Mercurial > octave
annotate libinterp/corefcn/qz.cc @ 20892:c07bee629973
2015 Code Sprint: use ovl ().
author | Rik <rik@octave.org> |
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date | Mon, 14 Dec 2015 11:28:48 -0800 |
parents | 1142cf6abc0d |
children | 48b2ad5ee801 |
rev | line source |
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3183 | 1 /* |
2 | |
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3 Copyright (C) 1998-2015 A. S. Hodel |
3183 | 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. | |
3183 | 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/>. | |
3183 | 20 |
21 */ | |
22 | |
23 // Generalized eigenvalue balancing via LAPACK | |
3911 | 24 |
25 // Author: A. S. Hodel <scotte@eng.auburn.edu> | |
3183 | 26 |
27 #undef DEBUG | |
28 #undef DEBUG_SORT | |
29 #undef DEBUG_EIG | |
30 | |
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31 #ifdef HAVE_CONFIG_H |
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32 #include <config.h> |
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33 #endif |
3183 | 34 |
35 #include <cfloat> | |
4051 | 36 |
3523 | 37 #include <iostream> |
4051 | 38 #include <iomanip> |
3183 | 39 |
40 #include "CmplxQRP.h" | |
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41 #include "CmplxQR.h" |
3183 | 42 #include "dbleQR.h" |
4153 | 43 #include "f77-fcn.h" |
7231 | 44 #include "lo-math.h" |
4153 | 45 #include "quit.h" |
46 | |
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47 #include "defun.h" |
3183 | 48 #include "error.h" |
49 #include "gripes.h" | |
50 #include "oct-obj.h" | |
51 #include "oct-map.h" | |
52 #include "ov.h" | |
53 #include "pager.h" | |
3185 | 54 #if defined (DEBUG) || defined (DEBUG_SORT) |
3183 | 55 #include "pr-output.h" |
56 #endif | |
57 #include "symtab.h" | |
58 #include "utils.h" | |
59 #include "variables.h" | |
60 | |
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61 typedef octave_idx_type (*sort_function) (const octave_idx_type& LSIZE, |
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62 const double& ALPHA, |
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63 const double& BETA, const double& S, |
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64 const double& P); |
3183 | 65 |
66 extern "C" | |
67 { | |
4552 | 68 F77_RET_T |
69 F77_FUNC (dggbal, DGGBAL) (F77_CONST_CHAR_ARG_DECL, | |
10311 | 70 const octave_idx_type& N, double* A, |
71 const octave_idx_type& LDA, double* B, | |
72 const octave_idx_type& LDB, octave_idx_type& ILO, | |
73 octave_idx_type& IHI, double* LSCALE, | |
74 double* RSCALE, double* WORK, | |
75 octave_idx_type& INFO | |
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76 F77_CHAR_ARG_LEN_DECL); |
3183 | 77 |
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78 F77_RET_T |
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79 F77_FUNC (zggbal, ZGGBAL) (F77_CONST_CHAR_ARG_DECL, |
10311 | 80 const octave_idx_type& N, Complex* A, |
81 const octave_idx_type& LDA, Complex* B, | |
82 const octave_idx_type& LDB, octave_idx_type& ILO, | |
83 octave_idx_type& IHI, double* LSCALE, | |
84 double* RSCALE, double* WORK, | |
85 octave_idx_type& INFO | |
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86 F77_CHAR_ARG_LEN_DECL); |
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87 |
4552 | 88 F77_RET_T |
89 F77_FUNC (dggbak, DGGBAK) (F77_CONST_CHAR_ARG_DECL, | |
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90 F77_CONST_CHAR_ARG_DECL, |
10311 | 91 const octave_idx_type& N, |
92 const octave_idx_type& ILO, | |
93 const octave_idx_type& IHI, | |
94 const double* LSCALE, const double* RSCALE, | |
95 octave_idx_type& M, double* V, | |
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96 const octave_idx_type& LDV, octave_idx_type& INFO |
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97 F77_CHAR_ARG_LEN_DECL |
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98 F77_CHAR_ARG_LEN_DECL); |
3183 | 99 |
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100 F77_RET_T |
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101 F77_FUNC (zggbak, ZGGBAK) (F77_CONST_CHAR_ARG_DECL, |
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102 F77_CONST_CHAR_ARG_DECL, |
10311 | 103 const octave_idx_type& N, |
104 const octave_idx_type& ILO, | |
105 const octave_idx_type& IHI, | |
106 const double* LSCALE, const double* RSCALE, | |
107 octave_idx_type& M, Complex* V, | |
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108 const octave_idx_type& LDV, octave_idx_type& INFO |
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109 F77_CHAR_ARG_LEN_DECL |
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110 F77_CHAR_ARG_LEN_DECL); |
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111 |
4552 | 112 F77_RET_T |
113 F77_FUNC (dgghrd, DGGHRD) (F77_CONST_CHAR_ARG_DECL, | |
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114 F77_CONST_CHAR_ARG_DECL, |
10311 | 115 const octave_idx_type& N, |
116 const octave_idx_type& ILO, | |
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117 const octave_idx_type& IHI, double* A, |
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118 const octave_idx_type& LDA, double* B, |
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119 const octave_idx_type& LDB, double* Q, |
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120 const octave_idx_type& LDQ, double* Z, |
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121 const octave_idx_type& LDZ, octave_idx_type& INFO |
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122 F77_CHAR_ARG_LEN_DECL |
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123 F77_CHAR_ARG_LEN_DECL); |
3183 | 124 |
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125 F77_RET_T |
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126 F77_FUNC (zgghrd, ZGGHRD) (F77_CONST_CHAR_ARG_DECL, |
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127 F77_CONST_CHAR_ARG_DECL, |
10311 | 128 const octave_idx_type& N, |
129 const octave_idx_type& ILO, | |
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130 const octave_idx_type& IHI, Complex* A, |
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131 const octave_idx_type& LDA, Complex* B, |
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132 const octave_idx_type& LDB, Complex* Q, |
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133 const octave_idx_type& LDQ, Complex* Z, |
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134 const octave_idx_type& LDZ, octave_idx_type& INFO |
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135 F77_CHAR_ARG_LEN_DECL |
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136 F77_CHAR_ARG_LEN_DECL); |
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137 |
4552 | 138 F77_RET_T |
139 F77_FUNC (dhgeqz, DHGEQZ) (F77_CONST_CHAR_ARG_DECL, | |
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140 F77_CONST_CHAR_ARG_DECL, |
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141 F77_CONST_CHAR_ARG_DECL, |
10311 | 142 const octave_idx_type& N, |
143 const octave_idx_type& ILO, | |
144 const octave_idx_type& IHI, | |
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145 double* A, const octave_idx_type& LDA, double* B, |
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146 const octave_idx_type& LDB, double* ALPHAR, |
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147 double* ALPHAI, double* BETA, double* Q, |
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148 const octave_idx_type& LDQ, double* Z, |
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149 const octave_idx_type& LDZ, double* WORK, |
10311 | 150 const octave_idx_type& LWORK, |
151 octave_idx_type& INFO | |
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152 F77_CHAR_ARG_LEN_DECL |
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153 F77_CHAR_ARG_LEN_DECL |
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154 F77_CHAR_ARG_LEN_DECL); |
3183 | 155 |
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156 F77_RET_T |
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157 F77_FUNC (zhgeqz, ZHGEQZ) (F77_CONST_CHAR_ARG_DECL, |
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158 F77_CONST_CHAR_ARG_DECL, |
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159 F77_CONST_CHAR_ARG_DECL, |
10311 | 160 const octave_idx_type& N, |
161 const octave_idx_type& ILO, | |
162 const octave_idx_type& IHI, | |
163 Complex* A, const octave_idx_type& LDA, | |
164 Complex* B, const octave_idx_type& LDB, | |
165 Complex* ALPHA, Complex* BETA, Complex* CQ, | |
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166 const octave_idx_type& LDQ, |
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167 Complex* CZ, const octave_idx_type& LDZ, |
10311 | 168 Complex* WORK, const octave_idx_type& LWORK, |
169 double* RWORK, octave_idx_type& INFO | |
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170 F77_CHAR_ARG_LEN_DECL |
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171 F77_CHAR_ARG_LEN_DECL |
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172 F77_CHAR_ARG_LEN_DECL); |
