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1 /* |
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2 |
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3 Copyright (C) 1996, 1997 John W. Eaton |
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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 the |
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9 Free Software Foundation; either version 2, or (at your option) any |
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10 later version. |
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11 |
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12 Octave is distributed in the hope that it will be useful, but WITHOUT |
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13 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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15 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, write to the Free |
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19 Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA |
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20 02110-1301, USA. |
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21 |
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22 */ |
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23 |
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24 #ifdef HAVE_CONFIG_H |
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25 #include <config.h> |
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26 #endif |
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27 |
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28 #include <string> |
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29 |
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30 #include "CmplxSCHUR.h" |
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31 #include "dbleSCHUR.h" |
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32 |
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33 #include "defun-dld.h" |
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34 #include "error.h" |
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35 #include "gripes.h" |
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36 #include "oct-obj.h" |
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37 #include "utils.h" |
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38 |
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39 DEFUN_DLD (schur, args, nargout, |
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40 "-*- texinfo -*-\n\ |
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41 @deftypefn {Loadable Function} {@var{s} =} schur (@var{a})\n\ |
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42 @deftypefnx {Loadable Function} {[@var{u}, @var{s}] =} schur (@var{a}, @var{opt})\n\ |
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43 @cindex Schur decomposition\n\ |
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44 The Schur decomposition is used to compute eigenvalues of a\n\ |
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45 square matrix, and has applications in the solution of algebraic\n\ |
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46 Riccati equations in control (see @code{are} and @code{dare}).\n\ |
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47 @code{schur} always returns\n\ |
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48 @iftex\n\ |
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49 @tex\n\ |
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50 $S = U^T A U$\n\ |
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51 @end tex\n\ |
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52 @end iftex\n\ |
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53 @ifinfo\n\ |
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54 @code{s = u' * a * u}\n\ |
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55 @end ifinfo\n\ |
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56 where\n\ |
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57 @iftex\n\ |
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58 @tex\n\ |
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59 $U$\n\ |
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60 @end tex\n\ |
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61 @end iftex\n\ |
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62 @ifinfo\n\ |
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63 @code{u}\n\ |
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64 @end ifinfo\n\ |
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65 is a unitary matrix\n\ |
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66 @iftex\n\ |
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67 @tex\n\ |
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68 ($U^T U$ is identity)\n\ |
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69 @end tex\n\ |
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70 @end iftex\n\ |
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71 @ifinfo\n\ |
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72 (@code{u'* u} is identity)\n\ |
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73 @end ifinfo\n\ |
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74 and\n\ |
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75 @iftex\n\ |
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76 @tex\n\ |
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77 $S$\n\ |
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78 @end tex\n\ |
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79 @end iftex\n\ |
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80 @ifinfo\n\ |
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81 @code{s}\n\ |
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82 @end ifinfo\n\ |
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83 is upper triangular. The eigenvalues of\n\ |
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84 @iftex\n\ |
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85 @tex\n\ |
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86 $A$ (and $S$)\n\ |
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87 @end tex\n\ |
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88 @end iftex\n\ |
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89 @ifinfo\n\ |
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90 @code{a} (and @code{s})\n\ |
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91 @end ifinfo\n\ |
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92 are the diagonal elements of\n\ |
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93 @iftex\n\ |
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94 @tex\n\ |
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95 $S$.\n\ |
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96 @end tex\n\ |
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97 @end iftex\n\ |
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98 @ifinfo\n\ |
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99 @code{s}.\n\ |
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100 @end ifinfo\n\ |
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101 If the matrix\n\ |
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102 @iftex\n\ |
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103 @tex\n\ |
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104 $A$\n\ |
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105 @end tex\n\ |
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106 @end iftex\n\ |
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107 @ifinfo\n\ |
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108 @code{a}\n\ |
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109 @end ifinfo\n\ |
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110 is real, then the real Schur decomposition is computed, in which the\n\ |
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111 matrix\n\ |
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112 @iftex\n\ |
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113 @tex\n\ |
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114 $U$\n\ |
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115 @end tex\n\ |
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116 @end iftex\n\ |
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117 @ifinfo\n\ |
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118 @code{u}\n\ |
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119 @end ifinfo\n\ |
