Mercurial > octave
annotate libinterp/corefcn/qz.cc @ 21751:b571fc85953f
maint: Use two spaces after period to indicate sentence break.
author | Rik <rik@octave.org> |
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date | Thu, 19 May 2016 18:48:52 -0700 |
parents | aba2e6293dd8 |
children | 112b20240c87 |
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 #if defined (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 |
4153 | 40 #include "f77-fcn.h" |
7231 | 41 #include "lo-math.h" |
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42 #include "qr.h" |
4153 | 43 #include "quit.h" |
44 | |
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45 #include "defun.h" |
3183 | 46 #include "error.h" |
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47 #include "errwarn.h" |
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48 #include "ovl.h" |
3183 | 49 #include "oct-map.h" |
50 #include "ov.h" | |
51 #include "pager.h" | |
3185 | 52 #if defined (DEBUG) || defined (DEBUG_SORT) |
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53 # include "pr-output.h" |
3183 | 54 #endif |
55 #include "symtab.h" | |
56 #include "utils.h" | |
57 #include "variables.h" | |
58 | |
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59 typedef octave_idx_type (*sort_function) (const octave_idx_type& LSIZE, |
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60 const double& ALPHA, |
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61 const double& BETA, const double& S, |
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62 const double& P); |
3183 | 63 |
64 extern "C" | |
65 { | |
4552 | 66 F77_RET_T |
67 F77_FUNC (dggbal, DGGBAL) (F77_CONST_CHAR_ARG_DECL, | |
10311 | 68 const octave_idx_type& N, double* A, |
69 const octave_idx_type& LDA, double* B, | |
70 const octave_idx_type& LDB, octave_idx_type& ILO, | |
71 octave_idx_type& IHI, double* LSCALE, | |
72 double* RSCALE, double* WORK, | |
73 octave_idx_type& INFO | |
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74 F77_CHAR_ARG_LEN_DECL); |
3183 | 75 |
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76 F77_RET_T |
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77 F77_FUNC (zggbal, ZGGBAL) (F77_CONST_CHAR_ARG_DECL, |
10311 | 78 const octave_idx_type& N, Complex* A, |
79 const octave_idx_type& LDA, Complex* B, | |
80 const octave_idx_type& LDB, octave_idx_type& ILO, | |
81 octave_idx_type& IHI, double* LSCALE, | |
82 double* RSCALE, double* WORK, | |
83 octave_idx_type& INFO | |
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84 F77_CHAR_ARG_LEN_DECL); |
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85 |
4552 | 86 F77_RET_T |
87 F77_FUNC (dggbak, DGGBAK) (F77_CONST_CHAR_ARG_DECL, | |
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88 F77_CONST_CHAR_ARG_DECL, |
10311 | 89 const octave_idx_type& N, |
90 const octave_idx_type& ILO, | |
91 const octave_idx_type& IHI, | |
92 const double* LSCALE, const double* RSCALE, | |
93 octave_idx_type& M, double* V, | |
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94 const octave_idx_type& LDV, octave_idx_type& INFO |
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95 F77_CHAR_ARG_LEN_DECL |
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96 F77_CHAR_ARG_LEN_DECL); |
3183 | 97 |
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98 F77_RET_T |
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99 F77_FUNC (zggbak, ZGGBAK) (F77_CONST_CHAR_ARG_DECL, |
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100 F77_CONST_CHAR_ARG_DECL, |
10311 | 101 const octave_idx_type& N, |
102 const octave_idx_type& ILO, | |
103 const octave_idx_type& IHI, | |
104 const double* LSCALE, const double* RSCALE, | |
105 octave_idx_type& M, Complex* V, | |
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106 const octave_idx_type& LDV, octave_idx_type& INFO |
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107 F77_CHAR_ARG_LEN_DECL |
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108 F77_CHAR_ARG_LEN_DECL); |
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109 |
4552 | 110 F77_RET_T |
111 F77_FUNC (dgghrd, DGGHRD) (F77_CONST_CHAR_ARG_DECL, | |
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112 F77_CONST_CHAR_ARG_DECL, |
10311 | 113 const octave_idx_type& N, |
114 const octave_idx_type& ILO, | |
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115 const octave_idx_type& IHI, double* A, |
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116 const octave_idx_type& LDA, double* B, |
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117 const octave_idx_type& LDB, double* Q, |
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118 const octave_idx_type& LDQ, double* Z, |
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119 const octave_idx_type& LDZ, octave_idx_type& INFO |
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120 F77_CHAR_ARG_LEN_DECL |
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121 F77_CHAR_ARG_LEN_DECL); |
3183 | 122 |
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123 F77_RET_T |
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124 F77_FUNC (zgghrd, ZGGHRD) (F77_CONST_CHAR_ARG_DECL, |
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125 F77_CONST_CHAR_ARG_DECL, |
10311 | 126 const octave_idx_type& N, |
127 const octave_idx_type& ILO, | |
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128 const octave_idx_type& IHI, Complex* A, |
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129 const octave_idx_type& LDA, Complex* B, |
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130 const octave_idx_type& LDB, Complex* Q, |
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131 const octave_idx_type& LDQ, Complex* Z, |
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132 const octave_idx_type& LDZ, octave_idx_type& INFO |
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133 F77_CHAR_ARG_LEN_DECL |
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134 F77_CHAR_ARG_LEN_DECL); |
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135 |
4552 | 136 F77_RET_T |
137 F77_FUNC (dhgeqz, DHGEQZ) (F77_CONST_CHAR_ARG_DECL, | |
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138 F77_CONST_CHAR_ARG_DECL, |
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139 F77_CONST_CHAR_ARG_DECL, |
10311 | 140 const octave_idx_type& N, |
141 const octave_idx_type& ILO, | |
142 const octave_idx_type& IHI, | |
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143 double* A, const octave_idx_type& LDA, double* B, |
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144 const octave_idx_type& LDB, double* ALPHAR, |
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145 double* ALPHAI, double* BETA, double* Q, |
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146 const octave_idx_type& LDQ, double* Z, |
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147 const octave_idx_type& LDZ, double* WORK, |
10311 | 148 const octave_idx_type& LWORK, |
149 octave_idx_type& INFO | |
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150 F77_CHAR_ARG_LEN_DECL |
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151 F77_CHAR_ARG_LEN_DECL |
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152 F77_CHAR_ARG_LEN_DECL); |
3183 | 153 |
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154 F77_RET_T |
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155 F77_FUNC (zhgeqz, ZHGEQZ) (F77_CONST_CHAR_ARG_DECL, |
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156 F77_CONST_CHAR_ARG_DECL, |
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157 F77_CONST_CHAR_ARG_DECL, |
10311 | 158 const octave_idx_type& N, |
159 const octave_idx_type& ILO, | |
160 const octave_idx_type& IHI, | |
161 Complex* A, const octave_idx_type& LDA, | |
162 Complex* B, const octave_idx_type& LDB, | |
163 Complex* ALPHA, Complex* BETA, Complex* CQ, | |
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164 const octave_idx_type& LDQ, |
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165 Complex* CZ, const octave_idx_type& LDZ, |
10311 | 166 Complex* WORK, const octave_idx_type& LWORK, |
167 double* RWORK, octave_idx_type& INFO | |
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168 F77_CHAR_ARG_LEN_DECL |
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169 F77_CHAR_ARG_LEN_DECL |
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170 F77_CHAR_ARG_LEN_DECL); |
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171 |
4552 | 172 F77_RET_T |
10311 | 173 F77_FUNC (dlag2, DLAG2) (const double* A, const octave_idx_type& LDA, |
174 const double* B, const octave_idx_type& LDB, | |
175 const double& SAFMIN, double& SCALE1, | |
176 double& SCALE2, double& WR1, double& WR2, | |
177 double& WI); | |
3183 | 178 |
179 // Van Dooren's code (netlib.org: toms/590) for reordering | |
180 // GEP. Only processes Z, not Q. | |
4552 | 181 F77_RET_T |
10311 | 182 F77_FUNC (dsubsp, DSUBSP) (const octave_idx_type& NMAX, |
183 const octave_idx_type& N, double* A, | |
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184 double* B, double* Z, sort_function, |
10311 | 185 const double& EPS, octave_idx_type& NDIM, |
186 octave_idx_type& FAIL, octave_idx_type* IND); | |