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173 |
4552 | 174 F77_RET_T |
10311 | 175 F77_FUNC (dlag2, DLAG2) (const double* A, const octave_idx_type& LDA, |
176 const double* B, const octave_idx_type& LDB, | |
177 const double& SAFMIN, double& SCALE1, | |
178 double& SCALE2, double& WR1, double& WR2, | |
179 double& WI); | |
3183 | 180 |
181 // Van Dooren's code (netlib.org: toms/590) for reordering | |
182 // GEP. Only processes Z, not Q. | |
4552 | 183 F77_RET_T |
10311 | 184 F77_FUNC (dsubsp, DSUBSP) (const octave_idx_type& NMAX, |
185 const octave_idx_type& N, double* A, | |
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186 double* B, double* Z, sort_function, |
10311 | 187 const double& EPS, octave_idx_type& NDIM, |
188 octave_idx_type& FAIL, octave_idx_type* IND); | |
3183 | 189 |
10311 | 190 // Documentation for DTGEVC incorrectly states that VR, VL are |
3183 | 191 // complex*16; they are declared in DTGEVC as double precision |
10311 | 192 // (probably a cut and paste problem fro ZTGEVC). |
4552 | 193 F77_RET_T |
194 F77_FUNC (dtgevc, DTGEVC) (F77_CONST_CHAR_ARG_DECL, | |
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195 F77_CONST_CHAR_ARG_DECL, |
10311 | 196 octave_idx_type* SELECT, |
197 const octave_idx_type& N, double* A, | |
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198 const octave_idx_type& LDA, double* B, |
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199 const octave_idx_type& LDB, double* VL, |
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200 const octave_idx_type& LDVL, double* VR, |
10311 | 201 const octave_idx_type& LDVR, |
202 const octave_idx_type& MM, octave_idx_type& M, | |
203 double* WORK, octave_idx_type& INFO | |
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204 F77_CHAR_ARG_LEN_DECL |
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205 F77_CHAR_ARG_LEN_DECL); |
3183 | 206 |
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207 F77_RET_T |
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208 F77_FUNC (ztgevc, ZTGEVC) (F77_CONST_CHAR_ARG_DECL, |
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209 F77_CONST_CHAR_ARG_DECL, |
10311 | 210 octave_idx_type* SELECT, |
211 const octave_idx_type& N, const Complex* A, | |
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212 const octave_idx_type& LDA,const Complex* B, |
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213 const octave_idx_type& LDB, Complex* xVL, |
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214 const octave_idx_type& LDVL, Complex* xVR, |
10311 | 215 const octave_idx_type& LDVR, |
216 const octave_idx_type& MM, octave_idx_type& M, | |
217 Complex* CWORK, double* RWORK, | |
218 octave_idx_type& INFO | |
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219 F77_CHAR_ARG_LEN_DECL |
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220 F77_CHAR_ARG_LEN_DECL); |
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221 |
4552 | 222 F77_RET_T |
223 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG_DECL, | |
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224 double& retval |
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225 F77_CHAR_ARG_LEN_DECL); |
3185 | 226 |
4552 | 227 F77_RET_T |
228 F77_FUNC (xdlange, XDLANGE) (F77_CONST_CHAR_ARG_DECL, | |
10311 | 229 const octave_idx_type&, |
230 const octave_idx_type&, const double*, | |
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231 const octave_idx_type&, double*, double& |
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232 F77_CHAR_ARG_LEN_DECL); |
3183 | 233 } |
234 | |
235 // fcrhp, fin, fout, folhp: | |
236 // routines for ordering of generalized eigenvalues | |
237 // return 1 if test is passed, 0 otherwise | |
238 // fin: |lambda| < 1 | |
239 // fout: |lambda| >= 1 | |
240 // fcrhp: real(lambda) >= 0 | |
241 // folhp: real(lambda) < 0 | |
242 | |
5275 | 243 static octave_idx_type |
244 fcrhp (const octave_idx_type& lsize, const double& alpha, | |
3185 | 245 const double& beta, const double& s, const double&) |
3183 | 246 { |
3185 | 247 if (lsize == 1) |
10311 | 248 return (alpha * beta >= 0 ? 1 : -1); |
3185 | 249 else |
3183 | 250 return (s >= 0 ? 1 : -1); |
251 } | |
3185 | 252 |
5275 | 253 static octave_idx_type |
254 fin (const octave_idx_type& lsize, const double& alpha, | |
3185 | 255 const double& beta, const double&, const double& p) |
3183 | 256 { |
5275 | 257 octave_idx_type retval; |
3183 | 258 |
3185 | 259 if (lsize == 1) |
260 retval = (fabs (alpha) < fabs (beta) ? 1 : -1); | |
261 else | |
262 retval = (fabs (p) < 1 ? 1 : -1); | |
263 | |
264 #ifdef DEBUG | |
3538 | 265 std::cout << "qz: fin: retval=" << retval << std::endl; |
3185 | 266 #endif |
267 | |
3183 | 268 return retval; |
269 } | |
3185 | 270 |
5275 | 271 static octave_idx_type |
272 folhp (const octave_idx_type& lsize, const double& alpha, | |
3185 | 273 const double& beta, const double& s, const double&) |
3183 | 274 { |
3185 | 275 if (lsize == 1) |
10311 | 276 return (alpha * beta < 0 ? 1 : -1); |
3185 | 277 else |
3183 | 278 return (s < 0 ? 1 : -1); |
279 } | |
3185 | 280 |
5275 | 281 static octave_idx_type |
282 fout (const octave_idx_type& lsize, const double& alpha, | |
3185 | 283 const double& beta, const double&, const double& p) |
3183 | 284 { |
3185 | 285 if (lsize == 1) |
286 return (fabs (alpha) >= fabs (beta) ? 1 : -1); | |
287 else | |
288 return (fabs (p) >= 1 ? 1 : -1); | |
3183 | 289 } |
290 | |
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291 |
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292 //FIXME: Matlab does not produce lambda as the first output argument. |
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293 // Compatibility problem? |
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294 DEFUN (qz, args, nargout, |
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295 "-*- texinfo -*-\n\ |
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296 @deftypefn {} {@var{lambda} =} qz (@var{A}, @var{B})\n\ |
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297 @deftypefnx {} {@var{lambda} =} qz (@var{A}, @var{B}, @var{opt})\n\ |
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298 QZ@tie{}decomposition of the generalized eigenvalue problem\n\ |
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299 (@math{A x = s B x}).\n\ |
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300 \n\ |
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301 There are three ways to call this function:\n\ |
3372 | 302 @enumerate\n\ |
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303 @item @code{@var{lambda} = qz (@var{A}, @var{B})}\n\ |
3185 | 304 \n\ |
5016 | 305 Computes the generalized eigenvalues\n\ |
306 @tex\n\ | |
307 $\\lambda$\n\ | |
308 @end tex\n\ | |
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309 @ifnottex\n\ |
5016 | 310 @var{lambda}\n\ |
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311 @end ifnottex\n\ |
5016 | 312 of @math{(A - s B)}.\n\ |
10840 | 313 \n\ |
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314 @item @code{[AA, BB, Q, Z, V, W, @var{lambda}] = qz (@var{A}, @var{B})}\n\ |
3185 | 315 \n\ |
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316 Computes QZ@tie{}decomposition, generalized eigenvectors, and generalized\n\ |
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317 eigenvalues of @math{(A - s B)}\n\ |
5016 | 318 @tex\n\ |
319 $$ AV = BV{ \\rm diag }(\\lambda) $$\n\ | |
320 $$ W^T A = { \\rm diag }(\\lambda)W^T B $$\n\ | |
321 $$ AA = Q^T AZ, BB = Q^T BZ $$\n\ | |
322 @end tex\n\ | |
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323 @ifnottex\n\ |
10840 | 324 \n\ |
3372 | 325 @example\n\ |
326 @group\n\ | |
5481 | 327 \n\ |
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328 A * V = B * V * diag (@var{lambda})\n\ |
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329 W' * A = diag (@var{lambda}) * W' * B\n\ |
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330 AA = Q * A * Z, BB = Q * B * Z\n\ |
5481 | 331 \n\ |
3372 | 332 @end group\n\ |
333 @end example\n\ | |
10840 | 334 \n\ |
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335 @end ifnottex\n\ |
5016 | 336 with @var{Q} and @var{Z} orthogonal (unitary)= @var{I}\n\ |
3185 | 337 \n\ |
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338 @item @code{[AA,BB,Z@{, @var{lambda}@}] = qz (@var{A}, @var{B}, @var{opt})}\n\ |
3185 | 339 \n\ |
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340 As in form [2], but allows ordering of generalized eigenpairs for, e.g.,\n\ |
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341 solution of discrete time algebraic Riccati equations. Form 3 is not\n\ |
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342 available for complex matrices, and does not compute the generalized\n\ |
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343 eigenvectors @var{V}, @var{W}, nor the orthogonal matrix @var{Q}.\n\ |
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344 \n\ |
3372 | 345 @table @var\n\ |