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120 is orthogonal and\n\ |
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121 @iftex\n\ |
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122 @tex\n\ |
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123 $S$\n\ |
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124 @end tex\n\ |
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125 @end iftex\n\ |
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126 @ifinfo\n\ |
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127 @code{s}\n\ |
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128 @end ifinfo\n\ |
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129 is block upper triangular\n\ |
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130 with blocks of size at most\n\ |
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131 @iftex\n\ |
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132 @tex\n\ |
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133 $2\\times 2$\n\ |
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134 @end tex\n\ |
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135 @end iftex\n\ |
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136 @ifinfo\n\ |
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137 @code{2 x 2}\n\ |
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138 @end ifinfo\n\ |
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139 along the diagonal. The diagonal elements of\n\ |
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140 @iftex\n\ |
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141 @tex\n\ |
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142 $S$\n\ |
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143 @end tex\n\ |
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144 @end iftex\n\ |
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145 @ifinfo\n\ |
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146 @code{s}\n\ |
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147 @end ifinfo\n\ |
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148 (or the eigenvalues of the\n\ |
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149 @iftex\n\ |
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150 @tex\n\ |
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151 $2\\times 2$\n\ |
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152 @end tex\n\ |
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153 @end iftex\n\ |
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154 @ifinfo\n\ |
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155 @code{2 x 2}\n\ |
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156 @end ifinfo\n\ |
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157 blocks, when\n\ |
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158 appropriate) are the eigenvalues of\n\ |
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159 @iftex\n\ |
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160 @tex\n\ |
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161 $A$\n\ |
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162 @end tex\n\ |
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163 @end iftex\n\ |
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164 @ifinfo\n\ |
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165 @code{a}\n\ |
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166 @end ifinfo\n\ |
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167 and\n\ |
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168 @iftex\n\ |
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169 @tex\n\ |
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170 $S$.\n\ |
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171 @end tex\n\ |
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172 @end iftex\n\ |
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173 @ifinfo\n\ |
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174 @code{s}.\n\ |
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175 @end ifinfo\n\ |
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176 \n\ |
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177 The eigenvalues are optionally ordered along the diagonal according to\n\ |
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178 the value of @code{opt}. @code{opt = \"a\"} indicates that all\n\ |
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179 eigenvalues with negative real parts should be moved to the leading\n\ |
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180 block of\n\ |
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181 @iftex\n\ |
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182 @tex\n\ |
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183 $S$\n\ |
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184 @end tex\n\ |
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185 @end iftex\n\ |
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186 @ifinfo\n\ |
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187 @code{s}\n\ |
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188 @end ifinfo\n\ |
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189 (used in @code{are}), @code{opt = \"d\"} indicates that all eigenvalues\n\ |
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190 with magnitude less than one should be moved to the leading block of\n\ |
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191 @iftex\n\ |
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192 @tex\n\ |
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193 $S$\n\ |
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194 @end tex\n\ |
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195 @end iftex\n\ |
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196 @ifinfo\n\ |
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197 @code{s}\n\ |
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198 @end ifinfo\n\ |
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199 (used in @code{dare}), and @code{opt = \"u\"}, the default, indicates that\n\ |
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200 no ordering of eigenvalues should occur. The leading\n\ |
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201 @iftex\n\ |
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202 @tex\n\ |
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203 $k$\n\ |
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204 @end tex\n\ |
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205 @end iftex\n\ |
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206 @ifinfo\n\ |
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207 @code{k}\n\ |
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208 @end ifinfo\n\ |
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209 columns of\n\ |
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210 @iftex\n\ |
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211 @tex\n\ |
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212 $U$\n\ |
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213 @end tex\n\ |
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214 @end iftex\n\ |
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215 @ifinfo\n\ |
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216 @code{u}\n\ |
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217 @end ifinfo\n\ |
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218 always span the\n\ |
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219 @iftex\n\ |
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220 @tex\n\ |
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221 $A$-invariant\n\ |
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222 @end tex\n\ |
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223 @end iftex\n\ |
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224 @ifinfo\n\ |
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225 @code{a}-invariant\n\ |
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226 @end ifinfo\n\ |
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227 subspace corresponding to the\n\ |
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228 @iftex\n\ |
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229 @tex\n\ |
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230 $k$\n\ |
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231 @end tex\n\ |
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232 @end iftex\n\ |
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233 @ifinfo\n\ |
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234 @code{k}\n\ |
