The Matrix Computation Toolbox
11 Sep 2002
11 Sep 2002)
A collection of M-files for carrying out various numerical linear algebra tasks.
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.
% 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);
U = P*Q';
if nargout == 2
H = Q*S*Q';
H = (H + H')/2; % Force Hermitian by taking nearest Hermitian matrix.