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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.
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1 function [X, iter] = logm_isst(T, prnt)
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2 %LOGM_ISST Log of triangular matrix by Schur-Pade method with scaling.
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3 % X = LOGM_ISST(A) computes the logarithm of an upper triangular
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4 % matrix A, for a matrix with no nonpositive real eigenvalues,
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5 % using the inverse scaling and squaring method with Pade
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6 % approximation. TOL is an error tolerance.
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7 % [X, ITER] = LOGM_ISST(A, PRNT) returns the number ITER of square
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8 % roots computed and prints this information if PRNT is nonzero.
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9
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10 % References:
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11 % S. H. Cheng, N. J. Higham, C. S. Kenney, and A. J. Laub, Approximating the
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12 % logarithm of a matrix to specified accuracy, SIAM J. Matrix Anal. Appl.,
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13 % 22(4):1112-1125, 2001.
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14 % N. J. Higham, Evaluating Pade approximants of the matrix logarithm,
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15 % SIAM J. Matrix Anal. Appl., 22(4):1126-1135, 2001.
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16
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17 if nargin < 2, prnt = 0; end
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18 n = length(T);
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19
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20 if any( imag(diag(T)) == 0 & real(diag(T)) <= 0 )
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21 error('A must not have nonpositive real eigenvalues!')
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22 end
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23
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24 if n == 1, X = log(T); iter = 0; return, end
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25
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26 R = T;
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27 maxlogiter = 50;
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28
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29 for iter = 0:maxlogiter
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30
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31 phi = norm(T-eye(n),'fro');
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32
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33 if phi <= 0.25
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34 if prnt, fprintf('LOGM_ISST computed %g square roots. \n', iter), end
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35 break
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36 end
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37 if iter == maxlogiter, error('Too many square roots in LOGM_ISST.\n'), end
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38
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39 % Compute upper triangular square root R of T, a column at a time.
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40 for j=1:n
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41 R(j,j) = sqrt(T(j,j));
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42 for i=j-1:-1:1
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43 R(i,j) = (T(i,j) - R(i,i+1:j-1)*R(i+1:j-1,j))/(R(i,i) + R(j,j));
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44 end
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45 end
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46 T = R;
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47 end
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48
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49 X = 2^(iter)*logm_pf(T-eye(n),8);
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50
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51 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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52 function S = logm_pf(A,m)
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53 %LOGM_PF Pade approximation to matrix log by partial fraction expansion.
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54 % Y = LOGM_PF(A,m) approximates LOG(I+A).
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55
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56 [nodes,wts] = gauss_legendre(m);
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57 % Convert from [-1,1] to [0,1].
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58 nodes = (nodes + 1)/2;
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59 wts = wts/2;
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60
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61 n = length(A);
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62 S = zeros(n);
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63
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64 for j=1:m
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65 S = S + wts(j)*(A/(eye(n) + nodes(j)*A));
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66 end
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67
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68 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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69 function [x,w] = gauss_legendre(n)
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70 %GAUSS_LEGENDRE Nodes and weights for Gauss-Legendre quadrature.
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71
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72 % Reference:
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73 % G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature
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74 % rules, Math. Comp., 23(106):221-230, 1969.
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75
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76 i = 1:n-1;
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77 v = i./sqrt((2*i).^2-1);
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78 [V,D] = eig( diag(v,-1)+diag(v,1) );
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79 x = diag(D);
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80 w = 2*(V(1,:)'.^2);