3183 | 187 |
10311 | 188 // Documentation for DTGEVC incorrectly states that VR, VL are |
3183 | 189 // complex*16; they are declared in DTGEVC as double precision |
10311 | 190 // (probably a cut and paste problem fro ZTGEVC). |
4552 | 191 F77_RET_T |
192 F77_FUNC (dtgevc, DTGEVC) (F77_CONST_CHAR_ARG_DECL, | |
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193 F77_CONST_CHAR_ARG_DECL, |
10311 | 194 octave_idx_type* SELECT, |
195 const octave_idx_type& N, double* A, | |
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196 const octave_idx_type& LDA, double* B, |
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197 const octave_idx_type& LDB, double* VL, |
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198 const octave_idx_type& LDVL, double* VR, |
10311 | 199 const octave_idx_type& LDVR, |
200 const octave_idx_type& MM, octave_idx_type& M, | |
201 double* WORK, octave_idx_type& INFO | |
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202 F77_CHAR_ARG_LEN_DECL |
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203 F77_CHAR_ARG_LEN_DECL); |
3183 | 204 |
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205 F77_RET_T |
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206 F77_FUNC (ztgevc, ZTGEVC) (F77_CONST_CHAR_ARG_DECL, |
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207 F77_CONST_CHAR_ARG_DECL, |
10311 | 208 octave_idx_type* SELECT, |
209 const octave_idx_type& N, const Complex* A, | |
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210 const octave_idx_type& LDA,const Complex* B, |
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211 const octave_idx_type& LDB, Complex* xVL, |
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212 const octave_idx_type& LDVL, Complex* xVR, |
10311 | 213 const octave_idx_type& LDVR, |
214 const octave_idx_type& MM, octave_idx_type& M, | |
215 Complex* CWORK, double* RWORK, | |
216 octave_idx_type& INFO | |
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217 F77_CHAR_ARG_LEN_DECL |
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218 F77_CHAR_ARG_LEN_DECL); |
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219 |
4552 | 220 F77_RET_T |
221 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG_DECL, | |
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222 double& retval |
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223 F77_CHAR_ARG_LEN_DECL); |
3185 | 224 |
4552 | 225 F77_RET_T |
226 F77_FUNC (xdlange, XDLANGE) (F77_CONST_CHAR_ARG_DECL, | |
10311 | 227 const octave_idx_type&, |
228 const octave_idx_type&, const double*, | |
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229 const octave_idx_type&, double*, double& |
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230 F77_CHAR_ARG_LEN_DECL); |
3183 | 231 } |
232 | |
233 // fcrhp, fin, fout, folhp: | |
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234 // Routines for ordering of generalized eigenvalues. |
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235 // Return 1 if test is passed, 0 otherwise. |
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236 // fin: |lambda| < 1 |
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237 // fout: |lambda| >= 1 |
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238 // fcrhp: real(lambda) >= 0 |
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239 // folhp: real(lambda) < 0 |
3183 | 240 |
5275 | 241 static octave_idx_type |
242 fcrhp (const octave_idx_type& lsize, const double& alpha, | |
3185 | 243 const double& beta, const double& s, const double&) |
3183 | 244 { |
3185 | 245 if (lsize == 1) |
10311 | 246 return (alpha * beta >= 0 ? 1 : -1); |
3185 | 247 else |
3183 | 248 return (s >= 0 ? 1 : -1); |
249 } | |
3185 | 250 |
5275 | 251 static octave_idx_type |
252 fin (const octave_idx_type& lsize, const double& alpha, | |
3185 | 253 const double& beta, const double&, const double& p) |
3183 | 254 { |
5275 | 255 octave_idx_type retval; |
3183 | 256 |
3185 | 257 if (lsize == 1) |
258 retval = (fabs (alpha) < fabs (beta) ? 1 : -1); | |
259 else | |
260 retval = (fabs (p) < 1 ? 1 : -1); | |
261 | |
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262 #if defined (DEBUG) |
3538 | 263 std::cout << "qz: fin: retval=" << retval << std::endl; |
3185 | 264 #endif |
265 | |
3183 | 266 return retval; |
267 } | |
3185 | 268 |
5275 | 269 static octave_idx_type |
270 folhp (const octave_idx_type& lsize, const double& alpha, | |
3185 | 271 const double& beta, const double& s, const double&) |
3183 | 272 { |
3185 | 273 if (lsize == 1) |
10311 | 274 return (alpha * beta < 0 ? 1 : -1); |
3185 | 275 else |
3183 | 276 return (s < 0 ? 1 : -1); |
277 } | |
3185 | 278 |
5275 | 279 static octave_idx_type |
280 fout (const octave_idx_type& lsize, const double& alpha, | |
3185 | 281 const double& beta, const double&, const double& p) |
3183 | 282 { |
3185 | 283 if (lsize == 1) |
284 return (fabs (alpha) >= fabs (beta) ? 1 : -1); | |
285 else | |
286 return (fabs (p) >= 1 ? 1 : -1); | |
3183 | 287 } |
288 | |
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289 |
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290 //FIXME: Matlab does not produce lambda as the first output argument. |
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291 // Compatibility problem? |
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292 DEFUN (qz, args, nargout, |
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293 "-*- texinfo -*-\n\ |
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294 @deftypefn {} {@var{lambda} =} qz (@var{A}, @var{B})\n\ |
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295 @deftypefnx {} {@var{lambda} =} qz (@var{A}, @var{B}, @var{opt})\n\ |
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296 QZ@tie{}decomposition of the generalized eigenvalue problem\n\ |
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297 (@math{A x = s B x}).\n\ |
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298 \n\ |
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299 There are three ways to call this function:\n\ |
3372 | 300 @enumerate\n\ |
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301 @item @code{@var{lambda} = qz (@var{A}, @var{B})}\n\ |
3185 | 302 \n\ |
5016 | 303 Computes the generalized eigenvalues\n\ |
304 @tex\n\ | |
305 $\\lambda$\n\ | |
306 @end tex\n\ | |
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307 @ifnottex\n\ |
5016 | 308 @var{lambda}\n\ |
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309 @end ifnottex\n\ |
5016 | 310 of @math{(A - s B)}.\n\ |
10840 | 311 \n\ |
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312 @item @code{[AA, BB, Q, Z, V, W, @var{lambda}] = qz (@var{A}, @var{B})}\n\ |
3185 | 313 \n\ |
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314 Computes QZ@tie{}decomposition, generalized eigenvectors, and generalized\n\ |
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315 eigenvalues of @math{(A - s B)}\n\ |
5016 | 316 @tex\n\ |
317 $$ AV = BV{ \\rm diag }(\\lambda) $$\n\ | |
318 $$ W^T A = { \\rm diag }(\\lambda)W^T B $$\n\ | |
319 $$ AA = Q^T AZ, BB = Q^T BZ $$\n\ | |
320 @end tex\n\ | |
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321 @ifnottex\n\ |
10840 | 322 \n\ |
3372 | 323 @example\n\ |
324 @group\n\ | |
5481 | 325 \n\ |
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326 A * V = B * V * diag (@var{lambda})\n\ |
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327 W' * A = diag (@var{lambda}) * W' * B\n\ |
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328 AA = Q * A * Z, BB = Q * B * Z\n\ |
5481 | 329 \n\ |
3372 | 330 @end group\n\ |
331 @end example\n\ | |
10840 | 332 \n\ |
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333 @end ifnottex\n\ |
5016 | 334 with @var{Q} and @var{Z} orthogonal (unitary)= @var{I}\n\ |
3185 | 335 \n\ |
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336 @item @code{[AA,BB,Z@{, @var{lambda}@}] = qz (@var{A}, @var{B}, @var{opt})}\n\ |
3185 | 337 \n\ |
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338 As in form [2], but allows ordering of generalized eigenpairs for, e.g.,\n\ |
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339 solution of discrete time algebraic Riccati equations. Form 3 is not\n\ |
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340 available for complex matrices, and does not compute the generalized\n\ |
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341 eigenvectors @var{V}, @var{W}, nor the orthogonal matrix @var{Q}.\n\ |
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342 \n\ |
3372 | 343 @table @var\n\ |
344 @item opt\n\ | |
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345 for ordering eigenvalues of the @nospell{GEP} pencil. The leading block of\n\ |
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346 the revised pencil contains all eigenvalues that satisfy:\n\ |
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347 \n\ |
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348 @table @asis\n\ |
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349 @item @qcode{\"N\"}\n\ |
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350 = unordered (default)\n\ |
3183 | 351 \n\ |
