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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, 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. |
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20 |
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21 */ |
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22 |
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23 #ifdef HAVE_CONFIG_H |
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24 #include <config.h> |
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25 #endif |
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26 |
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27 #include <string> |
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28 |
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29 #include "CmplxSCHUR.h" |
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30 #include "dbleSCHUR.h" |
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31 |
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32 #include "defun-dld.h" |
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33 #include "error.h" |
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34 #include "gripes.h" |
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35 #include "oct-obj.h" |
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36 #include "utils.h" |
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37 |
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38 DEFUN_DLD (schur, args, nargout, |
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39 "-*- texinfo -*-\n\ |
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40 @deftypefn {Loadable Function} {@var{s} =} schur (@var{a})\n\ |
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41 @deftypefnx {Loadable Function} {[@var{u}, @var{s}] =} schur (@var{a}, @var{opt})\n\ |
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42 @cindex Schur decomposition\n\ |
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43 The Schur decomposition is used to compute eigenvalues of a\n\ |
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44 square matrix, and has applications in the solution of algebraic\n\ |
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45 Riccati equations in control (see @code{are} and @code{dare}).\n\ |
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46 @code{schur} always returns\n\ |
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47 @iftex\n\ |
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48 @tex\n\ |
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49 $S = U^T A U$\n\ |
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50 @end tex\n\ |
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51 @end iftex\n\ |
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52 @ifinfo\n\ |
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53 @code{s = u' * a * u}\n\ |
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54 @end ifinfo\n\ |
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55 where\n\ |
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56 @iftex\n\ |
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57 @tex\n\ |
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58 $U$\n\ |
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59 @end tex\n\ |
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60 @end iftex\n\ |
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61 @ifinfo\n\ |
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62 @code{u}\n\ |
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63 @end ifinfo\n\ |
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64 is a unitary matrix\n\ |
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65 @iftex\n\ |
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66 @tex\n\ |
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67 ($U^T U$ is identity)\n\ |
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68 @end tex\n\ |
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69 @end iftex\n\ |
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70 @ifinfo\n\ |
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71 (@code{u'* u} is identity)\n\ |
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72 @end ifinfo\n\ |
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73 and\n\ |
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74 @iftex\n\ |
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75 @tex\n\ |
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76 $S$\n\ |
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77 @end tex\n\ |
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78 @end iftex\n\ |
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79 @ifinfo\n\ |
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80 @code{s}\n\ |
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81 @end ifinfo\n\ |
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82 is upper triangular. The eigenvalues of\n\ |
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83 @iftex\n\ |
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84 @tex\n\ |
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85 $A$ (and $S$)\n\ |
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86 @end tex\n\ |
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87 @end iftex\n\ |
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88 @ifinfo\n\ |
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89 @code{a} (and @code{s})\n\ |
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90 @end ifinfo\n\ |
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91 are the diagonal elements of\n\ |
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92 @iftex\n\ |
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93 @tex\n\ |
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94 $S$\n\ |
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95 @end tex\n\ |
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96 @end iftex\n\ |
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97 @ifinfo\n\ |
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98 @code{s}\n\ |
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99 @end ifinfo\n\ |
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100 If the matrix\n\ |
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101 @iftex\n\ |
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102 @tex\n\ |
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103 $A$\n\ |
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104 @end tex\n\ |
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105 @end iftex\n\ |
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106 @ifinfo\n\ |
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107 @code{a}\n\ |
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108 @end ifinfo\n\ |
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109 is real, then the real Schur decomposition is computed, in which the\n\ |
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110 matrix\n\ |
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111 @iftex\n\ |
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112 @tex\n\ |
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113 $U$\n\ |
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114 @end tex\n\ |
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115 @end iftex\n\ |
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116 @ifinfo\n\ |
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117 @code{u}\n\ |
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118 @end ifinfo\n\ |
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119 is orthogonal and\n\ |
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120 @iftex\n\ |
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121 @tex\n\ |
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122 $S$\n\ |
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123 @end tex\n\ |
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124 @end iftex\n\ |
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125 @ifinfo\n\ |
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126 @code{s}\n\ |
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127 @end ifinfo\n\ |
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128 is block upper triangular\n\ |
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129 with blocks of size at most\n\ |
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130 @iftex\n\ |
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131 @tex\n\ |
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132 $2\\times 2$\n\ |
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133 @end tex\n\ |
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134 @end iftex\n\ |
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135 @ifinfo\n\ |
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136 @code{2 x 2}\n\ |
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137 @end ifinfo\n\ |
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138 along the diagonal. The diagonal elements of\n\ |
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139 @iftex\n\ |
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140 @tex\n\ |
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141 $S$\n\ |
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142 @end tex\n\ |
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143 @end iftex\n\ |
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144 @ifinfo\n\ |
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145 @code{s}\n\ |
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146 @end ifinfo\n\ |
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147 (or the eigenvalues of the\n\ |
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148 @iftex\n\ |
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149 @tex\n\ |
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150 $2\\times 2$\n\ |
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151 @end tex\n\ |
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152 @end iftex\n\ |
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153 @ifinfo\n\ |
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154 @code{2 x 2}\n\ |
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155 @end ifinfo\n\ |
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156 blocks, when\n\ |
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157 appropriate) are the eigenvalues of\n\ |
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158 @iftex\n\ |
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159 @tex\n\ |
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160 $A$\n\ |
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161 @end tex\n\ |
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162 @end iftex\n\ |
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163 @ifinfo\n\ |
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164 @code{a}\n\ |
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165 @end ifinfo\n\ |
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166 and\n\ |
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167 @iftex\n\ |
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168 @tex\n\ |
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169 $S$.\n\ |
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170 @end tex\n\ |
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171 @end iftex\n\ |
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172 @ifinfo\n\ |
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173 @code{s}.\n\ |
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174 @end ifinfo\n\ |
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175 \n\ |
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176 The eigenvalues are optionally ordered along the diagonal according to\n\ |
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177 the value of @code{opt}. @code{opt = \"a\"} indicates that all\n\ |
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178 eigenvalues with negative real parts should be moved to the leading\n\ |