346 @item opt\n\ | |
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347 for ordering eigenvalues of the @nospell{GEP} pencil. The leading block of\n\ |
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348 the revised pencil contains all eigenvalues that satisfy:\n\ |
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349 \n\ |
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350 @table @asis\n\ |
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351 @item @qcode{\"N\"}\n\ |
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352 = unordered (default)\n\ |
3183 | 353 \n\ |
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354 @item @qcode{\"S\"}\n\ |
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355 = small: leading block has all |lambda| @leq{} 1\n\ |
3185 | 356 \n\ |
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357 @item @qcode{\"B\"}\n\ |
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358 = big: leading block has all |lambda| @geq{} 1\n\ |
3372 | 359 \n\ |
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360 @item @qcode{\"-\"}\n\ |
5481 | 361 = negative real part: leading block has all eigenvalues\n\ |
362 in the open left half-plane\n\ | |
3372 | 363 \n\ |
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364 @item @qcode{\"+\"}\n\ |
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365 = non-negative real part: leading block has all eigenvalues\n\ |
5481 | 366 in the closed right half-plane\n\ |
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367 @end table\n\ |
3372 | 368 @end table\n\ |
369 @end enumerate\n\ | |
3183 | 370 \n\ |
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371 Note: @code{qz} performs permutation balancing, but not scaling\n\ |
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372 (@pxref{XREFbalance}). The order of output arguments was selected for\n\ |
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373 compatibility with @sc{matlab}.\n\ |
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374 @seealso{eig, balance, lu, chol, hess, qr, qzhess, schur, svd}\n\ |
3372 | 375 @end deftypefn") |
3183 | 376 { |
377 int nargin = args.length (); | |
378 | |
3185 | 379 #ifdef DEBUG |
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380 std::cout << "qz: nargin = " << nargin |
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381 << ", nargout = " << nargout << std::endl; |
3185 | 382 #endif |
3183 | 383 |
3185 | 384 if (nargin < 2 || nargin > 3 || nargout > 7) |
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385 print_usage (); |
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386 |
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387 if (nargin == 3 && (nargout < 3 || nargout > 4)) |
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388 error ("qz: invalid number of output arguments for form [3] call"); |
3183 | 389 |
3185 | 390 #ifdef DEBUG |
3538 | 391 std::cout << "qz: determine ordering option" << std::endl; |
3185 | 392 #endif |
3183 | 393 |
10311 | 394 // Determine ordering option. |
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395 volatile char ord_job = 0; |
3183 | 396 static double safmin; |
3185 | 397 |
398 if (nargin == 2) | |
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399 ord_job = 'N'; |
3185 | 400 else |
401 { | |
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402 std::string tmp = args(2).xstring_value ("qz: OPT must be a string"); |
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403 |
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404 if (! tmp.empty ()) |
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405 ord_job = tmp[0]; |
3183 | 406 |
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407 if (! (ord_job == 'N' || ord_job == 'n' |
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408 || ord_job == 'S' || ord_job == 's' |
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409 || ord_job == 'B' || ord_job == 'b' |
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410 || ord_job == '+' || ord_job == '-')) |
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411 error ("qz: invalid order option"); |
3185 | 412 |
413 // overflow constant required by dlag2 | |
4552 | 414 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG2 ("S", 1), |
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415 safmin |
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416 F77_CHAR_ARG_LEN (1)); |
3183 | 417 |
3185 | 418 #ifdef DEBUG_EIG |
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419 std::cout << "qz: initial value of safmin=" |
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420 << setiosflags (std::ios::scientific) |
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421 << safmin << std::endl; |
3185 | 422 #endif |
3183 | 423 |
10311 | 424 // Some machines (e.g., DEC alpha) get safmin = 0; |
425 // for these, use eps instead to avoid problems in dlag2. | |
3185 | 426 if (safmin == 0) |
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427 { |
3185 | 428 #ifdef DEBUG_EIG |
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429 std::cout << "qz: DANGER WILL ROBINSON: safmin is 0!" << std::endl; |
3185 | 430 #endif |
3183 | 431 |
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432 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG2 ("E", 1), |
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433 safmin |
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434 F77_CHAR_ARG_LEN (1)); |
3185 | 435 |
436 #ifdef DEBUG_EIG | |
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437 std::cout << "qz: safmin set to " |
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438 << setiosflags (std::ios::scientific) |
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439 << safmin << std::endl; |
3185 | 440 #endif |
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441 } |
3183 | 442 } |
443 | |
3185 | 444 #ifdef DEBUG |
3538 | 445 std::cout << "qz: check argument 1" << std::endl; |
3185 | 446 #endif |
3183 | 447 |
10311 | 448 // Argument 1: check if it's o.k. dimensioned. |
5275 | 449 octave_idx_type nn = args(0).rows (); |
3185 | 450 |
451 #ifdef DEBUG | |
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452 std::cout << "argument 1 dimensions: (" |
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453 << nn << "," << args(0).columns () << ")" |
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454 << std::endl; |
3185 | 455 #endif |
456 | |
20892 | 457 octave_value_list retval; |
458 | |
3185 | 459 int arg_is_empty = empty_arg ("qz", nn, args(0).columns ()); |
460 | |
3183 | 461 if (arg_is_empty < 0) |
3185 | 462 { |
463 gripe_empty_arg ("qz: parameter 1", 0); | |
464 return retval; | |
465 } | |
3183 | 466 else if (arg_is_empty > 0) |
3185 | 467 { |
468 gripe_empty_arg ("qz: parameter 1; continuing", 0); | |
469 return octave_value_list (2, Matrix ()); | |
470 } | |
471 else if (args(0).columns () != nn) | |
472 { | |
473 gripe_square_matrix_required ("qz"); | |
474 return retval; | |
475 } | |
3183 | 476 |
10311 | 477 // Argument 1: dimensions look good; get the value. |
3183 | 478 Matrix aa; |
479 ComplexMatrix caa; | |
3185 | 480 |
481 if (args(0).is_complex_type ()) | |
3183 | 482 caa = args(0).complex_matrix_value (); |
3185 | 483 else |
3183 | 484 aa = args(0).matrix_value (); |
3185 | 485 |
486 #ifdef DEBUG | |
3538 | 487 std::cout << "qz: check argument 2" << std::endl; |
3185 | 488 #endif |
3183 | 489 |
10311 | 490 // Extract argument 2 (bb, or cbb if complex). |
3185 | 491 if ((nn != args(1).columns ()) || (nn != args(1).rows ())) |
492 { | |
493 gripe_nonconformant (); | |
494 return retval; | |
495 } | |
496 | |
3183 | 497 Matrix bb; |
498 ComplexMatrix cbb; | |
3185 | 499 |
500 if (args(1).is_complex_type ()) | |
3183 | 501 cbb = args(1).complex_matrix_value (); |
502 else | |
503 bb = args(1).matrix_value (); | |
3185 | 504 |
3183 | 505 // Both matrices loaded, now let's check what kind of arithmetic: |
10311 | 506 // declared volatile to avoid compiler warnings about long jumps, |
507 // vforks. | |
3185 | 508 |
10311 | 509 volatile int complex_case |
3185 | 510 = (args(0).is_complex_type () || args(1).is_complex_type ()); |
10311 | 511 |
3185 | 512 if (nargin == 3 && complex_case) |
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513 error ("qz: cannot re-order complex qz decomposition"); |
3183 | 514 |
10311 | 515 // First, declare variables used in both the real and complex case. |
3183 | 516 Matrix QQ(nn,nn), ZZ(nn,nn), VR(nn,nn), VL(nn,nn); |
517 RowVector alphar(nn), alphai(nn), betar(nn); | |
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518 ComplexRowVector xalpha(nn), xbeta(nn); |
3185 | 519 ComplexMatrix CQ(nn,nn), CZ(nn,nn), CVR(nn,nn), CVL(nn,nn); |
5275 | 520 octave_idx_type ilo, ihi, info; |
3185 | 521 char compq = (nargout >= 3 ? 'V' : 'N'); |
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522 char compz = ((nargout >= 4 || nargin == 3)? 'V' : 'N'); |
3183 | 523 |
10311 | 524 // Initialize Q, Z to identity if we need either of them. |