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352 @item @qcode{\"S\"}\n\ |
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353 = small: leading block has all |lambda| @leq{} 1\n\ |
3185 | 354 \n\ |
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355 @item @qcode{\"B\"}\n\ |
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356 = big: leading block has all |lambda| @geq{} 1\n\ |
3372 | 357 \n\ |
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358 @item @qcode{\"-\"}\n\ |
5481 | 359 = negative real part: leading block has all eigenvalues\n\ |
360 in the open left half-plane\n\ | |
3372 | 361 \n\ |
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362 @item @qcode{\"+\"}\n\ |
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363 = non-negative real part: leading block has all eigenvalues\n\ |
5481 | 364 in the closed right half-plane\n\ |
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365 @end table\n\ |
3372 | 366 @end table\n\ |
367 @end enumerate\n\ | |
3183 | 368 \n\ |
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369 Note: @code{qz} performs permutation balancing, but not scaling\n\ |
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370 (@pxref{XREFbalance}). The order of output arguments was selected for\n\ |
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371 compatibility with @sc{matlab}.\n\ |
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372 @seealso{eig, balance, lu, chol, hess, qr, qzhess, schur, svd}\n\ |
3372 | 373 @end deftypefn") |
3183 | 374 { |
375 int nargin = args.length (); | |
376 | |
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377 #if defined (DEBUG) |
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378 std::cout << "qz: nargin = " << nargin |
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379 << ", nargout = " << nargout << std::endl; |
3185 | 380 #endif |
3183 | 381 |
3185 | 382 if (nargin < 2 || nargin > 3 || nargout > 7) |
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383 print_usage (); |
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384 |
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385 if (nargin == 3 && (nargout < 3 || nargout > 4)) |
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386 error ("qz: invalid number of output arguments for form [3] call"); |
3183 | 387 |
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388 #if defined (DEBUG) |
3538 | 389 std::cout << "qz: determine ordering option" << std::endl; |
3185 | 390 #endif |
3183 | 391 |
10311 | 392 // Determine ordering option. |
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393 volatile char ord_job = 0; |
3183 | 394 static double safmin; |
3185 | 395 |
396 if (nargin == 2) | |
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397 ord_job = 'N'; |
3185 | 398 else |
399 { | |
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400 std::string tmp = args(2).xstring_value ("qz: OPT must be a string"); |
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401 |
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402 if (! tmp.empty ()) |
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403 ord_job = tmp[0]; |
3183 | 404 |
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405 if (! (ord_job == 'N' || ord_job == 'n' |
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406 || ord_job == 'S' || ord_job == 's' |
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407 || ord_job == 'B' || ord_job == 'b' |
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408 || ord_job == '+' || ord_job == '-')) |
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409 error ("qz: invalid order option"); |
3185 | 410 |
411 // overflow constant required by dlag2 | |
4552 | 412 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG2 ("S", 1), |
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413 safmin |
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414 F77_CHAR_ARG_LEN (1)); |
3183 | 415 |
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416 #if defined (DEBUG_EIG) |
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417 std::cout << "qz: initial value of safmin=" |
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418 << setiosflags (std::ios::scientific) |
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419 << safmin << std::endl; |
3185 | 420 #endif |
3183 | 421 |
10311 | 422 // Some machines (e.g., DEC alpha) get safmin = 0; |
423 // for these, use eps instead to avoid problems in dlag2. | |
3185 | 424 if (safmin == 0) |
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425 { |
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426 #if defined (DEBUG_EIG) |
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427 std::cout << "qz: DANGER WILL ROBINSON: safmin is 0!" << std::endl; |
3185 | 428 #endif |
3183 | 429 |
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430 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG2 ("E", 1), |
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431 safmin |
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432 F77_CHAR_ARG_LEN (1)); |
3185 | 433 |
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434 #if defined (DEBUG_EIG) |
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435 std::cout << "qz: safmin set to " |
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436 << setiosflags (std::ios::scientific) |
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437 << safmin << std::endl; |
3185 | 438 #endif |
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439 } |
3183 | 440 } |
441 | |
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442 #if defined (DEBUG) |
3538 | 443 std::cout << "qz: check argument 1" << std::endl; |
3185 | 444 #endif |
3183 | 445 |
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446 // Argument 1: check if it's okay dimensioned. |
5275 | 447 octave_idx_type nn = args(0).rows (); |
3185 | 448 |
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449 #if defined (DEBUG) |
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450 std::cout << "argument 1 dimensions: (" |
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451 << nn << "," << args(0).columns () << ")" |
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452 << std::endl; |
3185 | 453 #endif |
454 | |
20892 | 455 octave_value_list retval; |
456 | |
3185 | 457 int arg_is_empty = empty_arg ("qz", nn, args(0).columns ()); |
458 | |
3183 | 459 if (arg_is_empty < 0) |
3185 | 460 { |
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461 warn_empty_arg ("qz: parameter 1"); |
3185 | 462 return retval; |
463 } | |
3183 | 464 else if (arg_is_empty > 0) |
3185 | 465 { |
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466 warn_empty_arg ("qz: parameter 1; continuing"); |
3185 | 467 return octave_value_list (2, Matrix ()); |
468 } | |
469 else if (args(0).columns () != nn) | |
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470 err_square_matrix_required ("qz", "A"); |
3183 | 471 |
10311 | 472 // Argument 1: dimensions look good; get the value. |
3183 | 473 Matrix aa; |
474 ComplexMatrix caa; | |
3185 | 475 |
476 if (args(0).is_complex_type ()) | |
3183 | 477 caa = args(0).complex_matrix_value (); |
3185 | 478 else |
3183 | 479 aa = args(0).matrix_value (); |
3185 | 480 |
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481 #if defined (DEBUG) |
3538 | 482 std::cout << "qz: check argument 2" << std::endl; |
3185 | 483 #endif |
3183 | 484 |
10311 | 485 // Extract argument 2 (bb, or cbb if complex). |
3185 | 486 if ((nn != args(1).columns ()) || (nn != args(1).rows ())) |
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487 err_nonconformant (); |
3185 | 488 |
3183 | 489 Matrix bb; |
490 ComplexMatrix cbb; | |
3185 | 491 |
492 if (args(1).is_complex_type ()) | |
3183 | 493 cbb = args(1).complex_matrix_value (); |
494 else | |
495 bb = args(1).matrix_value (); | |
3185 | 496 |
3183 | 497 // Both matrices loaded, now let's check what kind of arithmetic: |
10311 | 498 // declared volatile to avoid compiler warnings about long jumps, |
499 // vforks. | |
3185 | 500 |
10311 | 501 volatile int complex_case |
3185 | 502 = (args(0).is_complex_type () || args(1).is_complex_type ()); |
10311 | 503 |
3185 | 504 if (nargin == 3 && complex_case) |
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505 error ("qz: cannot re-order complex qz decomposition"); |
3183 | 506 |
10311 | 507 // First, declare variables used in both the real and complex case. |
3183 | 508 Matrix QQ(nn,nn), ZZ(nn,nn), VR(nn,nn), VL(nn,nn); |
509 RowVector alphar(nn), alphai(nn), betar(nn); | |
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510 ComplexRowVector xalpha(nn), xbeta(nn); |
3185 | 511 ComplexMatrix CQ(nn,nn), CZ(nn,nn), CVR(nn,nn), CVL(nn,nn); |
5275 | 512 octave_idx_type ilo, ihi, info; |
3185 | 513 char compq = (nargout >= 3 ? 'V' : 'N'); |
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514 char compz = ((nargout >= 4 || nargin == 3)? 'V' : 'N'); |
3183 | 515 |
10311 | 516 // Initialize Q, Z to identity if we need either of them. |
3185 | 517 if (compq == 'V' || compz == 'V') |
5275 | 518 for (octave_idx_type ii = 0; ii < nn; ii++) |
519 for (octave_idx_type jj = 0; jj < nn; jj++) | |
