Mercurial > octave
annotate src/DLD-FUNCTIONS/schur.cc @ 13224:0a67c717c652 stable
Add support for additional argument "real" to schur() (Bug #34012).
* schur.cc: Adjust input validation to accept "real" argument.
Improve docstring.
author | Rik <octave@nomad.inbox5.com> |
---|---|
date | Sun, 25 Sep 2011 16:58:02 -0700 |
parents | f96b9b9f141b |
children | 990762e784fe |
rev | line source |
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2928 | 1 /* |
2 | |
11523 | 3 Copyright (C) 1996-2011 John W. Eaton |
2928 | 4 |
5 This file is part of Octave. | |
6 | |
7 Octave is free software; you can redistribute it and/or modify it | |
8 under the terms of the GNU General Public License as published by the | |
7016 | 9 Free Software Foundation; either version 3 of the License, or (at your |
10 option) any later version. | |
2928 | 11 |
12 Octave is distributed in the hope that it will be useful, but WITHOUT | |
13 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or | |
14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License | |
15 for more details. | |
16 | |
17 You should have received a copy of the GNU General Public License | |
7016 | 18 along with Octave; see the file COPYING. If not, see |
19 <http://www.gnu.org/licenses/>. | |
2928 | 20 |
21 */ | |
22 | |
23 #ifdef HAVE_CONFIG_H | |
24 #include <config.h> | |
25 #endif | |
26 | |
27 #include <string> | |
28 | |
29 #include "CmplxSCHUR.h" | |
30 #include "dbleSCHUR.h" | |
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31 #include "fCmplxSCHUR.h" |
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32 #include "floatSCHUR.h" |
2928 | 33 |
34 #include "defun-dld.h" | |
35 #include "error.h" | |
36 #include "gripes.h" | |
37 #include "oct-obj.h" | |
38 #include "utils.h" | |
39 | |
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40 template <class Matrix> |
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41 static octave_value |
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42 mark_upper_triangular (const Matrix& a) |
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43 { |
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44 octave_value retval = a; |
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45 |
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46 octave_idx_type n = a.rows (); |
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47 assert (a.columns () == n); |
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48 |
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49 const typename Matrix::element_type zero = typename Matrix::element_type (); |
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50 |
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51 for (octave_idx_type i = 0; i < n; i++) |
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52 if (a(i,i) == zero) |
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53 return retval; |
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54 |
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55 retval.matrix_type (MatrixType::Upper); |
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56 |
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57 return retval; |
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58 } |
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59 |
2928 | 60 DEFUN_DLD (schur, args, nargout, |
3548 | 61 "-*- texinfo -*-\n\ |
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62 @deftypefn {Loadable Function} {@var{S} =} schur (@var{A})\n\ |
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63 @deftypefnx {Loadable Function} {@var{S} =} schur (@var{A}, \"real\")\n\ |
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64 @deftypefnx {Loadable Function} {@var{S} =} schur (@var{A}, \"complex\")\n\ |
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65 @deftypefnx {Loadable Function} {@var{S} =} schur (@var{A}, @var{opt})\n\ |
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66 @deftypefnx {Loadable Function} {[@var{U}, @var{S}] =} schur (@var{A}, @dots{})\n\ |
3372 | 67 @cindex Schur decomposition\n\ |
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68 Compute the Schur@tie{}decomposition of @var{A}\n\ |
3372 | 69 @tex\n\ |
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70 $$\n\ |
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71 S = U^T A U\n\ |
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72 $$\n\ |
3372 | 73 @end tex\n\ |
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74 @ifnottex\n\ |
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75 \n\ |
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76 @example\n\ |
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77 @code{@var{S} = @var{U}' * @var{A} * @var{U}}\n\ |
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78 @end example\n\ |
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79 \n\ |
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80 @end ifnottex\n\ |
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81 where @var{U} is a unitary matrix\n\ |
3372 | 82 @tex\n\ |
83 ($U^T U$ is identity)\n\ | |
84 @end tex\n\ | |
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85 @ifnottex\n\ |
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86 (@code{@var{U}'* @var{U}} is identity)\n\ |
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87 @end ifnottex\n\ |
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88 and @var{S} is upper triangular. The eigenvalues of @var{A} (and @var{S})\n\ |
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89 are the diagonal elements of @var{S}. If the matrix @var{A}\n\ |
