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annotate mftoolbox/polar_newton.m @ 0:8f23314345f4 draft

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Create local repository for matrix toolboxes. Step #0 done.
author Antonio Pino Robles <data.script93@gmail.com>
date Wed, 06 May 2015 14:56:53 +0200
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8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
Antonio Pino Robles <data.script93@gmail.com>
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1 function [U,H,k] = polar_newton(A)
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2 %POLAR_NEWTON Polar decomposition by scaled Newton iteration.
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3 % [U,H,k] = POLAR_NEWTON(A), where the matrix A is square and
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4 % nonsingular, computes a unitary U and a Hermitian positive
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5 % definite H such that A = U*H. k is the number of iterations.
8f23314345f4 Create local repository for matrix toolboxes. Step #0 done.
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6 % A Newton iteration with acceleration parameters is used.
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7
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8 accel_tol = 1e-2; % Precise value not important.
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9 need_accel = 1;
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10
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11 tol = mft_tolerance(A);
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12
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13 X = A;
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14 reldiff = inf;
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15 maxit = 16;
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16
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17 for k = 1:maxit
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18
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19 Xold = X;
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20 Xinv = inv(X);
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21 if need_accel
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22 g = ( norm(Xinv,1)*norm(Xinv,inf) / (norm(X,1)*norm(X,inf)) )^(1/4);
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23 else
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24 g = 1;
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25 end
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26 X = 0.5*(g*X + Xinv'/g);
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27 reldiff_old = reldiff;
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28 diff_F = norm(X-Xold,'fro');
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29 reldiff = diff_F/norm(X,'fro');
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30
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31 if need_accel && (reldiff < accel_tol), need_accel = false; end
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32 cged = (diff_F <= sqrt(tol)) || (reldiff > reldiff_old/2 && ~need_accel);
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33 if cged, break, end
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34
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35 if k == maxit, error('Not converged after %2.0f iterations', maxit), end
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36
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37 end
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38
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39 U = X;
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40 if nargout >= 2
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41 H = U'*A;
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42 H = (H + H')/2; % Force Hermitian by taking nearest Hermitian matrix.
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43 end