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1 /* |
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2 |
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3 Copyright (C) 1998 A. S. Hodel |
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4 |
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5 This file is part of Octave. |
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6 |
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7 Octave is free software; you can redistribute it and/or modify it |
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8 under the terms of the GNU General Public License as published by the |
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9 Free Software Foundation; either version 3 of the License, or (at your |
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10 option) any later version. |
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11 |
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12 Octave is distributed in the hope that it will be useful, but WITHOUT |
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13 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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15 for more details. |
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16 |
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17 You should have received a copy of the GNU General Public License |
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18 along with Octave; see the file COPYING. If not, see |
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19 <http://www.gnu.org/licenses/>. |
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20 |
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21 */ |
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22 |
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23 // Generalized eigenvalue balancing via LAPACK |
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24 |
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25 // Author: A. S. Hodel <scotte@eng.auburn.edu> |
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26 |
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27 #undef DEBUG |
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28 #undef DEBUG_SORT |
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29 #undef DEBUG_EIG |
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30 |
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31 #include "config.h" |
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32 |
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33 #include <cfloat> |
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34 #include <cmath> |
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35 |
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36 #include <iostream> |
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37 #include <iomanip> |
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38 |
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39 #include "CmplxQRP.h" |
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40 #include "dbleQR.h" |
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41 #include "f77-fcn.h" |
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42 #include "quit.h" |
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43 |
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44 #include "defun-dld.h" |
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45 #include "error.h" |
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46 #include "gripes.h" |
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47 #include "oct-obj.h" |
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48 #include "oct-map.h" |
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49 #include "ov.h" |
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50 #include "pager.h" |
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51 #if defined (DEBUG) || defined (DEBUG_SORT) |
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52 #include "pr-output.h" |
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53 #endif |
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54 #include "symtab.h" |
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55 #include "utils.h" |
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56 #include "variables.h" |
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57 |
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58 typedef octave_idx_type (*sort_function) (const octave_idx_type& LSIZE, const double& ALPHA, |
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59 const double& BETA, const double& S, |
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60 const double& P); |
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61 |
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62 extern "C" |
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63 { |
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64 F77_RET_T |
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65 F77_FUNC (dggbal, DGGBAL) (F77_CONST_CHAR_ARG_DECL, |
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66 const octave_idx_type& N, double* A, const octave_idx_type& LDA, |
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67 double* B, const octave_idx_type& LDB, octave_idx_type& ILO, |
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68 octave_idx_type& IHI, double* LSCALE, double* RSCALE, |
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69 double* WORK, octave_idx_type& INFO |
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70 F77_CHAR_ARG_LEN_DECL); |
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71 |
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72 F77_RET_T |
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73 F77_FUNC (dggbak, DGGBAK) (F77_CONST_CHAR_ARG_DECL, |
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74 F77_CONST_CHAR_ARG_DECL, |
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75 const octave_idx_type& N, const octave_idx_type& ILO, |
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76 const octave_idx_type& IHI, const double* LSCALE, |
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77 const double* RSCALE, octave_idx_type& M, double* V, |
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78 const octave_idx_type& LDV, octave_idx_type& INFO |
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79 F77_CHAR_ARG_LEN_DECL |