3185 | 525 if (compq == 'V' || compz == 'V') |
5275 | 526 for (octave_idx_type ii = 0; ii < nn; ii++) |
527 for (octave_idx_type jj = 0; jj < nn; jj++) | |
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528 { |
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529 OCTAVE_QUIT; |
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530 QQ(ii,jj) = ZZ(ii,jj) = (ii == jj ? 1.0 : 0.0); |
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531 } |
3183 | 532 |
10311 | 533 // Always perform permutation balancing. |
4552 | 534 const char bal_job = 'P'; |
10311 | 535 RowVector lscale (nn), rscale (nn), work (6 * nn), rwork (nn); |
3183 | 536 |
3185 | 537 if (complex_case) |
538 { | |
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539 #ifdef DEBUG |
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540 if (compq == 'V') |
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541 std::cout << "qz: performing balancing; CQ=" << std::endl |
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542 << CQ << std::endl; |
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543 #endif |
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544 if (args(0).is_real_type ()) |
10311 | 545 caa = ComplexMatrix (aa); |
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546 |
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547 if (args(1).is_real_type ()) |
10311 | 548 cbb = ComplexMatrix (bb); |
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549 |
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550 if (compq == 'V') |
10311 | 551 CQ = ComplexMatrix (QQ); |
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552 |
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553 if (compz == 'V') |
10311 | 554 CZ = ComplexMatrix (ZZ); |
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555 |
10311 | 556 F77_XFCN (zggbal, ZGGBAL, |
557 (F77_CONST_CHAR_ARG2 (&bal_job, 1), | |
558 nn, caa.fortran_vec (), nn, cbb.fortran_vec (), | |
559 nn, ilo, ihi, lscale.fortran_vec (), | |
560 rscale.fortran_vec (), work.fortran_vec (), info | |
561 F77_CHAR_ARG_LEN (1))); | |
3185 | 562 } |
3183 | 563 else |
3185 | 564 { |
565 #ifdef DEBUG | |
566 if (compq == 'V') | |
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567 std::cout << "qz: performing balancing; QQ=" << std::endl |
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568 << QQ << std::endl; |
3185 | 569 #endif |
3183 | 570 |
3185 | 571 F77_XFCN (dggbal, DGGBAL, |
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572 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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573 nn, aa.fortran_vec (), nn, bb.fortran_vec (), |
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574 nn, ilo, ihi, lscale.fortran_vec (), |
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575 rscale.fortran_vec (), work.fortran_vec (), info |
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576 F77_CHAR_ARG_LEN (1))); |
3185 | 577 } |
3183 | 578 |
579 // Since we just want the balancing matrices, we can use dggbal | |
10311 | 580 // for both the real and complex cases; left first |
3185 | 581 |
10311 | 582 #if 0 |
583 if (compq == 'V') | |
3185 | 584 { |
585 F77_XFCN (dggbak, DGGBAK, | |
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586 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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587 F77_CONST_CHAR_ARG2 ("L", 1), |
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588 nn, ilo, ihi, lscale.data (), rscale.data (), |
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589 nn, QQ.fortran_vec (), nn, info |
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590 F77_CHAR_ARG_LEN (1) |
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591 F77_CHAR_ARG_LEN (1))); |
3183 | 592 |
3185 | 593 #ifdef DEBUG |
594 if (compq == 'V') | |
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595 std::cout << "qz: balancing done; QQ=" << std::endl << QQ << std::endl; |
3185 | 596 #endif |
3183 | 597 } |
598 | |
599 // then right | |
3185 | 600 if (compz == 'V') |
601 { | |
4552 | 602 F77_XFCN (dggbak, DGGBAK, |
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603 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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604 F77_CONST_CHAR_ARG2 ("R", 1), |
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605 nn, ilo, ihi, lscale.data (), rscale.data (), |
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606 nn, ZZ.fortran_vec (), nn, info |
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607 F77_CHAR_ARG_LEN (1) |
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608 F77_CHAR_ARG_LEN (1))); |
3183 | 609 |
3185 | 610 #ifdef DEBUG |
611 if (compz == 'V') | |
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612 std::cout << "qz: balancing done; ZZ=" << std::endl << ZZ << std::endl; |
3185 | 613 #endif |
10311 | 614 } |
615 #endif | |
3183 | 616 |
617 static char qz_job; | |
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618 qz_job = (nargout < 2 ? 'E' : 'S'); |
3185 | 619 |
3183 | 620 if (complex_case) |
3185 | 621 { |
10311 | 622 // Complex case. |
623 | |
624 // The QR decomposition of cbb. | |
625 ComplexQR cbqr (cbb); | |
626 // The R matrix of QR decomposition for cbb. | |
627 cbb = cbqr.R (); | |
628 // (Q*)caa for following work. | |
629 caa = (cbqr.Q ().hermitian ()) * caa; | |
630 CQ = CQ * cbqr.Q (); | |
631 | |
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632 F77_XFCN (zgghrd, ZGGHRD, |
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633 (F77_CONST_CHAR_ARG2 (&compq, 1), |
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634 F77_CONST_CHAR_ARG2 (&compz, 1), |
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635 nn, ilo, ihi, caa.fortran_vec (), |
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636 nn, cbb.fortran_vec (), nn, CQ.fortran_vec (), nn, |
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637 CZ.fortran_vec (), nn, info |
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638 F77_CHAR_ARG_LEN (1) |
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639 F77_CHAR_ARG_LEN (1))); |
10311 | 640 |
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641 ComplexRowVector cwork (1 * nn); |
10311 | 642 |
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643 F77_XFCN (zhgeqz, ZHGEQZ, |
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644 (F77_CONST_CHAR_ARG2 (&qz_job, 1), |
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645 F77_CONST_CHAR_ARG2 (&compq, 1), |
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646 F77_CONST_CHAR_ARG2 (&compz, 1), |
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647 nn, ilo, ihi, |
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648 caa.fortran_vec (), nn, |
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649 cbb.fortran_vec (),nn, |
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650 xalpha.fortran_vec (), xbeta.fortran_vec (), |
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651 CQ.fortran_vec (), nn, |
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652 CZ.fortran_vec (), nn, |
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653 cwork.fortran_vec (), nn, rwork.fortran_vec (), info |
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654 F77_CHAR_ARG_LEN (1) |
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655 F77_CHAR_ARG_LEN (1) |
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656 F77_CHAR_ARG_LEN (1))); |
3185 | 657 |
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658 if (compq == 'V') |
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659 { |
10311 | 660 // Left eigenvector. |
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661 F77_XFCN (zggbak, ZGGBAK, |
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662 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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663 F77_CONST_CHAR_ARG2 ("L", 1), |
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664 nn, ilo, ihi, lscale.data (), rscale.data (), |
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665 nn, CQ.fortran_vec (), nn, info |
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666 F77_CHAR_ARG_LEN (1) |
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667 F77_CHAR_ARG_LEN (1))); |
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668 } |
3185 | 669 |
10311 | 670 // Right eigenvector. |
3185 | 671 if (compz == 'V') |
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672 { |
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673 F77_XFCN (zggbak, ZGGBAK, |
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674 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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675 F77_CONST_CHAR_ARG2 ("R", 1), |
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676 nn, ilo, ihi, lscale.data (), rscale.data (), |
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677 nn, CZ.fortran_vec (), nn, info |