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90 is real, then the real Schur@tie{}decomposition is computed, in which the\n\ |
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91 matrix @var{U} is orthogonal and @var{S} is block upper triangular\n\ |
3372 | 92 with blocks of size at most\n\ |
93 @tex\n\ | |
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94 $2 \\times 2$\n\ |
3372 | 95 @end tex\n\ |
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96 @ifnottex\n\ |
3372 | 97 @code{2 x 2}\n\ |
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98 @end ifnottex\n\ |
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99 along the diagonal. The diagonal elements of @var{S}\n\ |
3372 | 100 (or the eigenvalues of the\n\ |
101 @tex\n\ | |
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102 $2 \\times 2$\n\ |
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104 @ifnottex\n\ |
3372 | 105 @code{2 x 2}\n\ |
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106 @end ifnottex\n\ |
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107 blocks, when appropriate) are the eigenvalues of @var{A} and @var{S}.\n\ |
2928 | 108 \n\ |
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109 The default for real matrices is a real Schur@tie{}decomposition.\n\ |
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110 A complex decomposition may be forced by passing the flag \"complex\".\n\ |
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111 \n\ |
3372 | 112 The eigenvalues are optionally ordered along the diagonal according to\n\ |
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113 the value of @var{opt}. @code{@var{opt} = \"a\"} indicates that all\n\ |
3372 | 114 eigenvalues with negative real parts should be moved to the leading\n\ |
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115 block of @var{S}\n\ |
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116 (used in @code{are}), @code{@var{opt} = \"d\"} indicates that all eigenvalues\n\ |
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117 with magnitude less than one should be moved to the leading block of @var{S}\n\ |
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118 (used in @code{dare}), and @code{@var{opt} = \"u\"}, the default, indicates\n\ |
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119 that no ordering of eigenvalues should occur. The leading @var{k}\n\ |
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120 columns of @var{U} always span the @var{A}-invariant\n\ |
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121 subspace corresponding to the @var{k} leading eigenvalues of @var{S}.\n\ |
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122 \n\ |
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123 The Schur@tie{}decomposition is used to compute eigenvalues of a\n\ |
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124 square matrix, and has applications in the solution of algebraic\n\ |
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125 Riccati equations in control (see @code{are} and @code{dare}).\n\ |
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126 @seealso{rsf2csf}\n\ |
3372 | 127 @end deftypefn") |
2928 | 128 { |
129 octave_value_list retval; | |
130 | |
131 int nargin = args.length (); | |
132 | |
133 if (nargin < 1 || nargin > 2 || nargout > 2) | |
134 { | |
5823 | 135 print_usage (); |
2928 | 136 return retval; |
137 } | |
138 | |
139 octave_value arg = args(0); | |
140 | |
3523 | 141 std::string ord; |
2928 | 142 |
143 if (nargin == 2) | |
144 { | |
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145 ord = args(1).string_value (); |
2928 | 146 |
147 if (error_state) | |
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148 { |
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149 error ("schur: second argument must be a string"); |
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150 return retval; |
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151 } |
2928 | 152 } |
153 | |
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154 bool force_complex = false; |
2928 | 155 |
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156 if (ord == "real") |
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157 { |
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158 ord = std::string (); |
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159 } |
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160 else if (ord == "complex") |
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161 { |
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162 force_complex = true; |
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163 ord = std::string (); |
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164 } |
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165 else |
2928 | 166 { |
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167 char ord_char = ord.empty () ? 'U' : ord[0]; |
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168 |
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169 if (ord_char != 'U' && ord_char != 'A' && ord_char != 'D' |
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170 && ord_char != 'u' && ord_char != 'a' && ord_char != 'd') |
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171 { |
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172 warning ("schur: incorrect ordered schur argument `%c'", |
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173 ord.c_str ()); |
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174 return retval; |
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175 } |
2928 | 176 } |
177 | |
5275 | 178 octave_idx_type nr = arg.rows (); |