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520 { |
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521 OCTAVE_QUIT; |
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522 QQ(ii,jj) = ZZ(ii,jj) = (ii == jj ? 1.0 : 0.0); |
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523 } |
3183 | 524 |
10311 | 525 // Always perform permutation balancing. |
4552 | 526 const char bal_job = 'P'; |
10311 | 527 RowVector lscale (nn), rscale (nn), work (6 * nn), rwork (nn); |
3183 | 528 |
3185 | 529 if (complex_case) |
530 { | |
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531 #if defined (DEBUG) |
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532 if (compq == 'V') |
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533 std::cout << "qz: performing balancing; CQ=" << std::endl |
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534 << CQ << std::endl; |
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535 #endif |
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536 if (args(0).is_real_type ()) |
10311 | 537 caa = ComplexMatrix (aa); |
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538 |
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539 if (args(1).is_real_type ()) |
10311 | 540 cbb = ComplexMatrix (bb); |
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541 |
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542 if (compq == 'V') |
10311 | 543 CQ = ComplexMatrix (QQ); |
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544 |
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545 if (compz == 'V') |
10311 | 546 CZ = ComplexMatrix (ZZ); |
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547 |
10311 | 548 F77_XFCN (zggbal, ZGGBAL, |
549 (F77_CONST_CHAR_ARG2 (&bal_job, 1), | |
550 nn, caa.fortran_vec (), nn, cbb.fortran_vec (), | |
551 nn, ilo, ihi, lscale.fortran_vec (), | |
552 rscale.fortran_vec (), work.fortran_vec (), info | |
553 F77_CHAR_ARG_LEN (1))); | |
3185 | 554 } |
3183 | 555 else |
3185 | 556 { |
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557 #if defined (DEBUG) |
3185 | 558 if (compq == 'V') |
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559 std::cout << "qz: performing balancing; QQ=" << std::endl |
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560 << QQ << std::endl; |
3185 | 561 #endif |
3183 | 562 |
3185 | 563 F77_XFCN (dggbal, DGGBAL, |
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564 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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565 nn, aa.fortran_vec (), nn, bb.fortran_vec (), |
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566 nn, ilo, ihi, lscale.fortran_vec (), |
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567 rscale.fortran_vec (), work.fortran_vec (), info |
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568 F77_CHAR_ARG_LEN (1))); |
3185 | 569 } |
3183 | 570 |
571 // Since we just want the balancing matrices, we can use dggbal | |
10311 | 572 // for both the real and complex cases; left first |
3185 | 573 |
10311 | 574 #if 0 |
575 if (compq == 'V') | |
3185 | 576 { |
577 F77_XFCN (dggbak, DGGBAK, | |
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578 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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579 F77_CONST_CHAR_ARG2 ("L", 1), |
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580 nn, ilo, ihi, lscale.data (), rscale.data (), |
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581 nn, QQ.fortran_vec (), nn, info |
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582 F77_CHAR_ARG_LEN (1) |
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583 F77_CHAR_ARG_LEN (1))); |
3183 | 584 |
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585 #if defined (DEBUG) |
3185 | 586 if (compq == 'V') |
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587 std::cout << "qz: balancing done; QQ=" << std::endl << QQ << std::endl; |
3185 | 588 #endif |
3183 | 589 } |
590 | |
591 // then right | |
3185 | 592 if (compz == 'V') |
593 { | |
4552 | 594 F77_XFCN (dggbak, DGGBAK, |
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595 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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596 F77_CONST_CHAR_ARG2 ("R", 1), |
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597 nn, ilo, ihi, lscale.data (), rscale.data (), |
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598 nn, ZZ.fortran_vec (), nn, info |
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599 F77_CHAR_ARG_LEN (1) |
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600 F77_CHAR_ARG_LEN (1))); |
3183 | 601 |
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602 #if defined (DEBUG) |
3185 | 603 if (compz == 'V') |
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604 std::cout << "qz: balancing done; ZZ=" << std::endl << ZZ << std::endl; |
3185 | 605 #endif |
10311 | 606 } |
607 #endif | |
3183 | 608 |
609 static char qz_job; | |
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610 qz_job = (nargout < 2 ? 'E' : 'S'); |
3185 | 611 |
3183 | 612 if (complex_case) |
3185 | 613 { |
10311 | 614 // Complex case. |
615 | |
616 // The QR decomposition of cbb. | |
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617 qr<ComplexMatrix> cbqr (cbb); |
10311 | 618 // The R matrix of QR decomposition for cbb. |
619 cbb = cbqr.R (); | |
620 // (Q*)caa for following work. | |
621 caa = (cbqr.Q ().hermitian ()) * caa; | |
622 CQ = CQ * cbqr.Q (); | |
623 | |
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624 F77_XFCN (zgghrd, ZGGHRD, |
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625 (F77_CONST_CHAR_ARG2 (&compq, 1), |
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626 F77_CONST_CHAR_ARG2 (&compz, 1), |
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627 nn, ilo, ihi, caa.fortran_vec (), |
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628 nn, cbb.fortran_vec (), nn, CQ.fortran_vec (), nn, |
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629 CZ.fortran_vec (), nn, info |
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630 F77_CHAR_ARG_LEN (1) |
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631 F77_CHAR_ARG_LEN (1))); |
10311 | 632 |
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633 ComplexRowVector cwork (1 * nn); |
10311 | 634 |
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635 F77_XFCN (zhgeqz, ZHGEQZ, |
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636 (F77_CONST_CHAR_ARG2 (&qz_job, 1), |
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637 F77_CONST_CHAR_ARG2 (&compq, 1), |
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638 F77_CONST_CHAR_ARG2 (&compz, 1), |
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639 nn, ilo, ihi, |
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640 caa.fortran_vec (), nn, |
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641 cbb.fortran_vec (),nn, |
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642 xalpha.fortran_vec (), xbeta.fortran_vec (), |
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643 CQ.fortran_vec (), nn, |
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644 CZ.fortran_vec (), nn, |
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645 cwork.fortran_vec (), nn, rwork.fortran_vec (), info |
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646 F77_CHAR_ARG_LEN (1) |
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647 F77_CHAR_ARG_LEN (1) |
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648 F77_CHAR_ARG_LEN (1))); |
3185 | 649 |
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650 if (compq == 'V') |
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651 { |
10311 | 652 // Left eigenvector. |
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653 F77_XFCN (zggbak, ZGGBAK, |
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654 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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655 F77_CONST_CHAR_ARG2 ("L", 1), |
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656 nn, ilo, ihi, lscale.data (), rscale.data (), |
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657 nn, CQ.fortran_vec (), nn, info |
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658 F77_CHAR_ARG_LEN (1) |
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659 F77_CHAR_ARG_LEN (1))); |
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660 } |
3185 | 661 |
10311 | 662 // Right eigenvector. |
3185 | 663 if (compz == 'V') |
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664 { |
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665 F77_XFCN (zggbak, ZGGBAK, |
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666 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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667 F77_CONST_CHAR_ARG2 ("R", 1), |
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668 nn, ilo, ihi, lscale.data (), rscale.data (), |
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669 nn, CZ.fortran_vec (), nn, info |
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670 F77_CHAR_ARG_LEN (1) |
