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author Antonio Pino Robles <data.script93@gmail.com>
date Fri, 29 May 2015 09:48:36 +0200
parents 8f23314345f4
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1 function [X,k] = signm_newton(A,scal)
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2 %SIGNM_NEWTON Matrix sign function by Newton iteration.
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3 % [S,k] = SIGNM_NEWTON(A,SCAL) computes the matrix sign
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4 % function S of A using the scaled Newton iteration, with
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5 % scale factors specified by SCAL:
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6 % SCAL = 0: no scaling.
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7 % SCAL = 1: determinantal scaling (default).
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8 % SCAL = 2: spectral scaling.
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9 % SCAL = 3: norm scaling.
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10 % k is the number of iterations.
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11
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12 if nargin < 2, scal = 1; end
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13
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14 tol = mft_tolerance(A);
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15 accel_tol = 1e-2; % Precise value not important.
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16 need_accel = 1;
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17 n = length(A);
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18 maxit = 16;
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19
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20 X = A;
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21 reldiff = inf;
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22
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23 for k = 1:maxit
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24
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25 Xold = X;
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26 Xinv = inv(X);
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27 if need_accel && scal > 0
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28 % In practice should estimate spectral radius and 2-norms;
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29 % here they are computed exactly.
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30 switch scal
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31 case 1
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32 g = abs(det(X))^(-1/n);
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33 case 2
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34 s1 = max(abs(eig(Xinv)));
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35 s2 = max(abs(eig(X)));
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36 g = sqrt(s1/s2);
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37 case 3
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38 g = sqrt( norm(Xinv) / (norm(X)) );
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39 end
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40 X = g*X; Xinv = Xinv/g;
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41 end
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42 X = 0.5*(X + Xinv);
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43 diff_F = norm(X-Xold,'fro');
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44 reldiff_old = reldiff;
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45 reldiff = diff_F/norm(X,'fro');
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46
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47 if need_accel && (reldiff < accel_tol), need_accel = false; end
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48 cged = (diff_F <= sqrt( tol*norm(X)/norm(Xinv) ) || ...
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49 reldiff > reldiff_old/2 && ~need_accel);
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50 if cged, return, end
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51
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52 end
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53 error('Not converged after %2.0f iterations', maxit)