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80 F77_CHAR_ARG_LEN_DECL); |
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81 |
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82 F77_RET_T |
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83 F77_FUNC (dgghrd, DGGHRD) (F77_CONST_CHAR_ARG_DECL, |
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84 F77_CONST_CHAR_ARG_DECL, |
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85 const octave_idx_type& N, const octave_idx_type& ILO, |
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86 const octave_idx_type& IHI, double* A, |
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87 const octave_idx_type& LDA, double* B, |
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88 const octave_idx_type& LDB, double* Q, |
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89 const octave_idx_type& LDQ, double* Z, |
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90 const octave_idx_type& LDZ, octave_idx_type& INFO |
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91 F77_CHAR_ARG_LEN_DECL |
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92 F77_CHAR_ARG_LEN_DECL); |
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93 |
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94 F77_RET_T |
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95 F77_FUNC (dhgeqz, DHGEQZ) (F77_CONST_CHAR_ARG_DECL, |
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96 F77_CONST_CHAR_ARG_DECL, |
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97 F77_CONST_CHAR_ARG_DECL, |
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98 const octave_idx_type& N, const octave_idx_type& ILO, const octave_idx_type& IHI, |
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99 double* A, const octave_idx_type& LDA, double* B, |
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100 const octave_idx_type& LDB, double* ALPHAR, |
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101 double* ALPHAI, double* BETA, double* Q, |
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102 const octave_idx_type& LDQ, double* Z, |
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103 const octave_idx_type& LDZ, double* WORK, |
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104 const octave_idx_type& LWORK, octave_idx_type& INFO |
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105 F77_CHAR_ARG_LEN_DECL |
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106 F77_CHAR_ARG_LEN_DECL |
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107 F77_CHAR_ARG_LEN_DECL); |
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108 |
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109 F77_RET_T |
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110 F77_FUNC (dlag2, DLAG2) (const double* A, const octave_idx_type& LDA, const double* B, |
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111 const octave_idx_type& LDB, const double& SAFMIN, |
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112 double& SCALE1, double& SCALE2, |
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113 double& WR1, double& WR2, double& WI); |
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114 |
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115 // Van Dooren's code (netlib.org: toms/590) for reordering |
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116 // GEP. Only processes Z, not Q. |
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117 F77_RET_T |
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118 F77_FUNC (dsubsp, DSUBSP) (const octave_idx_type& NMAX, const octave_idx_type& N, double* A, |
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119 double* B, double* Z, sort_function, |
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120 const double& EPS, octave_idx_type& NDIM, octave_idx_type& FAIL, |
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121 octave_idx_type* IND); |
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122 |
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123 // documentation for DTGEVC incorrectly states that VR, VL are |
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124 // complex*16; they are declared in DTGEVC as double precision |
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125 // (probably a cut and paste problem fro ZTGEVC) |
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126 F77_RET_T |
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127 F77_FUNC (dtgevc, DTGEVC) (F77_CONST_CHAR_ARG_DECL, |
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128 F77_CONST_CHAR_ARG_DECL, |
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129 octave_idx_type* SELECT, const octave_idx_type& N, double* A, |
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130 const octave_idx_type& LDA, double* B, |
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131 const octave_idx_type& LDB, double* VL, |
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132 const octave_idx_type& LDVL, double* VR, |
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133 const octave_idx_type& LDVR, const octave_idx_type& MM, |
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134 octave_idx_type& M, double* WORK, 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 |
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138 F77_RET_T |
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139 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG_DECL, |
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140 double& retval |
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141 F77_CHAR_ARG_LEN_DECL); |
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142 |
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143 F77_RET_T |
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144 F77_FUNC (xdlange, XDLANGE) (F77_CONST_CHAR_ARG_DECL, |
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145 const octave_idx_type&, const octave_idx_type&, const double*, |
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146 const octave_idx_type&, double*, double& |
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147 F77_CHAR_ARG_LEN_DECL); |
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148 } |
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149 |
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150 // fcrhp, fin, fout, folhp: |