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678 F77_CHAR_ARG_LEN (1) |
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679 F77_CHAR_ARG_LEN (1))); |
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680 } |
3185 | 681 |
682 } | |
10311 | 683 else |
3185 | 684 { |
685 #ifdef DEBUG | |
3538 | 686 std::cout << "qz: peforming qr decomposition of bb" << std::endl; |
3185 | 687 #endif |
3183 | 688 |
10311 | 689 // Compute the QR factorization of bb. |
3185 | 690 QR bqr (bb); |
691 | |
692 #ifdef DEBUG | |
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693 std::cout << "qz: qr (bb) done; now peforming qz decomposition" |
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694 << std::endl; |
3185 | 695 #endif |
3183 | 696 |
3185 | 697 bb = bqr.R (); |
698 | |
699 #ifdef DEBUG | |
3538 | 700 std::cout << "qz: extracted bb" << std::endl; |
3185 | 701 #endif |
3183 | 702 |
10311 | 703 aa = (bqr.Q ()).transpose () * aa; |
3185 | 704 |
705 #ifdef DEBUG | |
3538 | 706 std::cout << "qz: updated aa " << std::endl; |
707 std::cout << "bqr.Q () = " << std::endl << bqr.Q () << std::endl; | |
3183 | 708 |
3185 | 709 if (compq == 'V') |
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710 std::cout << "QQ =" << QQ << std::endl; |
3185 | 711 #endif |
3183 | 712 |
3185 | 713 if (compq == 'V') |
10311 | 714 QQ = QQ * bqr.Q (); |
3183 | 715 |
3185 | 716 #ifdef DEBUG |
3538 | 717 std::cout << "qz: precursors done..." << std::endl; |
3185 | 718 #endif |
3183 | 719 |
3185 | 720 #ifdef DEBUG |
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721 std::cout << "qz: compq = " << compq << ", compz = " << compz |
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722 << std::endl; |
3185 | 723 #endif |
3183 | 724 |
10311 | 725 // Reduce to generalized hessenberg form. |
3185 | 726 F77_XFCN (dgghrd, DGGHRD, |
10154
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727 (F77_CONST_CHAR_ARG2 (&compq, 1), |
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728 F77_CONST_CHAR_ARG2 (&compz, 1), |
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729 nn, ilo, ihi, aa.fortran_vec (), |
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730 nn, bb.fortran_vec (), nn, QQ.fortran_vec (), nn, |
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731 ZZ.fortran_vec (), nn, info |
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732 F77_CHAR_ARG_LEN (1) |
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733 F77_CHAR_ARG_LEN (1))); |
3183 | 734 |
10311 | 735 // Check if just computing generalized eigenvalues or if we're |
736 // actually computing the decomposition. | |
3183 | 737 |
10311 | 738 // Reduce to generalized Schur form. |
3185 | 739 F77_XFCN (dhgeqz, DHGEQZ, |
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740 (F77_CONST_CHAR_ARG2 (&qz_job, 1), |
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741 F77_CONST_CHAR_ARG2 (&compq, 1), |
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742 F77_CONST_CHAR_ARG2 (&compz, 1), |
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743 nn, ilo, ihi, aa.fortran_vec (), nn, bb.fortran_vec (), |
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744 nn, alphar.fortran_vec (), alphai.fortran_vec (), |
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745 betar.fortran_vec (), QQ.fortran_vec (), nn, |
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746 ZZ.fortran_vec (), nn, work.fortran_vec (), nn, info |
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747 F77_CHAR_ARG_LEN (1) |
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748 F77_CHAR_ARG_LEN (1) |
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749 F77_CHAR_ARG_LEN (1))); |
10311 | 750 |
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751 if (compq == 'V') |
10311 | 752 { |
753 F77_XFCN (dggbak, DGGBAK, | |
754 (F77_CONST_CHAR_ARG2 (&bal_job, 1), | |
755 F77_CONST_CHAR_ARG2 ("L", 1), | |
756 nn, ilo, ihi, lscale.data (), rscale.data (), | |
757 nn, QQ.fortran_vec (), nn, info | |
758 F77_CHAR_ARG_LEN (1) | |
759 F77_CHAR_ARG_LEN (1))); | |
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760 |
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761 #ifdef DEBUG |
10311 | 762 if (compq == 'V') |
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763 std::cout << "qz: balancing done; QQ=" << std::endl |
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764 << QQ << std::endl; |
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765 #endif |
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766 } |
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767 |
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768 // then right |
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769 if (compz == 'V') |
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770 { |
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771 F77_XFCN (dggbak, DGGBAK, |
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772 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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773 F77_CONST_CHAR_ARG2 ("R", 1), |
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774 nn, ilo, ihi, lscale.data (), rscale.data (), |
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775 nn, ZZ.fortran_vec (), nn, info |
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776 F77_CHAR_ARG_LEN (1) |
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777 F77_CHAR_ARG_LEN (1))); |
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778 |
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779 #ifdef DEBUG |
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780 if (compz == 'V') |
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781 std::cout << "qz: balancing done; ZZ=" << std::endl |
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782 << ZZ << std::endl; |
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783 #endif |
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784 } |
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|
785 |
3185 | 786 } |
3183 | 787 |
10311 | 788 // Order the QZ decomposition? |
8927
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789 if (! (ord_job == 'N' || ord_job == 'n')) |
3183 | 790 { |
3185 | 791 if (complex_case) |
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792 // Probably not needed, but better be safe. |
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793 error ("qz: cannot re-order complex qz decomposition"); |
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794 |
3185 | 795 #ifdef DEBUG_SORT |
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796 std::cout << "qz: ordering eigenvalues: ord_job = " |
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797 << ord_job << std::endl; |
3185 | 798 #endif |
3183 | 799 |
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800 // Declared static to avoid vfork/long jump compiler complaints. |
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801 static sort_function sort_test; |
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802 sort_test = 0; |
3183 | 803 |
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804 switch (ord_job) |
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805 { |
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806 case 'S': |
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807 case 's': |
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808 sort_test = &fin; |
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809 break; |
3183 | 810 |
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811 case 'B': |
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812 case 'b': |
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813 sort_test = &fout; |
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814 break; |
3183 | 815 |
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816 case '+': |
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817 sort_test = &fcrhp; |
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818 break; |
3183 | 819 |
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820 case '-': |
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821 sort_test = &folhp; |
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822 break; |
3185 | 823 |
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824 default: |
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825 // Invalid order option (should never happen, since we |
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826 // checked the options at the top). |
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827 panic_impossible (); |
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828 break; |
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829 } |
3183 | 830 |
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831 octave_idx_type ndim, fail; |
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832 double inf_norm; |
3185 | 833 |
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834 F77_XFCN (xdlange, XDLANGE, |