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671 F77_CHAR_ARG_LEN (1))); |
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672 } |
3185 | 673 |
674 } | |
10311 | 675 else |
3185 | 676 { |
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677 #if defined (DEBUG) |
3538 | 678 std::cout << "qz: peforming qr decomposition of bb" << std::endl; |
3185 | 679 #endif |
3183 | 680 |
10311 | 681 // Compute the QR factorization of bb. |
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682 qr<Matrix> bqr (bb); |
3185 | 683 |
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684 #if defined (DEBUG) |
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685 std::cout << "qz: qr (bb) done; now peforming qz decomposition" |
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686 << std::endl; |
3185 | 687 #endif |
3183 | 688 |
3185 | 689 bb = bqr.R (); |
690 | |
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691 #if defined (DEBUG) |
3538 | 692 std::cout << "qz: extracted bb" << std::endl; |
3185 | 693 #endif |
3183 | 694 |
10311 | 695 aa = (bqr.Q ()).transpose () * aa; |
3185 | 696 |
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697 #if defined (DEBUG) |
3538 | 698 std::cout << "qz: updated aa " << std::endl; |
699 std::cout << "bqr.Q () = " << std::endl << bqr.Q () << std::endl; | |
3183 | 700 |
3185 | 701 if (compq == 'V') |
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702 std::cout << "QQ =" << QQ << std::endl; |
3185 | 703 #endif |
3183 | 704 |
3185 | 705 if (compq == 'V') |
10311 | 706 QQ = QQ * bqr.Q (); |
3183 | 707 |
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708 #if defined (DEBUG) |
3538 | 709 std::cout << "qz: precursors done..." << std::endl; |
3185 | 710 #endif |
3183 | 711 |
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712 #if defined (DEBUG) |
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713 std::cout << "qz: compq = " << compq << ", compz = " << compz |
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714 << std::endl; |
3185 | 715 #endif |
3183 | 716 |
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717 // Reduce to generalized Hessenberg form. |
3185 | 718 F77_XFCN (dgghrd, DGGHRD, |
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719 (F77_CONST_CHAR_ARG2 (&compq, 1), |
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720 F77_CONST_CHAR_ARG2 (&compz, 1), |
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721 nn, ilo, ihi, aa.fortran_vec (), |
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722 nn, bb.fortran_vec (), nn, QQ.fortran_vec (), nn, |
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723 ZZ.fortran_vec (), nn, info |
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724 F77_CHAR_ARG_LEN (1) |
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725 F77_CHAR_ARG_LEN (1))); |
3183 | 726 |
10311 | 727 // Check if just computing generalized eigenvalues or if we're |
728 // actually computing the decomposition. | |
3183 | 729 |
10311 | 730 // Reduce to generalized Schur form. |
3185 | 731 F77_XFCN (dhgeqz, DHGEQZ, |
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732 (F77_CONST_CHAR_ARG2 (&qz_job, 1), |
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733 F77_CONST_CHAR_ARG2 (&compq, 1), |
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734 F77_CONST_CHAR_ARG2 (&compz, 1), |
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735 nn, ilo, ihi, aa.fortran_vec (), nn, bb.fortran_vec (), |
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736 nn, alphar.fortran_vec (), alphai.fortran_vec (), |
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737 betar.fortran_vec (), QQ.fortran_vec (), nn, |
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738 ZZ.fortran_vec (), nn, work.fortran_vec (), nn, info |
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739 F77_CHAR_ARG_LEN (1) |
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740 F77_CHAR_ARG_LEN (1) |
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741 F77_CHAR_ARG_LEN (1))); |
10311 | 742 |
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743 if (compq == 'V') |
10311 | 744 { |
745 F77_XFCN (dggbak, DGGBAK, | |
746 (F77_CONST_CHAR_ARG2 (&bal_job, 1), | |
747 F77_CONST_CHAR_ARG2 ("L", 1), | |
748 nn, ilo, ihi, lscale.data (), rscale.data (), | |
749 nn, QQ.fortran_vec (), nn, info | |
750 F77_CHAR_ARG_LEN (1) | |
751 F77_CHAR_ARG_LEN (1))); | |
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752 |
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753 #if defined (DEBUG) |
10311 | 754 if (compq == 'V') |
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755 std::cout << "qz: balancing done; QQ=" << std::endl |
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756 << QQ << std::endl; |
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757 #endif |
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758 } |
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|
759 |
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760 // then right |
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761 if (compz == 'V') |
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762 { |
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763 F77_XFCN (dggbak, DGGBAK, |
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764 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
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765 F77_CONST_CHAR_ARG2 ("R", 1), |
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|
766 nn, ilo, ihi, lscale.data (), rscale.data (), |
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767 nn, ZZ.fortran_vec (), nn, info |
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768 F77_CHAR_ARG_LEN (1) |
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|
769 F77_CHAR_ARG_LEN (1))); |
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770 |
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771 #if defined (DEBUG) |
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772 if (compz == 'V') |
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773 std::cout << "qz: balancing done; ZZ=" << std::endl |
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774 << ZZ << std::endl; |
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775 #endif |
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776 } |
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777 |
3185 | 778 } |
3183 | 779 |
10311 | 780 // Order the QZ decomposition? |
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781 if (! (ord_job == 'N' || ord_job == 'n')) |
3183 | 782 { |
3185 | 783 if (complex_case) |
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784 // Probably not needed, but better be safe. |
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785 error ("qz: cannot re-order complex qz decomposition"); |
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786 |
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787 #if defined (DEBUG_SORT) |
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788 std::cout << "qz: ordering eigenvalues: ord_job = " |
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789 << ord_job << std::endl; |
3185 | 790 #endif |
3183 | 791 |
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792 // Declared static to avoid vfork/long jump compiler complaints. |
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793 static sort_function sort_test; |
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794 sort_test = 0; |
3183 | 795 |
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796 switch (ord_job) |
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797 { |
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798 case 'S': |
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799 case 's': |
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800 sort_test = &fin; |
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801 break; |
3183 | 802 |
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803 case 'B': |
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804 case 'b': |
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805 sort_test = &fout; |
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806 break; |
3183 | 807 |
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808 case '+': |
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|
809 sort_test = &fcrhp; |
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810 break; |
3183 | 811 |
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812 case '-': |
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813 sort_test = &folhp; |
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814 break; |
3185 | 815 |
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816 default: |
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817 // Invalid order option (should never happen, since we |
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818 // checked the options at the top). |
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819 panic_impossible (); |
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820 break; |
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821 } |
