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annotate mftoolbox/logm_frechet_pade.m @ 8:a587712dcf5f draft default tip
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author | Antonio Pino Robles <data.script93@gmail.com> |
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date | Fri, 29 May 2015 09:48:36 +0200 |
parents | 8f23314345f4 |
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rev | line source |
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1 function L = logm_frechet_pade(A,E) |
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2 %LOGM_FRECHET_PADE Frechet derivative of matrix logarithm via Pade approx. |
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3 % L = LOGM_FRECHET_PADE(A,E) evaluates the Frechet |
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4 % derivative of the matrix logarithm at A in the direction E via the |
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5 % inverse scaling and squaring and a Pade approximant of the function |
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6 % tanh(x)/x. A must have no eigenvalues on the negative real axis. |
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7 |
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8 real_data = isreal(A) && isreal(E); |
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9 % Form complex Schur form if A not already upper triangular. |
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10 use_Schur = false; |
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11 if ~isequal(A,triu(A)) |
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12 use_Schur = true; |
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13 [Q,T] = schur(A,'complex'); A = T; E = Q'*E*Q; |
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14 end |
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15 |
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16 I = eye(size(A)); |
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17 B = A; |
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18 for i = 0:inf |
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19 if norm(B-I,1) <= 1-1/exp(1), s = i; break, end |
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20 B = sqrtm(B); |
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21 Aroot{i+1} = B; |
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22 end |
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23 |
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24 % Positive zeros of p8 and q8 in r8 = p8/q8 Pade approximant. |
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25 load tau_r8_zeros |
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26 % Zeros come in \pm pairs. |
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27 a = complex(0, [pzero; -pzero]); |
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28 b = complex(0, [qzero; -qzero]); |
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29 |
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30 E = 2^s*E; |
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31 for i = 1:s |
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32 E = sylvsol(Aroot{i},Aroot{i},E); |
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33 end |
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34 |
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35 G = sylvsol(B,B,E); |
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36 |
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37 X = logm(B); |
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38 |
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39 for i=8:-1:1 |
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40 rhs = (I + X/b(i)) * G + G * (I - X/b(i)); |
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41 AA = I + X/a(i); BB = I - X/a(i); |
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42 G = sylvsol(AA, BB, rhs); |
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43 end |
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44 |
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45 L = 2*G; |
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46 if use_Schur, L = Q*L*Q'; end |
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47 if real_data, L = real(L); end |