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179 block of\n\ |
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180 @iftex\n\ |
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181 @tex\n\ |
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182 $S$\n\ |
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183 @end tex\n\ |
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184 @end iftex\n\ |
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185 @ifinfo\n\ |
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186 @code{s}\n\ |
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187 @end ifinfo\n\ |
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188 (used in @code{are}), @code{opt = \"d\"} indicates that all eigenvalues\n\ |
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189 with magnitude less than one should be moved to the leading block of\n\ |
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190 @iftex\n\ |
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191 @tex\n\ |
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192 $S$\n\ |
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193 @end tex\n\ |
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194 @end iftex\n\ |
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195 @ifinfo\n\ |
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196 @code{s}\n\ |
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197 @end ifinfo\n\ |
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198 (used in @code{dare}), and @code{opt = \"u\"}, the default, indicates that\n\ |
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199 no ordering of eigenvalues should occur. The leading\n\ |
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200 @iftex\n\ |
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201 @tex\n\ |
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202 $k$\n\ |
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203 @end tex\n\ |
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204 @end iftex\n\ |
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205 @ifinfo\n\ |
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206 @code{k}\n\ |
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207 @end ifinfo\n\ |
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208 columns of\n\ |
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209 @iftex\n\ |
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210 @tex\n\ |
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211 $U$\n\ |
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212 @end tex\n\ |
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213 @end iftex\n\ |
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214 @ifinfo\n\ |
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215 @code{u}\n\ |
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216 @end ifinfo\n\ |
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217 always span the\n\ |
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218 @iftex\n\ |
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219 @tex\n\ |
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220 $A$-invariant\n\ |
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221 @end tex\n\ |
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222 @end iftex\n\ |
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223 @ifinfo\n\ |
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224 @code{a}-invariant\n\ |
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225 @end ifinfo\n\ |
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226 subspace corresponding to the\n\ |
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227 @iftex\n\ |
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228 @tex\n\ |
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229 $k$\n\ |
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230 @end tex\n\ |
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231 @end iftex\n\ |
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232 @ifinfo\n\ |
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233 @code{k}\n\ |
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234 @end ifinfo\n\ |
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235 leading eigenvalues of\n\ |
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236 @iftex\n\ |
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237 @tex\n\ |
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238 $S$.\n\ |
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239 @end tex\n\ |
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240 @end iftex\n\ |
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241 @ifinfo\n\ |
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242 @code{s}.\n\ |
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243 @end ifinfo\n\ |
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244 @end deftypefn") |
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245 { |
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246 octave_value_list retval; |
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247 |
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248 int nargin = args.length (); |
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249 |
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250 if (nargin < 1 || nargin > 2 || nargout > 2) |
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251 { |
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252 print_usage ("schur"); |
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253 return retval; |
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254 } |
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255 |
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256 octave_value arg = args(0); |
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257 |
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258 std::string ord; |
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259 |
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260 if (nargin == 2) |
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261 { |
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262 ord = args(1).string_value (); |
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263 |
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264 if (error_state) |
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265 { |
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266 error ("schur: expecting string as second argument"); |
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267 return retval; |
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268 } |
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269 } |
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270 |
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271 char ord_char = ord.empty () ? 'U' : ord[0]; |
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272 |
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273 if (ord_char != 'U' && ord_char != 'A' && ord_char != 'D' |
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274 && ord_char != 'u' && ord_char != 'a' && ord_char != 'd') |
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275 { |
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276 warning ("schur: incorrect ordered schur argument `%c'", |
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277 ord.c_str ()); |
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278 return retval; |
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279 } |
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280 |
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281 int nr = arg.rows (); |
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282 int nc = arg.columns (); |
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283 |
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284 int arg_is_empty = empty_arg ("schur", nr, nc); |
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285 |
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286 if (arg_is_empty < 0) |
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287 return retval; |
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288 else if (arg_is_empty > 0) |
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289 return octave_value_list (2, Matrix ()); |
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290 |
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291 if (nr != nc) |
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292 { |
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293 gripe_square_matrix_required ("schur"); |
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294 return retval; |
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295 } |
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296 |
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297 if (arg.is_real_type ()) |
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298 { |
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299 Matrix tmp = arg.matrix_value (); |
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300 |
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301 if (! error_state) |
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302 { |
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303 SCHUR result (tmp, ord); |
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304 |
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305 if (nargout == 0 || nargout == 1) |
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306 { |
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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 retval(1) = result.schur_matrix (); |
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312 retval(0) = result.unitary_matrix (); |
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313 } |
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314 } |
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315 } |
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316 else if (arg.is_complex_type ()) |
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317 { |
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318 ComplexMatrix ctmp = arg.complex_matrix_value (); |
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319 |
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320 if (! error_state) |
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321 { |
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322 ComplexSCHUR result (ctmp, ord); |
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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 retval(0) = result.schur_matrix (); |
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327 } |
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328 else |
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329 { |
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330 retval(1) = result.schur_matrix (); |
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331 retval(0) = result.unitary_matrix (); |
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332 } |
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333 } |
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334 } |
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335 else |
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336 { |
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337 gripe_wrong_type_arg ("schur", arg); |
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338 } |
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339 |
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340 return retval; |
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341 } |
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342 |
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343 /* |
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344 ;;; Local Variables: *** |
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345 ;;; mode: C++ *** |
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346 ;;; End: *** |
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347 */ |