179 octave_idx_type nc = arg.columns (); | |
2928 | 180 |
181 if (nr != nc) | |
182 { | |
183 gripe_square_matrix_required ("schur"); | |
184 return retval; | |
185 } | |
186 | |
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187 if (! arg.is_numeric_type ()) |
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188 gripe_wrong_type_arg ("schur", arg); |
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189 else if (arg.is_single_type ()) |
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191 if (! force_complex && arg.is_real_type ()) |
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192 { |
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193 FloatMatrix tmp = arg.float_matrix_value (); |
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195 if (! error_state) |
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196 { |
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197 if (nargout == 0 || nargout == 1) |
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198 { |
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199 FloatSCHUR result (tmp, ord, false); |
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200 retval(0) = result.schur_matrix (); |
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201 } |
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202 else |
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203 { |
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204 FloatSCHUR result (tmp, ord, true); |
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205 retval(1) = result.schur_matrix (); |
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206 retval(0) = result.unitary_matrix (); |
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207 } |
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208 } |
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209 } |
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210 else |
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211 { |
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212 FloatComplexMatrix ctmp = arg.float_complex_matrix_value (); |
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213 |
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214 if (! error_state) |
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215 { |
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216 |
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217 if (nargout == 0 || nargout == 1) |
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218 { |
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219 FloatComplexSCHUR result (ctmp, ord, false); |
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220 retval(0) = mark_upper_triangular (result.schur_matrix ()); |
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221 } |
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222 else |
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223 { |
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224 FloatComplexSCHUR result (ctmp, ord, true); |
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225 retval(1) = mark_upper_triangular (result.schur_matrix ()); |
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226 retval(0) = result.unitary_matrix (); |
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227 } |
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228 } |
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229 } |
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231 else |
2928 | 232 { |
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233 if (! force_complex && arg.is_real_type ()) |
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234 { |
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235 Matrix tmp = arg.matrix_value (); |
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237 if (! error_state) |
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238 { |
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239 if (nargout == 0 || nargout == 1) |
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240 { |
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241 SCHUR result (tmp, ord, false); |
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242 retval(0) = result.schur_matrix (); |
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243 } |
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244 else |
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245 { |
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246 SCHUR result (tmp, ord, true); |
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247 retval(1) = result.schur_matrix (); |
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248 retval(0) = result.unitary_matrix (); |
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249 } |
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250 } |
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251 } |
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252 else |
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253 { |
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254 ComplexMatrix ctmp = arg.complex_matrix_value (); |
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255 |
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256 if (! error_state) |
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257 { |
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258 |
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259 if (nargout == 0 || nargout == 1) |
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260 { |
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261 ComplexSCHUR result (ctmp, ord, false); |
