Mercurial > octave
annotate scripts/linear-algebra/logm.m @ 31551:fd29c7a50a78 stable
maint: use commas, semicolons consistently with Octave conventions.
* makeValidName.m: Remove %!test and move BIST %!asserts to column 1.
* base64decode.m, base64encode.m, which.m, logm.m, uniquetol.m, perms.m:
Delete semicolon (';') at end of %!assert BIST.
* lin2mu.m, interp2.m, interpn.m, lsqnonneg.m, pqpnonneg.m, uniquetol.m,
betainc.m, normalize.m: Add semicolon (';') to end of assert statement within
%!test BIST.
* __memoize__.m, tar_is_bsd.m, publish.m: Add semicolon (';') to line with
keyword "persistent".
* stft.m: Use comma (',') after "case" keyword when code immediately follows.
* gallery.m: Align commas used in case statements in massive switch block.
Remove unnecessary parentheses around a numeric case argument.
* ranks.m: Remove semicolon (';') from case statemnt argument.
author | Rik <rik@octave.org> |
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date | Sat, 26 Nov 2022 06:32:08 -0800 |
parents | a40c0b7aa376 |
children | 597f3ee61a48 |
rev | line source |
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1 ######################################################################## |
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2 ## |
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3 ## Copyright (C) 2008-2022 The Octave Project Developers |
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4 ## |
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5 ## See the file COPYRIGHT.md in the top-level directory of this |
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6 ## distribution or <https://octave.org/copyright/>. |
4334 | 7 ## |
8 ## This file is part of Octave. | |
9 ## | |
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10 ## Octave is free software: you can redistribute it and/or modify it |
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11 ## under the terms of the GNU General Public License as published by |
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12 ## the Free Software Foundation, either version 3 of the License, or |
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13 ## (at your option) any later version. |
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14 ## |
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15 ## Octave is distributed in the hope that it will be useful, but |
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16 ## WITHOUT ANY WARRANTY; without even the implied warranty of |
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17 ## MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
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18 ## GNU General Public License for more details. |
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19 ## |
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20 ## You should have received a copy of the GNU General Public License |
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21 ## along with Octave; see the file COPYING. If not, see |
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22 ## <https://www.gnu.org/licenses/>. |
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23 ## |
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24 ######################################################################## |
4334 | 25 |
26 ## -*- texinfo -*- | |
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27 ## @deftypefn {} {@var{s} =} logm (@var{A}) |
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28 ## @deftypefnx {} {@var{s} =} logm (@var{A}, @var{opt_iters}) |
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29 ## @deftypefnx {} {[@var{s}, @var{iters}] =} logm (@dots{}) |
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30 ## Compute the matrix logarithm of the square matrix @var{A}. |
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31 ## |
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32 ## The implementation utilizes a Pad@'e approximant and the identity |
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33 ## |
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34 ## @example |
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35 ## logm (@var{A}) = 2^k * logm (@var{A}^(1 / 2^k)) |
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36 ## @end example |
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37 ## |
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38 ## The optional input @var{opt_iters} is the maximum number of square roots |
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39 ## to compute and defaults to 100. |
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40 ## |
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41 ## The optional output @var{iters} is the number of square roots actually |
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42 ## computed. |
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43 ## @seealso{expm, sqrtm} |
4334 | 44 ## @end deftypefn |
45 | |
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46 ## Reference: N. J. Higham, Functions of Matrices: Theory and Computation |
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47 ## (SIAM, 2008.) |
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48 ## |
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16768 | 50 ## Author: N. J. Higham |
51 ## Author: Richard T. Guy <guyrt7@wfu.edu> | |
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52 |
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53 function [s, iters] = logm (A, opt_iters = 100) |
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54 |
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55 if (nargin == 0) |
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4334 | 57 endif |
58 | |
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59 if (! issquare (A)) |
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60 error ("logm: A must be a square matrix"); |
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61 endif |
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62 |
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63 if (isscalar (A)) |
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64 s = log (A); |
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65 return; |
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66 elseif (isdiag (A)) |
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67 s = diag (log (diag (A))); |
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68 return; |
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69 endif |
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70 |
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71 [u, s] = schur (A); |
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72 |
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73 if (isreal (A)) |
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74 [u, s] = rsf2csf (u, s); |
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75 endif |
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76 |
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77 eigv = diag (s); |
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78 n = rows (A); |
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79 tol = n * eps (max (abs (eigv))); |
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80 real_neg_eigv = (real (eigv) < -tol) & (imag (eigv) <= tol); |
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81 if (any (real_neg_eigv)) |
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82 warning ("Octave:logm:non-principal", |
11458 | 83 "logm: principal matrix logarithm is not defined for matrices with negative eigenvalues; computing non-principal logarithm"); |
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84 endif |
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85 |
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86 real_eig = ! any (real_neg_eigv); |
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87 |
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88 if (max (abs (triu (s,1))(:)) < tol) |
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89 ## Will run for Hermitian matrices as Schur decomposition is diagonal. |
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90 ## This way is faster and more accurate but only works on a diagonal matrix. |
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91 logeigv = log (eigv); |
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92 logeigv(isinf (logeigv)) = -log (realmax ()); |
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93 s = u * diag (logeigv) * u'; |
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94 iters = 0; |
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95 else |
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96 k = 0; |
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97 ## Algorithm 11.9 in "Function of matrices", by N. Higham |
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98 theta = [0, 0, 1.61e-2, 5.38e-2, 1.13e-1, 1.86e-1, 2.6429608311114350e-1]; |
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99 p = 0; |