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835 (F77_CONST_CHAR_ARG2 ("I", 1), |
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836 nn, nn, aa.data (), nn, work.fortran_vec (), inf_norm |
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837 F77_CHAR_ARG_LEN (1))); |
3185 | 838 |
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839 double eps = std::numeric_limits<double>::epsilon () * inf_norm * nn; |
3185 | 840 |
841 #ifdef DEBUG_SORT | |
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842 std::cout << "qz: calling dsubsp: aa=" << std::endl; |
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843 octave_print_internal (std::cout, aa, 0); |
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844 std::cout << std::endl << "bb=" << std::endl; |
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845 octave_print_internal (std::cout, bb, 0); |
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846 if (compz == 'V') |
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847 { |
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848 std::cout << std::endl << "ZZ=" << std::endl; |
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849 octave_print_internal (std::cout, ZZ, 0); |
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850 } |
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851 std::cout << std::endl; |
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852 std::cout << "alphar = " << std::endl; |
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853 octave_print_internal (std::cout, (Matrix) alphar, 0); |
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854 std::cout << std::endl << "alphai = " << std::endl; |
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855 octave_print_internal (std::cout, (Matrix) alphai, 0); |
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856 std::cout << std::endl << "beta = " << std::endl; |
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857 octave_print_internal (std::cout, (Matrix) betar, 0); |
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858 std::cout << std::endl; |
3185 | 859 #endif |
860 | |
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861 Array<octave_idx_type> ind (dim_vector (nn, 1)); |
3550 | 862 |
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863 F77_XFCN (dsubsp, DSUBSP, |
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864 (nn, nn, aa.fortran_vec (), bb.fortran_vec (), |
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865 ZZ.fortran_vec (), sort_test, eps, ndim, fail, |
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866 ind.fortran_vec ())); |
3185 | 867 |
868 #ifdef DEBUG | |
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869 std::cout << "qz: back from dsubsp: aa=" << std::endl; |
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870 octave_print_internal (std::cout, aa, 0); |
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871 std::cout << std::endl << "bb=" << std::endl; |
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872 octave_print_internal (std::cout, bb, 0); |
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873 if (compz == 'V') |
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874 { |
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875 std::cout << std::endl << "ZZ=" << std::endl; |
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876 octave_print_internal (std::cout, ZZ, 0); |
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877 } |
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878 std::cout << std::endl; |
3185 | 879 #endif |
880 | |
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881 // Manually update alphar, alphai, betar. |
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882 static int jj; |
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883 |
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884 jj = 0; |
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885 while (jj < nn) |
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886 { |
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887 #ifdef DEBUG_EIG |
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888 std::cout << "computing gen eig #" << jj << std::endl; |
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889 #endif |
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890 |
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891 // Number of zeros in this block. |
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892 static int zcnt; |
3185 | 893 |
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894 if (jj == (nn-1)) |
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895 zcnt = 1; |
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896 else if (aa(jj+1,jj) == 0) |
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897 zcnt = 1; |
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898 else zcnt = 2; |
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|
899 |
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900 if (zcnt == 1) |
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901 { |
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902 // Real zero. |
3185 | 903 #ifdef DEBUG_EIG |
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904 std::cout << " single gen eig:" << std::endl; |
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905 std::cout << " alphar(" << jj << ") = " << aa(jj,jj) |
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906 << std::endl; |
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907 std::cout << " betar(" << jj << ") = " << bb(jj,jj) |
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908 << std::endl; |
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909 std::cout << " alphai(" << jj << ") = 0" << std::endl; |
3185 | 910 #endif |
911 | |
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912 alphar(jj) = aa(jj,jj); |
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913 alphai(jj) = 0; |
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914 betar(jj) = bb(jj,jj); |
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915 } |
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|
916 else |
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|
917 { |
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|
918 // Complex conjugate pair. |
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|
919 #ifdef DEBUG_EIG |
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920 std::cout << "qz: calling dlag2:" << std::endl; |
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921 std::cout << "safmin=" |
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|
922 << setiosflags (std::ios::scientific) |
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|
923 << safmin << std::endl; |
3185 | 924 |
20831
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925 for (int idr = jj; idr <= jj+1; idr++) |
10154
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|
926 { |
20831
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|
927 for (int idc = jj; idc <= jj+1; idc++) |
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|
928 { |
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|
929 std::cout << "aa(" << idr << "," << idc << ")=" |
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|
930 << aa(idr,idc) << std::endl; |
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|
931 std::cout << "bb(" << idr << "," << idc << ")=" |
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932 << bb(idr,idc) << std::endl; |
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|
933 } |
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|
934 } |
3185 | 935 #endif |
936 | |
20831
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|
937 // FIXME: probably should be using |
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|
938 // fortran_vec instead of &aa(jj,jj) here. |
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|
939 |
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|
940 double scale1, scale2, wr1, wr2, wi; |
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|
941 const double *aa_ptr = aa.data () + jj * nn + jj; |
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|
942 const double *bb_ptr = bb.data () + jj * nn + jj; |
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|
943 F77_XFCN (dlag2, DLAG2, |
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|
944 (aa_ptr, nn, bb_ptr, nn, safmin, |
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|
945 scale1, scale2, wr1, wr2, wi)); |
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|
946 |
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diff
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|
947 #ifdef DEBUG_EIG |
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|
948 std::cout << "dlag2 returns: scale1=" << scale1 |
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|
949 << "\tscale2=" << scale2 << std::endl |
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|
950 << "\twr1=" << wr1 << "\twr2=" << wr2 |