3183 | 822 |
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823 octave_idx_type ndim, fail; |
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824 double inf_norm; |
3185 | 825 |
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826 F77_XFCN (xdlange, XDLANGE, |
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827 (F77_CONST_CHAR_ARG2 ("I", 1), |
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828 nn, nn, aa.data (), nn, work.fortran_vec (), inf_norm |
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829 F77_CHAR_ARG_LEN (1))); |
3185 | 830 |
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831 double eps = std::numeric_limits<double>::epsilon () * inf_norm * nn; |
3185 | 832 |
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833 #if defined (DEBUG_SORT) |
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834 std::cout << "qz: calling dsubsp: aa=" << std::endl; |
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835 octave_print_internal (std::cout, aa, 0); |
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836 std::cout << std::endl << "bb=" << std::endl; |
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837 octave_print_internal (std::cout, bb, 0); |
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838 if (compz == 'V') |
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839 { |
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840 std::cout << std::endl << "ZZ=" << std::endl; |
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841 octave_print_internal (std::cout, ZZ, 0); |
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842 } |
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843 std::cout << std::endl; |
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844 std::cout << "alphar = " << std::endl; |
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845 octave_print_internal (std::cout, (Matrix) alphar, 0); |
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846 std::cout << std::endl << "alphai = " << std::endl; |
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847 octave_print_internal (std::cout, (Matrix) alphai, 0); |
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848 std::cout << std::endl << "beta = " << std::endl; |
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849 octave_print_internal (std::cout, (Matrix) betar, 0); |
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850 std::cout << std::endl; |
3185 | 851 #endif |
852 | |
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853 Array<octave_idx_type> ind (dim_vector (nn, 1)); |
3550 | 854 |
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855 F77_XFCN (dsubsp, DSUBSP, |
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856 (nn, nn, aa.fortran_vec (), bb.fortran_vec (), |
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857 ZZ.fortran_vec (), sort_test, eps, ndim, fail, |
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858 ind.fortran_vec ())); |
3185 | 859 |
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860 #if defined (DEBUG) |
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861 std::cout << "qz: back from dsubsp: aa=" << std::endl; |
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862 octave_print_internal (std::cout, aa, 0); |
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863 std::cout << std::endl << "bb=" << std::endl; |
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864 octave_print_internal (std::cout, bb, 0); |
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865 if (compz == 'V') |
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866 { |
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867 std::cout << std::endl << "ZZ=" << std::endl; |
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868 octave_print_internal (std::cout, ZZ, 0); |
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869 } |
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870 std::cout << std::endl; |
3185 | 871 #endif |
872 | |
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873 // Manually update alphar, alphai, betar. |
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874 static int jj; |
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875 |
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876 jj = 0; |
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877 while (jj < nn) |
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878 { |
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879 #if defined (DEBUG_EIG) |
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880 std::cout << "computing gen eig #" << jj << std::endl; |
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881 #endif |
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882 |
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883 // Number of zeros in this block. |
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884 static int zcnt; |
3185 | 885 |
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886 if (jj == (nn-1)) |
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887 zcnt = 1; |
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888 else if (aa(jj+1,jj) == 0) |
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889 zcnt = 1; |
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890 else zcnt = 2; |
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891 |
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892 if (zcnt == 1) |
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893 { |
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894 // Real zero. |
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895 #if defined (DEBUG_EIG) |
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896 std::cout << " single gen eig:" << std::endl; |
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897 std::cout << " alphar(" << jj << ") = " << aa(jj,jj) |
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898 << std::endl; |
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899 std::cout << " betar(" << jj << ") = " << bb(jj,jj) |
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900 << std::endl; |
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901 std::cout << " alphai(" << jj << ") = 0" << std::endl; |
3185 | 902 #endif |
903 | |
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904 alphar(jj) = aa(jj,jj); |
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905 alphai(jj) = 0; |
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906 betar(jj) = bb(jj,jj); |
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907 } |
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908 else |
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909 { |
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910 // Complex conjugate pair. |
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911 #if defined (DEBUG_EIG) |
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912 std::cout << "qz: calling dlag2:" << std::endl; |
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913 std::cout << "safmin=" |
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914 << setiosflags (std::ios::scientific) |
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915 << safmin << std::endl; |
3185 | 916 |
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917 for (int idr = jj; idr <= jj+1; idr++) |
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918 { |
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919 for (int idc = jj; idc <= jj+1; idc++) |
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920 { |
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921 std::cout << "aa(" << idr << "," << idc << ")=" |
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922 << aa(idr,idc) << std::endl; |
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923 std::cout << "bb(" << idr << "," << idc << ")=" |
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924 << bb(idr,idc) << std::endl; |
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925 } |
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926 } |
3185 | 927 #endif |
928 | |
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929 // FIXME: probably should be using |
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930 // fortran_vec instead of &aa(jj,jj) here. |
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931 |
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932 double scale1, scale2, wr1, wr2, wi; |
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933 const double *aa_ptr = aa.data () + jj * nn + jj; |
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934 const double *bb_ptr = bb.data () + jj * nn + jj; |
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935 F77_XFCN (dlag2, DLAG2, |
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936 (aa_ptr, nn, bb_ptr, nn, safmin, |
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937 scale1, scale2, wr1, wr2, wi)); |
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938 |
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939 #if defined (DEBUG_EIG) |
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940 std::cout << "dlag2 returns: scale1=" << scale1 |