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100 m = 7; |
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101 while (k < opt_iters) |
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102 tau = norm (s - eye (n), 1); |
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103 if (tau <= theta (7)) |
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104 p += 1; |
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105 j(1) = find (tau <= theta, 1); |
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106 j(2) = find (tau / 2 <= theta, 1); |
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107 if (j(1) - j(2) <= 1 || p == 2) |
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108 m = j(1); |
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109 break; |
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110 endif |
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111 endif |
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112 k += 1; |
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113 s = sqrtm (s); |
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114 endwhile |
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115 |
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116 if (k >= opt_iters) |
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117 warning ("logm: maximum number of square roots exceeded; results may still be accurate"); |
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118 endif |
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119 |
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120 s -= eye (n); |
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121 |
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122 if (m > 1) |
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123 s = logm_pade_pf (s, m); |
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124 endif |
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125 |
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126 s = 2^k * u * s * u'; |
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127 |
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128 if (nargout == 2) |
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129 iters = k; |
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130 endif |
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131 endif |
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132 ## Remove small complex values (O(eps)) which may have entered calculation |
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133 if (real_eig && isreal (A)) |
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134 s = real (s); |
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135 endif |
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136 |
4334 | 137 endfunction |
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138 |
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139 ################## ANCILLARY FUNCTIONS ################################ |
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140 ###### Taken from the mfttoolbox (GPL 3) by D. Higham. |
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141 ###### Reference: |
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142 ###### D. Higham, Functions of Matrices: Theory and Computation |
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143 ###### (SIAM, 2008.). |
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144 ####################################################################### |
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145 |
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146 ## LOGM_PADE_PF Evaluate Pade approximant to matrix log by partial fractions. |
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147 ## Y = LOGM_PADE_PF(A,M) evaluates the [M/M] Pade approximation to |
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148 ## LOG(EYE(SIZE(A))+A) using a partial fraction expansion. |
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149 |
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150 function s = logm_pade_pf (A, m) |
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151 |
11458 | 152 [nodes, wts] = gauss_legendre (m); |
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153 ## Convert from [-1,1] to [0,1]. |
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155 wts /= 2; |
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156 |
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157 n = length (A); |
11458 | 158 s = zeros (n); |
159 for j = 1:m | |
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160 s += wts(j)*(A/(eye (n) + nodes(j)*A)); |
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161 endfor |
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162 |
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163 endfunction |
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164 |
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165 ###################################################################### |
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166 ## GAUSS_LEGENDRE Nodes and weights for Gauss-Legendre quadrature. |
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167 ## [X,W] = GAUSS_LEGENDRE(N) computes the nodes X and weights W |
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168 ## for N-point Gauss-Legendre quadrature. |
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169 |
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170 ## Reference: |
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171 ## G. H. Golub and J. H. Welsch, Calculation of Gauss quadrature |
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172 ## rules, Math. Comp., 23(106):221-230, 1969. |
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173 |
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175 |
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176 i = 1:n-1; |
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177 v = i./sqrt ((2*i).^2-1); |
11458 | 178 [V, D] = eig (diag (v, -1) + diag (v, 1)); |
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179 x = diag (D); |
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180 w = 2*(V(1,:)'.^2); |
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181 |
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182 endfunction |
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183 |
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184 |
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185 %!assert (norm (logm ([1 -1;0 1]) - [0 -1; 0 0]) < 1e-5) |
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186 %!test |
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187 %! warning ("off", "Octave:logm:non-principal", "local"); |
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188 %! assert (norm (expm (logm ([-1 2 ; 4 -1])) - [-1 2 ; 4 -1]) < 1e-5); |
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189 %!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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190 %!assert (logm (10), log (10)) |
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191 %!assert (full (logm (eye (3))), logm (full (eye (3)))) |
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192 %!assert (full (logm (10*eye (3))), logm (full (10*eye (3))), 8*eps) |
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193 %!assert (logm (expm ([0 1i; -1i 0])), [0 1i; -1i 0], 10 * eps) |
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194 %!test <*60738> |
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195 %! A = [0.2510, 1.2808, -1.2252; ... |
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196 %! 0.2015, 1.0766, 0.5630; ... |
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197 %! -1.9769, -1.0922, -0.5831]; |
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198 %! if (ismac ()) |
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199 %! ## The math libraries on macOS seem to require larger tolerances |
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200 %! tol = 60*eps; |
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201 %! else |
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202 %! tol = 40*eps; |
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203 %! endif |
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204 %! warning ("off", "Octave:logm:non-principal", "local"); |
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205 %! assert (expm (logm (A)), A, tol); |
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206 %!assert (expm (logm (eye (3))), eye (3)) |
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207 %!assert (expm (logm (zeros (3))), zeros (3)) |
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208 |
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209 ## Test input validation |
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210 %!error <Invalid call> logm () |
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211 %!error <logm: A must be a square matrix> logm ([1 0;0 1; 2 2]) |