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|
951 << "\twi=" << wi << std::endl; |
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|
952 #endif |
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|
953 |
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|
954 // Just to be safe, check if it's a real pair. |
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|
955 if (wi == 0) |
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|
956 { |
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|
957 alphar(jj) = wr1; |
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|
958 alphai(jj) = 0; |
20831
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|
959 betar(jj) = scale1; |
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|
960 alphar(jj+1) = wr2; |
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|
961 alphai(jj+1) = 0; |
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|
962 betar(jj+1) = scale2; |
10154
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|
963 } |
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|
964 else |
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|
965 { |
20831
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|
966 alphar(jj) = alphar(jj+1) = wr1; |
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|
967 alphai(jj) = -(alphai(jj+1) = wi); |
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|
968 betar(jj) = betar(jj+1) = scale1; |
10154
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|
969 } |
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|
970 } |
3185 | 971 |
20831
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|
972 // Advance past this block. |
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|
973 jj += zcnt; |
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|
974 } |
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|
975 |
3185 | 976 #ifdef DEBUG_SORT |
20831
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|
977 std::cout << "qz: back from dsubsp: aa=" << std::endl; |
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|
978 octave_print_internal (std::cout, aa, 0); |
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|
979 std::cout << std::endl << "bb=" << std::endl; |
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|
980 octave_print_internal (std::cout, bb, 0); |
3185 | 981 |
20831
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|
982 if (compz == 'V') |
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|
983 { |
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|
984 std::cout << std::endl << "ZZ=" << std::endl; |
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|
985 octave_print_internal (std::cout, ZZ, 0); |
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diff
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|
986 } |
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|
987 std::cout << std::endl << "qz: ndim=" << ndim << std::endl |
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988 << "fail=" << fail << std::endl; |
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989 std::cout << "alphar = " << std::endl; |
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990 octave_print_internal (std::cout, (Matrix) alphar, 0); |
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991 std::cout << std::endl << "alphai = " << std::endl; |
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992 octave_print_internal (std::cout, (Matrix) alphai, 0); |
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993 std::cout << std::endl << "beta = " << std::endl; |
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994 octave_print_internal (std::cout, (Matrix) betar, 0); |
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995 std::cout << std::endl; |
3185 | 996 #endif |
3183 | 997 } |
3185 | 998 |
10311 | 999 // Compute generalized eigenvalues? |
3183 | 1000 ComplexColumnVector gev; |
3185 | 1001 |
1002 if (nargout < 2 || nargout == 7 || (nargin == 3 && nargout == 4)) | |
3183 | 1003 { |
3185 | 1004 if (complex_case) |
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1005 { |
10311 | 1006 int cnt = 0; |
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1007 |
10311 | 1008 for (int ii = 0; ii < nn; ii++) |
1009 cnt++; | |
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1010 |
10311 | 1011 ComplexColumnVector tmp (cnt); |
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1012 |
10311 | 1013 cnt = 0; |
1014 for (int ii = 0; ii < nn; ii++) | |
1015 tmp(cnt++) = xalpha(ii) / xbeta(ii); | |
1016 | |
1017 gev = tmp; | |
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1018 } |
3185 | 1019 else |
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1020 { |
3185 | 1021 #ifdef DEBUG |
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1022 std::cout << "qz: computing generalized eigenvalues" << std::endl; |
3185 | 1023 #endif |
3183 | 1024 |
10311 | 1025 // Return finite generalized eigenvalues. |
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1026 int cnt = 0; |
3185 | 1027 |
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1028 for (int ii = 0; ii < nn; ii++) |
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1029 if (betar(ii) != 0) |
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1030 cnt++; |
3185 | 1031 |
10311 | 1032 ComplexColumnVector tmp (cnt); |
3185 | 1033 |
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1034 cnt = 0; |
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1035 for (int ii = 0; ii < nn; ii++) |
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1036 if (betar(ii) != 0) |
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1037 tmp(cnt++) = Complex(alphar(ii), alphai(ii))/betar(ii); |
10311 | 1038 |
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1039 gev = tmp; |
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1040 } |
3183 | 1041 } |
1042 | |
10311 | 1043 // Right, left eigenvector matrices. |
3185 | 1044 if (nargout >= 5) |
3183 | 1045 { |
10311 | 1046 // Which side to compute? |
1047 char side = (nargout == 5 ? 'R' : 'B'); | |
1048 // Compute all of them and backtransform | |
1049 char howmny = 'B'; | |
1050 // Dummy pointer; select is not used. | |
1051 octave_idx_type *select = 0; | |
3185 | 1052 |
1053 if (complex_case) | |
10154
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1054 { |
10311 | 1055 CVL = CQ; |
1056 CVR = CZ; | |
1057 ComplexRowVector cwork2 (2 * nn); | |
1058 RowVector rwork2 (8 * nn); | |
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1059 octave_idx_type m; |
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1060 |
10311 | 1061 F77_XFCN (ztgevc, ZTGEVC, |
1062 (F77_CONST_CHAR_ARG2 (&side, 1), | |
1063 F77_CONST_CHAR_ARG2 (&howmny, 1), | |
1064 select, nn, caa.fortran_vec (), nn, cbb.fortran_vec (), | |
1065 nn, CVL.fortran_vec (), nn, CVR.fortran_vec (), nn, nn, | |
1066 m, cwork2.fortran_vec (), rwork2.fortran_vec (), info | |
1067 F77_CHAR_ARG_LEN (1) | |
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1068 F77_CHAR_ARG_LEN (1))); |
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1069 } |
3185 | 1070 else |
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1071 { |
3185 | 1072 #ifdef DEBUG |
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1073 std::cout << "qz: computing generalized eigenvectors" << std::endl; |
3185 | 1074 #endif |
1075 | |
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1076 VL = QQ; |
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1077 VR = ZZ; |
10311 | 1078 octave_idx_type m; |
1079 | |
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1080 F77_XFCN (dtgevc, DTGEVC, |
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1081 (F77_CONST_CHAR_ARG2 (&side, 1), |
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1082 F77_CONST_CHAR_ARG2 (&howmny, 1), |
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1083 select, nn, aa.fortran_vec (), nn, bb.fortran_vec (), |
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1084 nn, VL.fortran_vec (), nn, VR.fortran_vec (), nn, nn, |
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1085 m, work.fortran_vec (), info |
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1086 F77_CHAR_ARG_LEN (1) |
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1087 F77_CHAR_ARG_LEN (1))); |
3185 | 1088 |
10311 | 1089 // Now construct the complex form of VV, WW. |
10154
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1090 int jj = 0; |
3185 | 1091 |
10154
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1092 while (jj < nn) |
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1093 { |
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1094 OCTAVE_QUIT; |
4153 | 1095 |
10311 | 1096 // See if real or complex eigenvalue. |
1097 | |
1098 // Column increment; assume complex eigenvalue. | |
1099 int cinc = 2; | |
3185 | 1100 |
10154
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1101 if (jj == (nn-1)) |
10311 | 1102 // Single column. |
1103 cinc = 1; | |