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151 // routines for ordering of generalized eigenvalues |
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152 // return 1 if test is passed, 0 otherwise |
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153 // fin: |lambda| < 1 |
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154 // fout: |lambda| >= 1 |
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155 // fcrhp: real(lambda) >= 0 |
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156 // folhp: real(lambda) < 0 |
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157 |
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158 static octave_idx_type |
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159 fcrhp (const octave_idx_type& lsize, const double& alpha, |
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160 const double& beta, const double& s, const double&) |
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161 { |
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162 if (lsize == 1) |
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163 return (alpha*beta >= 0 ? 1 : -1); |
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164 else |
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165 return (s >= 0 ? 1 : -1); |
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166 } |
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167 |
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168 static octave_idx_type |
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169 fin (const octave_idx_type& lsize, const double& alpha, |
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170 const double& beta, const double&, const double& p) |
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171 { |
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172 octave_idx_type retval; |
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173 |
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174 if (lsize == 1) |
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175 retval = (fabs (alpha) < fabs (beta) ? 1 : -1); |
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176 else |
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177 retval = (fabs (p) < 1 ? 1 : -1); |
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178 |
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179 #ifdef DEBUG |
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180 std::cout << "qz: fin: retval=" << retval << std::endl; |
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181 #endif |
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182 |
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183 return retval; |
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184 } |
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185 |
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186 static octave_idx_type |
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187 folhp (const octave_idx_type& lsize, const double& alpha, |
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188 const double& beta, const double& s, const double&) |
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189 { |
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190 if (lsize == 1) |
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191 return (alpha*beta < 0 ? 1 : -1); |
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192 else |
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193 return (s < 0 ? 1 : -1); |
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194 } |
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195 |
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196 static octave_idx_type |
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197 fout (const octave_idx_type& lsize, const double& alpha, |
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198 const double& beta, const double&, const double& p) |
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199 { |
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200 if (lsize == 1) |
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201 return (fabs (alpha) >= fabs (beta) ? 1 : -1); |
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202 else |
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203 return (fabs (p) >= 1 ? 1 : -1); |
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204 } |
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205 |
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206 DEFUN_DLD (qz, args, nargout, |
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207 "-*- texinfo -*-\n\ |
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208 @deftypefn {Loadable Function} {@var{lambda} =} qz (@var{a}, @var{b})\n\ |
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209 Generalized eigenvalue problem @math{A x = s B x},\n\ |
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210 @var{QZ} decomposition. There are three ways to call this function:\n\ |
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211 @enumerate\n\ |
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212 @item @code{lambda = qz(A,B)}\n\ |
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213 \n\ |
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214 Computes the generalized eigenvalues\n\ |
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215 @iftex\n\ |
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216 @tex\n\ |
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217 $\\lambda$\n\ |
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218 @end tex\n\ |
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219 @end iftex\n\ |
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220 @ifinfo\n\ |
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221 @var{lambda}\n\ |
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222 @end ifinfo\n\ |
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223 of @math{(A - s B)}.\n\ |
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224 @item @code{[AA, BB, Q, Z, V, W, lambda] = qz (A, B)}\n\ |
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225 \n\ |
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226 Computes qz decomposition, generalized eigenvectors, and \n\ |
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227 generalized eigenvalues of @math{(A - sB)}\n\ |
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228 @iftex\n\ |