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941 << "\tscale2=" << scale2 << std::endl |
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|
942 << "\twr1=" << wr1 << "\twr2=" << wr2 |
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943 << "\twi=" << wi << std::endl; |
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944 #endif |
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|
945 |
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946 // Just to be safe, check if it's a real pair. |
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947 if (wi == 0) |
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948 { |
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949 alphar(jj) = wr1; |
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950 alphai(jj) = 0; |
20831
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951 betar(jj) = scale1; |
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952 alphar(jj+1) = wr2; |
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953 alphai(jj+1) = 0; |
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954 betar(jj+1) = scale2; |
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955 } |
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956 else |
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957 { |
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958 alphar(jj) = alphar(jj+1) = wr1; |
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959 alphai(jj) = -(alphai(jj+1) = wi); |
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960 betar(jj) = betar(jj+1) = scale1; |
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961 } |
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962 } |
3185 | 963 |
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964 // Advance past this block. |
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965 jj += zcnt; |
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966 } |
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967 |
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968 #if defined (DEBUG_SORT) |
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969 std::cout << "qz: back from dsubsp: aa=" << std::endl; |
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970 octave_print_internal (std::cout, aa, 0); |
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971 std::cout << std::endl << "bb=" << std::endl; |
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972 octave_print_internal (std::cout, bb, 0); |
3185 | 973 |
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974 if (compz == 'V') |
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975 { |
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976 std::cout << std::endl << "ZZ=" << std::endl; |
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977 octave_print_internal (std::cout, ZZ, 0); |
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978 } |
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979 std::cout << std::endl << "qz: ndim=" << ndim << std::endl |
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980 << "fail=" << fail << std::endl; |
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981 std::cout << "alphar = " << std::endl; |
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982 octave_print_internal (std::cout, (Matrix) alphar, 0); |
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983 std::cout << std::endl << "alphai = " << std::endl; |
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984 octave_print_internal (std::cout, (Matrix) alphai, 0); |
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985 std::cout << std::endl << "beta = " << std::endl; |
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986 octave_print_internal (std::cout, (Matrix) betar, 0); |
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987 std::cout << std::endl; |
3185 | 988 #endif |
3183 | 989 } |
3185 | 990 |
10311 | 991 // Compute generalized eigenvalues? |
3183 | 992 ComplexColumnVector gev; |
3185 | 993 |
994 if (nargout < 2 || nargout == 7 || (nargin == 3 && nargout == 4)) | |
3183 | 995 { |
3185 | 996 if (complex_case) |
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997 { |
10311 | 998 int cnt = 0; |
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999 |
10311 | 1000 for (int ii = 0; ii < nn; ii++) |
1001 cnt++; | |
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1002 |
10311 | 1003 ComplexColumnVector tmp (cnt); |
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1004 |
10311 | 1005 cnt = 0; |
1006 for (int ii = 0; ii < nn; ii++) | |
1007 tmp(cnt++) = xalpha(ii) / xbeta(ii); | |
1008 | |
1009 gev = tmp; | |
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1010 } |
3185 | 1011 else |
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1012 { |
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1013 #if defined (DEBUG) |
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1014 std::cout << "qz: computing generalized eigenvalues" << std::endl; |
3185 | 1015 #endif |
3183 | 1016 |
10311 | 1017 // Return finite generalized eigenvalues. |
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1018 int cnt = 0; |
3185 | 1019 |
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1020 for (int ii = 0; ii < nn; ii++) |
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1021 if (betar(ii) != 0) |
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1022 cnt++; |
3185 | 1023 |
10311 | 1024 ComplexColumnVector tmp (cnt); |
3185 | 1025 |
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1026 cnt = 0; |
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1027 for (int ii = 0; ii < nn; ii++) |
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1028 if (betar(ii) != 0) |
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1029 tmp(cnt++) = Complex(alphar(ii), alphai(ii))/betar(ii); |
10311 | 1030 |
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1031 gev = tmp; |
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1032 } |
3183 | 1033 } |
1034 | |
10311 | 1035 // Right, left eigenvector matrices. |
3185 | 1036 if (nargout >= 5) |
3183 | 1037 { |
10311 | 1038 // Which side to compute? |
1039 char side = (nargout == 5 ? 'R' : 'B'); | |
1040 // Compute all of them and backtransform | |
1041 char howmny = 'B'; | |
1042 // Dummy pointer; select is not used. | |
1043 octave_idx_type *select = 0; | |
3185 | 1044 |
1045 if (complex_case) | |
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1046 { |
10311 | 1047 CVL = CQ; |
1048 CVR = CZ; | |
1049 ComplexRowVector cwork2 (2 * nn); | |
1050 RowVector rwork2 (8 * nn); | |
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1051 octave_idx_type m; |
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1052 |
10311 | 1053 F77_XFCN (ztgevc, ZTGEVC, |
1054 (F77_CONST_CHAR_ARG2 (&side, 1), | |
1055 F77_CONST_CHAR_ARG2 (&howmny, 1), | |
1056 select, nn, caa.fortran_vec (), nn, cbb.fortran_vec (), | |
1057 nn, CVL.fortran_vec (), nn, CVR.fortran_vec (), nn, nn, | |
1058 m, cwork2.fortran_vec (), rwork2.fortran_vec (), info | |
1059 F77_CHAR_ARG_LEN (1) | |
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1060 F77_CHAR_ARG_LEN (1))); |
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1061 } |
3185 | 1062 else |
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1063 { |
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1064 #if defined (DEBUG) |
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|
1065 std::cout << "qz: computing generalized eigenvectors" << std::endl; |
3185 | 1066 #endif |
1067 | |
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1068 VL = QQ; |
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1069 VR = ZZ; |
10311 | 1070 octave_idx_type m; |
1071 | |
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1072 F77_XFCN (dtgevc, DTGEVC, |
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1073 (F77_CONST_CHAR_ARG2 (&side, 1), |
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1074 F77_CONST_CHAR_ARG2 (&howmny, 1), |
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1075 select, nn, aa.fortran_vec (), nn, bb.fortran_vec (), |
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1076 nn, VL.fortran_vec (), nn, VR.fortran_vec (), nn, nn, |
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1077 m, work.fortran_vec (), info |
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1078 F77_CHAR_ARG_LEN (1) |
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1079 F77_CHAR_ARG_LEN (1))); |
3185 | 1080 |
10311 | 1081 // Now construct the complex form of VV, WW. |
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1082 int jj = 0; |
3185 | 1083 |
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1084 while (jj < nn) |
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1085 { |
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1086 OCTAVE_QUIT; |
4153 | 1087 |
10311 | 1088 // See if real or complex eigenvalue. |
1089 | |
1090 // Column increment; assume complex eigenvalue. | |
1091 int cinc = 2; | |
3185 | 1092 |
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1093 if (jj == (nn-1)) |
10311 | 1094 // Single column. |
1095 cinc = 1; | |
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1096 else if (aa(jj+1,jj) == 0) |