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1104 else if (aa(jj+1,jj) == 0) |
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1105 cinc = 1; |
3185 | 1106 |
10311 | 1107 // Now copy the eigenvector (s) to CVR, CVL. |
10154
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1108 if (cinc == 1) |
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1109 { |
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1110 for (int ii = 0; ii < nn; ii++) |
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|
1111 CVR(ii,jj) = VR(ii,jj); |
3185 | 1112 |
10154
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1113 if (side == 'B') |
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1114 for (int ii = 0; ii < nn; ii++) |
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1115 CVL(ii,jj) = VL(ii,jj); |
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1116 } |
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1117 else |
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1118 { |
10311 | 1119 // Double column; complex vector. |
3185 | 1120 |
10154
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1121 for (int ii = 0; ii < nn; ii++) |
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1122 { |
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1123 CVR(ii,jj) = Complex (VR(ii,jj), VR(ii,jj+1)); |
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1124 CVR(ii,jj+1) = Complex (VR(ii,jj), -VR(ii,jj+1)); |
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1125 } |
3183 | 1126 |
10154
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1127 if (side == 'B') |
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1128 for (int ii = 0; ii < nn; ii++) |
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1129 { |
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1130 CVL(ii,jj) = Complex (VL(ii,jj), VL(ii,jj+1)); |
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1131 CVL(ii,jj+1) = Complex (VL(ii,jj), -VL(ii,jj+1)); |
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1132 } |
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1133 } |
3185 | 1134 |
10311 | 1135 // Advance to next eigenvectors (if any). |
10154
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1136 jj += cinc; |
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1137 } |
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1138 } |
3183 | 1139 } |
3185 | 1140 |
1141 switch (nargout) | |
1142 { | |
1143 case 7: | |
1144 retval(6) = gev; | |
1145 | |
10311 | 1146 case 6: |
1147 // Return eigenvectors. | |
3185 | 1148 retval(5) = CVL; |
1149 | |
10311 | 1150 case 5: |
1151 // Return eigenvectors. | |
3185 | 1152 retval(4) = CVR; |
1153 | |
1154 case 4: | |
1155 if (nargin == 3) | |
10154
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1156 { |
3185 | 1157 #ifdef DEBUG |
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1158 std::cout << "qz: sort: retval(3) = gev = " << std::endl; |
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1159 octave_print_internal (std::cout, gev); |
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1160 std::cout << std::endl; |
3185 | 1161 #endif |
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1162 retval(3) = gev; |
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1163 } |
3185 | 1164 else |
10311 | 1165 { |
1166 if (complex_case) | |
1167 retval(3) = CZ; | |
1168 else | |
1169 retval(3) = ZZ; | |
1170 } | |
3185 | 1171 |
1172 case 3: | |
1173 if (nargin == 3) | |
14876
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1174 { |
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1175 if (complex_case) |
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1176 retval(2) = CZ; |
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1177 else |
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1178 retval(2) = ZZ; |
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1179 } |
3185 | 1180 else |
10311 | 1181 { |
1182 if (complex_case) | |
1183 retval(2) = CQ.hermitian (); | |
1184 else | |
1185 retval(2) = QQ.transpose (); | |
1186 } | |
1187 | |
3185 | 1188 case 2: |
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1189 { |
10311 | 1190 if (complex_case) |
1191 { | |
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1192 #ifdef DEBUG |
14844
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|
1193 std::cout << "qz: retval(1) = cbb = " << std::endl; |
10311 | 1194 octave_print_internal (std::cout, cbb, 0); |
1195 std::cout << std::endl << "qz: retval(0) = caa = " <<std::endl; | |
1196 octave_print_internal (std::cout, caa, 0); | |
1197 std::cout << std::endl; | |
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1198 #endif |
10311 | 1199 retval(1) = cbb; |
1200 retval(0) = caa; | |
1201 } | |
17787
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|
1202 else |
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1203 { |
3185 | 1204 #ifdef DEBUG |
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1205 std::cout << "qz: retval(1) = bb = " << std::endl; |
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1206 octave_print_internal (std::cout, bb, 0); |
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1207 std::cout << std::endl << "qz: retval(0) = aa = " <<std::endl; |
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1208 octave_print_internal (std::cout, aa, 0); |
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1209 std::cout << std::endl; |
3185 | 1210 #endif |
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1211 retval(1) = bb; |
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1212 retval(0) = aa; |
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1213 } |
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1214 } |
10311 | 1215 break; |
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1216 |
3185 | 1217 case 1: |
1218 case 0: | |
1219 #ifdef DEBUG | |
3538 | 1220 std::cout << "qz: retval(0) = gev = " << gev << std::endl; |
3185 | 1221 #endif |
1222 retval(0) = gev; | |
1223 break; | |
1224 | |
1225 default: | |
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1226 error ("qz: too many return arguments"); |
3185 | 1227 break; |
3183 | 1228 } |
1229 | |
3185 | 1230 #ifdef DEBUG |
3538 | 1231 std::cout << "qz: exiting (at long last)" << std::endl; |
3185 | 1232 #endif |
3183 | 1233 |
1234 return retval; | |
1235 } | |
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1236 |
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1237 /* |
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1238 %!shared a, b, c |
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1239 %! a = [1 2; 0 3]; |
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1240 %! b = [1 0; 0 0]; |
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1241 %! c = [0 1; 0 0]; |
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1242 %!assert (qz (a,b), 1) |
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1243 %!assert (isempty (qz (a,c))) |
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1244 |
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1245 ## Exaple 7.7.3 in Golub & Van Loan |
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1246 %!test |
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1247 %! a = [ 10 1 2; |
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1248 %! 1 2 -1; |
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1249 %! 1 1 2]; |
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1250 %! b = reshape (1:9,3,3); |
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1251 %! [aa, bb, q, z, v, w, lambda] = qz (a, b); |
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1252 %! sz = length (lambda); |
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1253 %! observed = (b * v * diag ([lambda;0])) (:, 1:sz); |
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1254 %! assert ((a*v)(:, 1:sz), observed, norm (observed) * 1e-14); |
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1255 %! observed = (diag ([lambda;0]) * w' * b) (1:sz, :); |
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1256 %! assert ((w'*a)(1:sz, :) , observed, norm (observed) * 1e-13); |
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1257 %! assert (q * a * z, aa, norm (aa) * 1e-14); |
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1258 %! assert (q * b * z, bb, norm (bb) * 1e-14); |
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1259 |
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1260 %!test |
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1261 %! A = [0, 0, -1, 0; 1, 0, 0, 0; -1, 0, -2, -1; 0, -1, 1, 0]; |
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1262 %! B = [0, 0, 0, 0; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]; |
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1263 %! [AA, BB, Q, Z1] = qz (A, B); |
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1264 %! [AA, BB, Z2] = qz (A, B, '-'); |
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1265 %! assert (Z1, Z2); |
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1266 */ |