s; + + % Apply H.T. to rest of matrix. + if j > 1 + y = x'*L(j:n, 1:j-1); + L(j:n, 1:j-1) = L(j:n, 1:j-1) - x*(y/beta); + end + + % Update H.T. product. + y = x'*Q(j:n,:); + Q(j:n,:) = Q(j:n,:) - x*(y/beta); + end + end +end + +T = L(1:r,:); % Rows r+1:n have been zeroed out.
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/tridiag.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,41 @@ +function T = tridiag(n, x, y, z) +%TRIDIAG Tridiagonal matrix (sparse). +% TRIDIAG(X, Y, Z) is the tridiagonal matrix with subdiagonal X, +% diagonal Y, and superdiagonal Z. +% X and Z must be vectors of dimension one less than Y. +% Alternatively TRIDIAG(N, C, D, E), where C, D, and E are all +% scalars, yields the Toeplitz tridiagonal matrix of order N +% with subdiagonal elements C, diagonal elements D, and superdiagonal +% elements E. This matrix has eigenvalues (Todd 1977) +% D + 2*SQRT(C*E)*COS(k*PI/(N+1)), k=1:N. +% TRIDIAG(N) is the same as TRIDIAG(N,-1,2,-1), which is +% a symmetric positive definite M-matrix (the negative of the +% second difference matrix). + +% References: +% J. Todd, Basic Numerical Mathematics, Vol. 2: Numerical Algebra, +% Birkhauser, Basel, and Academic Press, New York, 1977, p. 155. +% D.E. Rutherford, Some continuant determinants arising in physics and +% chemistry---II, Proc. Royal Soc. Edin., 63, A (1952), pp. 232-241. + +if nargin == 1, x = -1; y = 2; z = -1; end +if nargin == 3, z = y; y = x; x = n; end + +x = x(:); y = y(:); z = z(:); % Force column vectors. + +if max( [ size(x) size(y) size(z) ] ) == 1 + x = x*ones(n-1,1); + z = z*ones(n-1,1); + y = y*ones(n,1); +else + [nx, m] = size(x); + [ny, m] = size(y); + [nz, m] = size(z); + if (ny - nx - 1) | (ny - nz -1) + error('Dimensions of vector arguments are incorrect.') + end +end + +% T = diag(x, -1) + diag(y) + diag(z, 1); % For non-sparse matrix. +n = max(size(y)); +T = spdiags([ [x;0] y [0;z] ], -1:1, n, n);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/triw.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,41 @@ +function t = triw(n, alpha, k) +%TRIW Upper triangular matrix discussed by Wilkinson and others. +% TRIW(N, ALPHA, K) is the upper triangular matrix with ones on +% the diagonal and ALPHAs on the first K >= 0 superdiagonals. +% N may be a 2-vector, in which case the matrix is N(1)-by-N(2) and +% upper trapezoidal. +% Defaults: ALPHA = -1, +% K = N - 1 (full upper triangle). +% TRIW(N) is a matrix discussed by Kahan, Golub and Wilkinson. +% +% Ostrowski (1954) shows that +% COND(TRIW(N,2)) = COT(PI/(4*N))^2, +% and for large ABS(ALPHA), +% COND(TRIW(N,ALPHA)) is approximately ABS(ALPHA)^N*SIN(PI/(4*N-2)). +% +% Adding -2^(2-N) to the (N,1) element makes TRIW(N) singular, +% as does adding -2^(1-N) to all elements in the first column. + +% References: +% G.H. Golub and J.H. Wilkinson, Ill-conditioned eigensystems and the +% computation of the Jordan canonical form, SIAM Review, +% 18(4), 1976, pp. 578-619. +% W. Kahan, Numerical linear algebra, Canadian Math. Bulletin, +% 9 (1966), pp. 757-801. +% A.M. Ostrowski, On the spectrum of a one-parametric family of +% matrices, J. Reine Angew. Math., 193 (3/4), 1954, pp. 143-160. +% J.H. Wilkinson, Singular-value decomposition---basic aspects, +% in D.A.H. Jacobs, ed., Numerical Software---Needs and Availability, +% Academic Press, London, 1978, pp. 109-135. + +m = n(1); % Parameter n specifies dimension: m-by-n. +n = n(max(size(n))); + +if nargin < 3, k = n-1; end +if nargin < 2, alpha = -1; end + +if max(size(alpha)) ~= 1 + error('Second argument must be a scalar.') +end + +t = tril( eye(m,n) + alpha*triu(ones(m,n), 1), k);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/vand.