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229 @tex\n\ |
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230 $$ AV = BV{ \\rm diag }(\\lambda) $$\n\ |
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231 $$ W^T A = { \\rm diag }(\\lambda)W^T B $$\n\ |
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232 $$ AA = Q^T AZ, BB = Q^T BZ $$\n\ |
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233 @end tex\n\ |
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234 @end iftex\n\ |
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235 @ifinfo\n\ |
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236 @example\n\ |
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237 @group\n\ |
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238 \n\ |
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239 A*V = B*V*diag(lambda)\n\ |
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240 W'*A = diag(lambda)*W'*B\n\ |
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241 AA = Q'*A*Z, BB = Q'*B*Z\n\ |
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242 \n\ |
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243 @end group\n\ |
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244 @end example\n\ |
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245 @end ifinfo\n\ |
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246 with @var{Q} and @var{Z} orthogonal (unitary)= @var{I}\n\ |
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247 \n\ |
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248 @item @code{[AA,BB,Z@{, lambda@}] = qz(A,B,opt)}\n\ |
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249 \n\ |
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250 As in form [2], but allows ordering of generalized eigenpairs\n\ |
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251 for (e.g.) solution of discrete time algebraic Riccati equations.\n\ |
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252 Form 3 is not available for complex matrices, and does not compute\n\ |
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253 the generalized eigenvectors @var{V}, @var{W}, nor the orthogonal matrix @var{Q}.\n\ |
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254 @table @var\n\ |
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255 @item opt\n\ |
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256 for ordering eigenvalues of the GEP pencil. The leading block\n\ |
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257 of the revised pencil contains all eigenvalues that satisfy:\n\ |
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258 @table @code\n\ |
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259 @item \"N\"\n\ |
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260 = unordered (default) \n\ |
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261 \n\ |
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262 @item \"S\"\n\ |
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263 = small: leading block has all |lambda| <=1 \n\ |
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264 \n\ |
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265 @item \"B\"\n\ |
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266 = big: leading block has all |lambda| >= 1 \n\ |
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267 \n\ |
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268 @item \"-\"\n\ |
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269 = negative real part: leading block has all eigenvalues\n\ |
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270 in the open left half-plane\n\ |
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271 \n\ |
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272 @item \"+\"\n\ |
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273 = nonnegative real part: leading block has all eigenvalues\n\ |
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274 in the closed right half-plane\n\ |
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275 @end table\n\ |
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276 @end table\n\ |
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277 @end enumerate\n\ |
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278 \n\ |
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279 Note: qz performs permutation balancing, but not scaling (see balance).\n\ |
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280 Order of output arguments was selected for compatibility with MATLAB\n\ |
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281 \n\ |
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282 @seealso{balance, dare, eig, schur}\n\ |
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283 @end deftypefn") |
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284 { |
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285 octave_value_list retval; |
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286 int nargin = args.length (); |
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287 |
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288 #ifdef DEBUG |
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289 std::cout << "qz: nargin = " << nargin << ", nargout = " << nargout << std::endl; |
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290 #endif |
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291 |
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292 if (nargin < 2 || nargin > 3 || nargout > 7) |
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293 { |
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294 print_usage (); |
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295 return retval; |
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296 } |
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297 else if (nargin == 3 && (nargout < 3 || nargout > 4)) |
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298 { |
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299 error ("qz: invalid number of output arguments for form [3] call"); |
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300 return retval; |
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301 } |
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302 |
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303 #ifdef DEBUG |
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304 std::cout << "qz: determine ordering option" << std::endl; |