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1097 cinc = 1; |
3185 | 1098 |
10311 | 1099 // Now copy the eigenvector (s) to CVR, CVL. |
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1100 if (cinc == 1) |
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1101 { |
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1102 for (int ii = 0; ii < nn; ii++) |
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1103 CVR(ii,jj) = VR(ii,jj); |
3185 | 1104 |
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1105 if (side == 'B') |
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1106 for (int ii = 0; ii < nn; ii++) |
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1107 CVL(ii,jj) = VL(ii,jj); |
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1108 } |
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1109 else |
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1110 { |
10311 | 1111 // Double column; complex vector. |
3185 | 1112 |
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1113 for (int ii = 0; ii < nn; ii++) |
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1114 { |
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1115 CVR(ii,jj) = Complex (VR(ii,jj), VR(ii,jj+1)); |
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1116 CVR(ii,jj+1) = Complex (VR(ii,jj), -VR(ii,jj+1)); |
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1117 } |
3183 | 1118 |
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1119 if (side == 'B') |
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1120 for (int ii = 0; ii < nn; ii++) |
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1121 { |
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1122 CVL(ii,jj) = Complex (VL(ii,jj), VL(ii,jj+1)); |
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1123 CVL(ii,jj+1) = Complex (VL(ii,jj), -VL(ii,jj+1)); |
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1124 } |
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1125 } |
3185 | 1126 |
10311 | 1127 // Advance to next eigenvectors (if any). |
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1128 jj += cinc; |
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1129 } |
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1130 } |
3183 | 1131 } |
3185 | 1132 |
1133 switch (nargout) | |
1134 { | |
1135 case 7: | |
1136 retval(6) = gev; | |
1137 | |
10311 | 1138 case 6: |
1139 // Return eigenvectors. | |
3185 | 1140 retval(5) = CVL; |
1141 | |
10311 | 1142 case 5: |
1143 // Return eigenvectors. | |
3185 | 1144 retval(4) = CVR; |
1145 | |
1146 case 4: | |
1147 if (nargin == 3) | |
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1148 { |
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1149 #if defined (DEBUG) |
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1150 std::cout << "qz: sort: retval(3) = gev = " << std::endl; |
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1151 octave_print_internal (std::cout, gev); |
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1152 std::cout << std::endl; |
3185 | 1153 #endif |
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1154 retval(3) = gev; |
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1155 } |
3185 | 1156 else |
10311 | 1157 { |
1158 if (complex_case) | |
1159 retval(3) = CZ; | |
1160 else | |
1161 retval(3) = ZZ; | |
1162 } | |
3185 | 1163 |
1164 case 3: | |
1165 if (nargin == 3) | |
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1166 { |
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1167 if (complex_case) |
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1168 retval(2) = CZ; |
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1169 else |
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1170 retval(2) = ZZ; |
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1171 } |
3185 | 1172 else |
10311 | 1173 { |
1174 if (complex_case) | |
1175 retval(2) = CQ.hermitian (); | |
1176 else | |
1177 retval(2) = QQ.transpose (); | |
1178 } | |
1179 | |
3185 | 1180 case 2: |
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1181 { |
10311 | 1182 if (complex_case) |
1183 { | |
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1184 #if defined (DEBUG) |
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1185 std::cout << "qz: retval(1) = cbb = " << std::endl; |
10311 | 1186 octave_print_internal (std::cout, cbb, 0); |
1187 std::cout << std::endl << "qz: retval(0) = caa = " <<std::endl; | |
1188 octave_print_internal (std::cout, caa, 0); | |
1189 std::cout << std::endl; | |
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1190 #endif |
10311 | 1191 retval(1) = cbb; |
1192 retval(0) = caa; | |
1193 } | |
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1194 else |
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1195 { |
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1196 #if defined (DEBUG) |
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1197 std::cout << "qz: retval(1) = bb = " << std::endl; |
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1198 octave_print_internal (std::cout, bb, 0); |
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1199 std::cout << std::endl << "qz: retval(0) = aa = " <<std::endl; |
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1200 octave_print_internal (std::cout, aa, 0); |
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1201 std::cout << std::endl; |
3185 | 1202 #endif |
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1203 retval(1) = bb; |
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1204 retval(0) = aa; |
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1205 } |
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1206 } |
10311 | 1207 break; |
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1208 |
3185 | 1209 case 1: |
1210 case 0: | |
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1211 #if defined (DEBUG) |
3538 | 1212 std::cout << "qz: retval(0) = gev = " << gev << std::endl; |
3185 | 1213 #endif |
1214 retval(0) = gev; | |
1215 break; | |
1216 | |
1217 default: | |
11553
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1218 error ("qz: too many return arguments"); |
3185 | 1219 break; |
3183 | 1220 } |
1221 | |
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1222 #if defined (DEBUG) |
3538 | 1223 std::cout << "qz: exiting (at long last)" << std::endl; |
3185 | 1224 #endif |
3183 | 1225 |
1226 return retval; | |
1227 } | |
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1228 |
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1229 /* |
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1230 %!shared a, b, c |
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1231 %! a = [1 2; 0 3]; |
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1232 %! b = [1 0; 0 0]; |
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1233 %! c = [0 1; 0 0]; |
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1234 %!assert (qz (a,b), 1) |
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1235 %!assert (isempty (qz (a,c))) |
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1236 |
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1237 ## Exaple 7.7.3 in Golub & Van Loan |
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1238 %!test |
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1239 %! a = [ 10 1 2; |
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1240 %! 1 2 -1; |
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1241 %! 1 1 2]; |
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1242 %! b = reshape (1:9,3,3); |
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1243 %! [aa, bb, q, z, v, w, lambda] = qz (a, b); |
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1244 %! sz = length (lambda); |
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1245 %! observed = (b * v * diag ([lambda;0])) (:, 1:sz); |
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1246 %! assert ((a*v)(:, 1:sz), observed, norm (observed) * 1e-14); |
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1247 %! observed = (diag ([lambda;0]) * w' * b) (1:sz, :); |
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1248 %! assert ((w'*a)(1:sz, :) , observed, norm (observed) * 1e-13); |
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1249 %! assert (q * a * z, aa, norm (aa) * 1e-14); |
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1250 %! assert (q * b * z, bb, norm (bb) * 1e-14); |
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1251 |
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1252 %!test |
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1253 %! A = [0, 0, -1, 0; 1, 0, 0, 0; -1, 0, -2, -1; 0, -1, 1, 0]; |
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1254 %! B = [0, 0, 0, 0; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]; |
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1255 %! [AA, BB, Q, Z1] = qz (A, B); |
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1256 %! [AA, BB, Z2] = qz (A, B, '-'); |
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1257 %! assert (Z1, Z2); |
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1258 */ |