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,29 @@ +function V = vand(m, p) +%VAND Vandermonde matrix. +% V = VAND(P), where P is a vector, produces the (primal) +% Vandermonde matrix based on the points P, i.e. V(i,j) = P(j)^(i-1). +% VAND(M,P) is a rectangular version of VAND(P) with M rows. +% Special case: If P is a scalar then P equally spaced points on [0,1] +% are used. + +% Reference: +% N.J. Higham, Stability analysis of algorithms for solving +% confluent Vandermonde-like systems, SIAM J. Matrix Anal. Appl., +% 11 (1990), pp. 23-41. + +if nargin == 1, p = m; end +n = max(size(p)); + +% Handle scalar p. +if n == 1 + n = p; + p = seqa(0,1,n); +end + +if nargin == 1, m = n; end + +p = p(:).'; % Ensure p is a row vector. +V = ones(m,n); +for i=2:m + V(i,:) = p.*V(i-1,:); +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/wathen.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,54 @@ +function A = wathen(nx, ny, k) +%WATHEN Wathen matrix - a finite element matrix (sparse, random entries). +% A = WATHEN(NX,NY) is a sparse random N-by-N finite element matrix +% where N = 3*NX*NY + 2*NX + 2*NY + 1. +% A is precisely the `consistent mass matrix' for a regular NX-by-NY +% grid of 8-node (serendipity) elements in 2 space dimensions. +% A is symmetric positive definite for any (positive) values of +% the `density', RHO(NX,NY), which is chosen randomly in this routine. +% In particular, if D = DIAG(DIAG(A)), then +% 0.25 <= EIG(INV(D)*A) <= 4.5 +% for any positive integers NX and NY and any densities RHO(NX,NY). +% This diagonally scaled matrix is returned by WATHEN(NX,NY,1). + +% Reference: +% A.J. Wathen, Realistic eigenvalue bounds for the Galerkin +% mass matrix, IMA J. Numer. Anal., 7 (1987), pp. 449-457. + +if nargin < 2, error('Two dimensioning arguments must be specified.'), end +if nargin < 3, k = 0; end + +e1 = [6 -6 2 -8;-6 32 -6 20;2 -6 6 -6;-8 20 -6 32]; +e2 = [3 -8 2 -6;-8 16 -8 20;2 -8 3 -8;-6 20 -8 16]; +e = [e1 e2; e2' e1]/45; +n = 3*nx*ny+2*nx+2*ny+1; +A = sparse(n,n); + +RHO = 100*rand(nx,ny); + + for j=1:ny + for i=1:nx + + nn(1) = 3*j*nx+2*i+2*j+1; + nn(2) = nn(1)-1; + nn(3) = nn(2)-1; + nn(4) = (3*j-1)*nx+2*j+i-1; + nn(5) = 3*(j-1)*nx+2*i+2*j-3; + nn(6) = nn(5)+1; + nn(7) = nn(6)+1; + nn(8) = nn(4)+1; + + em = e*RHO(i,j); + + for krow=1:8 + for kcol=1:8 + A(nn(krow),nn(kcol)) = A(nn(krow),nn(kcol))+em(krow,kcol); + end + end + + end + end + +if k == 1 + A = diag(diag(A)) \ A; +end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/toolbox/wilk.m Wed May 06 14:56:53 2015 +0200 @@ -0,0 +1,45 @@ +function [A, b] = wilk(n) +%WILK Various specific matrices devised/discussed by Wilkinson. +% [A, b] = WILK(N) is the matrix or system of order N. +% N = 3: upper triangular system Ux=b illustrating inaccurate solution. +% N = 4: lower triangular system Lx=b, ill-conditioned. +% N = 5: HILB(6)(1:5,2:6)*1.8144. Symmetric positive definite. +% N = 21: W21+, tridiagonal. Eigenvalue problem. + +% References: +% J.H. Wilkinson, Error analysis of direct methods of matrix inversion, +% J. Assoc. Comput. Mach., 8 (1961), pp. 281-330. +% J.H. Wilkinson, Rounding Errors in Algebraic Processes, Notes on Applied +% Science No. 32, Her Majesty's Stationery Office, London, 1963. +% J.H. Wilkinson, The Algebraic Eigenvalue Problem, Oxford University +% Press, 1965. + +if n == 3 + % Wilkinson (1961) p.323. + A = [ 1e-10 .9 -.4 + 0 .9 -.4 + 0 0 1e-10]; + b = [ 0 0 1]'; + +elseif n == 4 + % Wilkinson (1963) p.105. + A = [0.9143e-4 0 0 0 + 0.8762 0.7156e-4 0 0 + 0.7943 0.8143 0.9504e-4 0 + 0.8017 0.6123 0.7165 0.7123e-4]; + b = [0.6524 0.3127 0.4186 0.7853]'; + +elseif n == 5 + % Wilkinson (1965), p.234. + A = hilb(6); + A = A(1:5, 2:6)*1.8144; + +elseif n == 21 + % Taken from gallery.m. Wilkinson (1965), p.308. + E = diag(ones(n-1,1),1); + m = (n-1)/2; + A = diag(abs(-m:m)) + E + E'; + +else + error('Sorry, that value of N is not available.') +end