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305 #endif |
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306 |
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307 // Determine ordering option |
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308 std::string ord_job; |
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309 static double safmin; |
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310 |
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311 if (nargin == 2) |
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312 ord_job = "N"; |
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313 else if (!args(2).is_string ()) |
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314 { |
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315 error ("qz: argument 3 must be a string"); |
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316 return retval; |
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317 } |
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318 else |
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319 { |
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320 ord_job = args(2).string_value (); |
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321 |
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322 if (ord_job[0] != 'N' |
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323 && ord_job[0] != 'S' |
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324 && ord_job[0] != 'B' |
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325 && ord_job[0] != '+' |
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326 && ord_job[0] != '-') |
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327 { |
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328 error ("qz: invalid order option"); |
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329 return retval; |
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330 } |
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331 |
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332 // overflow constant required by dlag2 |
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333 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG2 ("S", 1), |
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334 safmin |
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335 F77_CHAR_ARG_LEN (1)); |
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336 |
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337 #ifdef DEBUG_EIG |
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338 std::cout << "qz: initial value of safmin=" << setiosflags (std::ios::scientific) |
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339 << safmin << std::endl; |
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340 #endif |
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341 |
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342 // some machines (e.g., DEC alpha) get safmin = 0; |
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343 // for these, use eps instead to avoid problems in dlag2 |
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344 if (safmin == 0) |
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345 { |
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346 #ifdef DEBUG_EIG |
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347 std::cout << "qz: DANGER WILL ROBINSON: safmin is 0!" << std::endl; |
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348 #endif |
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349 |
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350 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG2 ("E", 1), |
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351 safmin |
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352 F77_CHAR_ARG_LEN (1)); |
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353 |
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354 #ifdef DEBUG_EIG |
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355 std::cout << "qz: safmin set to " << setiosflags (std::ios::scientific) |
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356 << safmin << std::endl; |
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357 #endif |
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358 } |
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359 } |
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360 |
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361 #ifdef DEBUG |
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362 std::cout << "qz: check argument 1" << std::endl; |
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363 #endif |
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364 |
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365 // Argument 1: check if it's o.k. dimensioned |
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366 octave_idx_type nn = args(0).rows (); |
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367 |
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368 #ifdef DEBUG |
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369 std::cout << "argument 1 dimensions: (" << nn << "," << args(0).columns () << ")" |
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370 << std::endl; |
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371 #endif |
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372 |
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373 int arg_is_empty = empty_arg ("qz", nn, args(0).columns ()); |
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374 |
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375 if (arg_is_empty < 0) |
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376 { |
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377 gripe_empty_arg ("qz: parameter 1", 0); |
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378 return retval; |
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379 } |
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380 else if (arg_is_empty > 0) |
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381 { |
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382 gripe_empty_arg ("qz: parameter 1; continuing", 0); |
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383 return octave_value_list (2, Matrix ()); |
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384 } |
