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update copyright notices
author John W. Eaton <jwe@octave.org>
date Fri, 14 Jan 2011 05:47:45 -0500
parents ff7e0776ba0f
children c792872f8942
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11523
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1 ## Copyright (C) 2008-2011 N.J. Higham
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2 ## Copyright (C) 2010 Richard T. Guy <guyrt7@wfu.edu>
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3 ## Copyright (C) 2010 Marco Caliari <marco.caliari@univr.it>
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4 ##
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5 ## This file is part of Octave.
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6 ##
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7 ## Octave is free software; you can redistribute it and/or modify it
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8 ## under the terms of the GNU General Public License as published by
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9 ## the Free Software Foundation; either version 3 of the License, or (at
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10 ## your option) any later version.
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11 ##
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12 ## Octave is distributed in the hope that it will be useful, but
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13 ## WITHOUT ANY WARRANTY; without even the implied warranty of
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14 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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15 ## General Public License for more details.
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16 ##
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17 ## You should have received a copy of the GNU General Public License
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18 ## along with Octave; see the file COPYING. If not, see
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19 ## <http://www.gnu.org/licenses/>.
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20
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21 ## -*- texinfo -*-
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22 ## @deftypefn {Function File} {@var{s} =} logm (@var{A})
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23 ## @deftypefnx {Function File} {@var{s} =} logm (@var{A}, @var{opt_iters})
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24 ## @deftypefnx {Function File} {[@var{s}, @var{iters}] =} logm (@dots{})
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25 ## Compute the matrix logarithm of the square matrix @var{A}. The
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26 ## implementation utilizes a Pad@'e approximant and the identity
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27 ##
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28 ## @example
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29 ## logm(@var{A}) = 2^k * logm(@var{A}^(1 / 2^k))
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30 ## @end example
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31 ##
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32 ## The optional argument @var{opt_iters} is the maximum number of square roots
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33 ## to compute and defaults to 100. The optional output @var{iters} is the
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34 ## number of square roots actually computed.
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35 ##
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36 ## @end deftypefn
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37
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38 ## Reference: N. J. Higham, Functions of Matrices: Theory and Computation
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39 ## (SIAM, 2008.)
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40 ##
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41
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42 function [s, iters] = logm (A, opt_iters = 100)
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43
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44 if (nargin == 0 || nargin > 2)
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45 print_usage ();
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46 endif
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47
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48 if (! issquare (A))
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49 error ("logm: A must be a square matrix");
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50 endif
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51
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52 if (isscalar (A))
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53 s = log (A);
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54 return;
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55 elseif (strfind (typeinfo (A), "diagonal matrix"))
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56 s = diag (log (diag (A)));
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57 return;
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58 endif
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59
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60 [u, s] = schur (A);
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61
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62 if (isreal (A))
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63 [u, s] = rsf2csf (u, s);
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64 endif
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65
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66 if (any (diag (s) < 0))
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67 warning ("Octave:logm:non-principal",
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68 "logm: principal matrix logarithm is not defined for matrices with negative eigenvalues; computing non-principal logarithm");
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69 endif
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70
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71 k = 0;
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72 ## Algorithm 11.9 in "Function of matrices", by N. Higham
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73 theta = [0, 0, 1.61e-2, 5.38e-2, 1.13e-1, 1.86e-1, 2.6429608311114350e-1];
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74 p = 0;
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75 m = 7;
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76 while (k < opt_iters)
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77 tau = norm (s - eye (size (s)),1);
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78 if (tau <= theta (7))
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79 p = p + 1;
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80 j(1) = find (tau <= theta, 1);
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81 j(2) = find (tau / 2 <= theta, 1);
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82 if (j(1) - j(2) <= 1 || p == 2)
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83 m = j(1);
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84 break
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85 endif
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86 endif
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87 k = k + 1;
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88 s = sqrtm (s);
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89 endwhile
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90
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91 if (k >= opt_iters)
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92 warning ("logm: maximum number of square roots exceeded; results may still be accurate");
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93 endif
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94
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95 s = s - eye (size (s));
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96
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97 if (m > 1)
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98 s = logm_pade_pf (s, m);
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99 endif
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100
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101 s = 2^k * u * s * u';
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102
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103 if (nargout == 2)
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104 iters = k;
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105 endif
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106
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107 endfunction
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108
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109 ################## ANCILLARY FUNCTIONS ################################
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110 ###### Taken from the mfttoolbox (GPL 3) by D. Higham.
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111 ###### Reference:
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112 ###### D. Higham, Functions of Matrices: Theory and Computation
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113 ###### (SIAM, 2008.).
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114 #######################################################################
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115
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116 ##LOGM_PADE_PF Evaluate Pade approximant to matrix log by partial fractions.
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117 ## Y = LOGM_PADE_PF(A,M) evaluates the [M/M] Pade approximation to
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118 ## LOG(EYE(SIZE(A))+A) using a partial fraction expansion.
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119
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120 function s = logm_pade_pf (A, m)
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121 [nodes, wts] = gauss_legendre (m);
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122 ## Convert from [-1,1] to [0,1].
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123 nodes = (nodes+1)/2;
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124 wts = wts/2;
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125
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126 n = length (A);
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127 s = zeros (n);
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128 for j = 1:m
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129 s += wts(j)*(A/(eye (n) + nodes(j)*A));
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130 endfor
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131 endfunction
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132
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133 ######################################################################
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134 ## GAUSS_LEGENDRE Nodes and weights for Gauss-Legendre quadrature.
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135 ## [X,W] = GAUSS_LEGENDRE(N) computes the nodes X and weights W
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136 ## for N-point Gauss-Legendre quadrature.
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137
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138 ## Reference:
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139 ## G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature
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140 ## rules, Math. Comp., 23(106):221-230, 1969.
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141
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142 function [x, w] = gauss_legendre (n)
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143 i = 1:n-1;
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144 v = i./sqrt ((2*i).^2-1);
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145 [V, D] = eig (diag (v, -1) + diag (v, 1));
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146 x = diag (D);
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147 w = 2*(V(1,:)'.^2);
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148 endfunction
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149
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150
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151 %!assert(norm(logm([1 -1;0 1]) - [0 -1; 0 0]) < 1e-5);
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152 %!assert(norm(expm(logm([-1 2 ; 4 -1])) - [-1 2 ; 4 -1]) < 1e-5);
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153 %!assert(logm([1 -1 -1;0 1 -1; 0 0 1]), [0 -1 -1.5; 0 0 -1; 0 0 0], 1e-5);
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154
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155 %% Test input validation
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156 %!error logm ();
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157 %!error logm (1, 2, 3);
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158 %!error <logm: A must be a square matrix> logm([1 0;0 1; 2 2]);
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159
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160 %!assert (logm (10), log (10))
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161 %!assert (full (logm (eye (3))), logm (full (eye (3))))
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162 %!assert (full (logm (10*eye (3))), logm (full (10*eye (3))), 8*eps)