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262 retval(0) = mark_upper_triangular (result.schur_matrix ()); |
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263 } |
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264 else |
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265 { |
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266 ComplexSCHUR result (ctmp, ord, true); |
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267 retval(1) = mark_upper_triangular (result.schur_matrix ()); |
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268 retval(0) = result.unitary_matrix (); |
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269 } |
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270 } |
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271 } |
2928 | 272 } |
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273 |
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274 return retval; |
2928 | 275 } |
276 | |
277 /* | |
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278 |
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279 %!test |
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280 %! a = [1, 2, 3; 4, 5, 9; 7, 8, 6]; |
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281 %! [u, s] = schur (a); |
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282 %! assert(u' * a * u, s, sqrt (eps)); |
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283 |
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284 %!test |
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285 %! a = single([1, 2, 3; 4, 5, 9; 7, 8, 6]); |
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286 %! [u, s] = schur (a); |
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287 %! assert(u' * a * u, s, sqrt (eps('single'))); |
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288 |
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289 %!test |
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290 %! fail("schur ([1, 2; 3, 4], 2)","warning"); |
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291 |
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292 %!error <Invalid call to schur.*> schur (); |
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293 %!error schur ([1, 2, 3; 4, 5, 6]); |
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294 |
2928 | 295 */ |
10822 | 296 |
297 DEFUN_DLD (rsf2csf, args, nargout, | |
298 "-*- texinfo -*-\n\ | |
299 @deftypefn {Function File} {[@var{U}, @var{T}] =} rsf2csf (@var{UR}, @var{TR})\n\ | |
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300 Convert a real, upper quasi-triangular Schur@tie{}form @var{TR} to a complex,\n\ |
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301 upper triangular Schur@tie{}form @var{T}.\n\ |
10822 | 302 \n\ |
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303 Note that the following relations hold:\n\ |
10822 | 304 \n\ |
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305 @tex\n\ |
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306 $UR \\cdot TR \\cdot {UR}^T = U T U^{\\dagger}$ and\n\ |
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307 $U^{\\dagger} U$ is the identity matrix I.\n\ |
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308 @end tex\n\ |
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309 @ifnottex\n\ |
10822 | 310 @code{@var{UR} * @var{TR} * @var{UR}' = @var{U} * @var{T} * @var{U}'} and\n\ |
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311 @code{@var{U}' * @var{U}} is the identity matrix I.\n\ |
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312 @end ifnottex\n\ |
10822 | 313 \n\ |
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314 Note also that @var{U} and @var{T} are not unique.\n\ |
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315 @seealso{schur}\n\ |
10822 | 316 @end deftypefn") |
317 { | |
318 octave_value_list retval; | |
319 | |
320 if (args.length () == 2 && nargout <= 2) | |
321 { | |
322 if (! args(0).is_numeric_type ()) | |
323 gripe_wrong_type_arg ("rsf2csf", args(0)); | |
324 else if (! args(1).is_numeric_type ()) | |
325 gripe_wrong_type_arg ("rsf2csf", args(1)); | |
326 else if (args(0).is_complex_type () || args(1).is_complex_type ()) | |
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327 error ("rsf2csf: UR and TR must be real matrices"); |
10822 | 328 else |
329 { | |
330 | |
331 if (args(0).is_single_type () || args(1).is_single_type ()) | |
332 { | |
333 FloatMatrix u = args(0).float_matrix_value (); | |
334 FloatMatrix t = args(1).float_matrix_value (); | |
335 if (! error_state) | |
336 { | |
337 FloatComplexSCHUR cs (FloatSCHUR (t, u)); | |
338 | |
339 retval(1) = cs.schur_matrix (); | |
340 retval(0) = cs.unitary_matrix (); | |
341 } | |
342 } | |
343 else | |
344 { | |
345 Matrix u = args(0).matrix_value (); | |
346 Matrix t = args(1).matrix_value (); | |
347 if (! error_state) | |
348 { | |
349 ComplexSCHUR cs (SCHUR (t, u)); | |
350 | |
351 retval(1) = cs.schur_matrix (); | |
352 retval(0) = cs.unitary_matrix (); | |
353 } | |
354 } | |
355 } | |
356 } | |
357 else | |
358 print_usage (); | |
359 | |
360 return retval; | |
361 } | |
362 | |
363 /* | |
364 | |
365 %!test | |
366 %! A = [1, 1, 1, 2; 1, 2, 1, 1; 1, 1, 3, 1; -2, 1, 1, 1]; | |
367 %! [u, t] = schur (A); | |
368 %! [U, T] = rsf2csf (u, t); | |
369 %! assert (norm (u * t * u' - U * T * U'), 0, 1e-12) | |
370 %! assert (norm (A - U * T * U'), 0, 1e-12) | |
371 | |
372 %!test | |
373 %! A = rand (10); | |
374 %! [u, t] = schur (A); | |
375 %! [U, T] = rsf2csf (u, t); | |
376 %! assert (norm (tril (T, -1)), 0) | |
377 %! assert (norm (U * U'), 1, 1e-14) | |
378 | |
379 */ |