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235 @end ifinfo\n\ |
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236 leading eigenvalues of\n\ |
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237 @iftex\n\ |
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238 @tex\n\ |
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239 $S$.\n\ |
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240 @end tex\n\ |
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241 @end iftex\n\ |
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242 @ifinfo\n\ |
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243 @code{s}.\n\ |
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244 @end ifinfo\n\ |
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245 @end deftypefn") |
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246 { |
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247 octave_value_list retval; |
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248 |
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249 int nargin = args.length (); |
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250 |
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251 if (nargin < 1 || nargin > 2 || nargout > 2) |
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252 { |
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253 print_usage ("schur"); |
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254 return retval; |
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255 } |
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256 |
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257 octave_value arg = args(0); |
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258 |
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259 std::string ord; |
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260 |
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261 if (nargin == 2) |
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262 { |
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263 ord = args(1).string_value (); |
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264 |
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265 if (error_state) |
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266 { |
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267 error ("schur: expecting string as second argument"); |
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268 return retval; |
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269 } |
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270 } |
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271 |
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272 char ord_char = ord.empty () ? 'U' : ord[0]; |
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273 |
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274 if (ord_char != 'U' && ord_char != 'A' && ord_char != 'D' |
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275 && ord_char != 'u' && ord_char != 'a' && ord_char != 'd') |
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276 { |
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277 warning ("schur: incorrect ordered schur argument `%c'", |
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278 ord.c_str ()); |
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279 return retval; |
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280 } |
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281 |
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282 octave_idx_type nr = arg.rows (); |
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283 octave_idx_type nc = arg.columns (); |
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284 |
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285 int arg_is_empty = empty_arg ("schur", nr, nc); |
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286 |
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287 if (arg_is_empty < 0) |
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288 return retval; |
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289 else if (arg_is_empty > 0) |
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290 return octave_value_list (2, Matrix ()); |
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291 |
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292 if (nr != nc) |
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293 { |
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294 gripe_square_matrix_required ("schur"); |
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295 return retval; |
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296 } |
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297 |
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298 if (arg.is_real_type ()) |
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299 { |
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300 Matrix tmp = arg.matrix_value (); |
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301 |
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302 if (! error_state) |
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303 { |
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304 if (nargout == 0 || nargout == 1) |
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305 { |
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306 SCHUR result (tmp, ord, false); |
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307 retval(0) = result.schur_matrix (); |
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308 } |
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309 else |
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310 { |
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311 SCHUR result (tmp, ord, true); |
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312 retval(1) = result.schur_matrix (); |
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313 retval(0) = result.unitary_matrix (); |
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314 } |
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315 } |
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316 } |
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317 else if (arg.is_complex_type ()) |
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318 { |
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319 ComplexMatrix ctmp = arg.complex_matrix_value (); |
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320 |
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321 if (! error_state) |
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322 { |
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323 |
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324 if (nargout == 0 || nargout == 1) |
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325 { |
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326 ComplexSCHUR result (ctmp, ord, false); |
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327 retval(0) = result.schur_matrix (); |
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328 } |
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329 else |
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330 { |
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331 ComplexSCHUR result (ctmp, ord, true); |
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332 retval(1) = result.schur_matrix (); |
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333 retval(0) = result.unitary_matrix (); |
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334 } |
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335 } |
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336 } |
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337 else |
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338 { |
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339 gripe_wrong_type_arg ("schur", arg); |
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340 } |
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341 |
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342 return retval; |
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343 } |
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344 |
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345 /* |
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346 ;;; Local Variables: *** |
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347 ;;; mode: C++ *** |
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348 ;;; End: *** |
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349 */ |