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385 else if (args(0).columns () != nn) |
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386 { |
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387 gripe_square_matrix_required ("qz"); |
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388 return retval; |
|
|
389 } |
|
3183
|
390 |
|
|
391 // Argument 1: dimensions look good; get the value |
|
|
392 Matrix aa; |
|
|
393 ComplexMatrix caa; |
|
3185
|
394 |
|
|
395 if (args(0).is_complex_type ()) |
|
3183
|
396 caa = args(0).complex_matrix_value (); |
|
3185
|
397 else |
|
3183
|
398 aa = args(0).matrix_value (); |
|
3185
|
399 |
|
|
400 if (error_state) |
|
3183
|
401 return retval; |
|
|
402 |
|
3185
|
403 #ifdef DEBUG |
|
3538
|
404 std::cout << "qz: check argument 2" << std::endl; |
|
3185
|
405 #endif |
|
3183
|
406 |
|
|
407 // Extract argument 2 (bb, or cbb if complex) |
|
3185
|
408 if ((nn != args(1).columns ()) || (nn != args(1).rows ())) |
|
|
409 { |
|
|
410 gripe_nonconformant (); |
|
|
411 return retval; |
|
|
412 } |
|
|
413 |
|
3183
|
414 Matrix bb; |
|
|
415 ComplexMatrix cbb; |
|
3185
|
416 |
|
|
417 if (args(1).is_complex_type ()) |
|
3183
|
418 cbb = args(1).complex_matrix_value (); |
|
|
419 else |
|
|
420 bb = args(1).matrix_value (); |
|
3185
|
421 |
|
|
422 if (error_state) |
|
3183
|
423 return retval; |
|
|
424 |
|
|
425 // Both matrices loaded, now let's check what kind of arithmetic: |
|
|
426 //declared static to avoid compiler warnings about long jumps, vforks. |
|
3185
|
427 |
|
|
428 static int complex_case |
|
|
429 = (args(0).is_complex_type () || args(1).is_complex_type ()); |
|
3183
|
430 |
|
3185
|
431 if (nargin == 3 && complex_case) |
|
|
432 { |
|
|
433 error ("qz: cannot re-order complex qz decomposition."); |
|
|
434 return retval; |
|
|
435 } |
|
3183
|
436 |
|
|
437 // first, declare variables used in both the real and complex case |
|
|
438 Matrix QQ(nn,nn), ZZ(nn,nn), VR(nn,nn), VL(nn,nn); |
|
|
439 RowVector alphar(nn), alphai(nn), betar(nn); |
|
|
440 |
|
3185
|
441 ComplexMatrix CQ(nn,nn), CZ(nn,nn), CVR(nn,nn), CVL(nn,nn); |
|
5275
|
442 octave_idx_type ilo, ihi, info; |
|
3185
|
443 char compq = (nargout >= 3 ? 'V' : 'N'); |
|
|
444 char compz = (nargout >= 4 ? 'V' : 'N'); |
|
3183
|
445 |
|
3185
|
446 // initialize Q, Z to identity if we need either of them |
|
|
447 if (compq == 'V' || compz == 'V') |
|
5275
|
448 for (octave_idx_type ii = 0; ii < nn; ii++) |
|
|
449 for (octave_idx_type jj = 0; jj < nn; jj++) |
|
4153
|
450 { |
|
|
451 OCTAVE_QUIT; |
|
|
452 QQ(ii,jj) = ZZ(ii,jj) = (ii == jj ? 1.0 : 0.0); |
|
|
453 } |
|
3183
|
454 |
|
3185
|
455 // always perform permutation balancing |
|
4552
|
456 const char bal_job = 'P'; |
|
3183
|
457 RowVector lscale(nn), rscale(nn), work(6*nn); |
|
|
458 |
|
3185
|
459 if (complex_case) |
|
|
460 { |
|
|
461 error ("Complex case not implemented yet"); |
|
|
462 return retval; |
|
|
463 } |
|
3183
|
464 else |
|
3185
|
465 { |
|
|
466 #ifdef DEBUG |
|
|
467 if (compq == 'V') |
|
3538
|
468 std::cout << "qz: performing balancing; QQ=" << std::endl << QQ << std::endl; |
|
3185
|
469 #endif |
|
3183
|
470 |
|
3185
|
471 F77_XFCN (dggbal, DGGBAL, |
|
4552
|
472 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
|
|
473 nn, aa.fortran_vec (), nn, bb.fortran_vec (), |
|
|
474 nn, ilo, ihi, lscale.fortran_vec (), |
|
|
475 rscale.fortran_vec (), work.fortran_vec (), info |
|
|
476 F77_CHAR_ARG_LEN (1))); |
|
3185
|
477 |
|
|
478 if (f77_exception_encountered) |
|
|
479 { |
|
|
480 error ("unrecoverable error in qz (bal)"); |
|
|
481 return retval; |
|
|
482 } |
|
|
483 } |
|
3183
|
484 |
|
|
485 // Since we just want the balancing matrices, we can use dggbal |
|
|
486 // for both the real and complex cases; |
|
|
487 // left first |
|
3185
|
488 |
|
|
489 if (compq == 'V') |
|
|
490 { |
|
|
491 F77_XFCN (dggbak, DGGBAK, |
|
4552
|
492 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
|
|
493 F77_CONST_CHAR_ARG2 ("L", 1), |
|
4566
|
494 nn, ilo, ihi, lscale.data (), rscale.data (), |
|
|
495 nn, QQ.fortran_vec (), nn, info |
|
4552
|
496 F77_CHAR_ARG_LEN (1) |
|
|
497 F77_CHAR_ARG_LEN (1))); |
|
3183
|
498 |
|
3185
|
499 #ifdef DEBUG |
|
|
500 if (compq == 'V') |
|
3538
|
501 std::cout << "qz: balancing done; QQ=" << std::endl << QQ << std::endl; |
|
3185
|
502 #endif |
|
3183
|
503 |
|
3185
|
504 if (f77_exception_encountered) |
|
|
505 { |
|
|
506 error ("unrecoverable error in qz (bal-L)"); |
|
|
507 return retval; |
|
|
508 } |
|
3183
|
509 } |
|
|
510 |
|
|
511 // then right |
|
3185
|
512 if (compz == 'V') |
|
|
513 { |
|
4552
|
514 F77_XFCN (dggbak, DGGBAK, |
|
|
515 (F77_CONST_CHAR_ARG2 (&bal_job, 1), |
|
|
516 F77_CONST_CHAR_ARG2 ("R", 1), |
|
4566
|
517 nn, ilo, ihi, lscale.data (), rscale.data (), |
|
|
518 nn, ZZ.fortran_vec (), nn, info |
|
4552
|
519 F77_CHAR_ARG_LEN (1) |
|
|
520 F77_CHAR_ARG_LEN (1))); |
|
3183
|
521 |
|
3185
|
522 #ifdef DEBUG |
|
|
523 if (compz == 'V') |
|
3538
|
524 std::cout << "qz: balancing done; ZZ=" << std::endl << ZZ << std::endl; |
|
3185
|
525 #endif |
|
3183
|
526 |
|
3185
|
527 if (f77_exception_encountered) |
|
|
528 { |
|
|
529 error ("unrecoverable error in qz (bal-R)"); |
|
|
530 return retval; |
|
|
531 } |
|
|
532 } |
|
3183
|
533 |
|
|
534 static char qz_job; |
|
|
535 qz_job = (nargout < 2 ? 'E' : 'S'); |
|
3185
|
536 |
|
3183
|
537 if (complex_case) |
|
3185
|
538 { |
|
|
539 // complex case |
|
|
540 if (args(0).is_real_type ()) |
|
3587
|
541 caa = ComplexMatrix (aa); |
|
3185
|
542 |
|
|
543 if (args(1).is_real_type ()) |
|
3587
|
544 cbb = ComplexMatrix (bb); |
|
3185
|
545 |
|
|
546 if (compq == 'V') |
|
3587
|
547 CQ = ComplexMatrix (QQ); |
|
3185
|
548 |
|
|
549 if (compz == 'V') |
|
3587
|
550 CZ = ComplexMatrix (ZZ); |
|
3185
|
551 |
|
|
552 error ("complex case not done yet"); |
|
|
553 return retval; |
|
|
554 } |
|
3183
|
555 else // real matrices case |
|
3185
|
556 { |
|
|
557 #ifdef DEBUG |
|
3538
|
558 std::cout << "qz: peforming qr decomposition of bb" << std::endl; |
|
3185
|
559 #endif |
|
3183
|
560 |
|
3185
|
561 // compute the QR factorization of bb |
|
|
562 QR bqr (bb); |
|
|
563 |
|
|
564 #ifdef DEBUG |
|
3538
|
565 std::cout << "qz: qr (bb) done; now peforming qz decomposition" << std::endl; |
|
3185
|
566 #endif |
|
3183
|
567 |
|
3185
|
568 bb = bqr.R (); |
|
|
569 |
|
|
570 #ifdef DEBUG |
|
3538
|
571 std::cout << "qz: extracted bb" << std::endl; |
|
3185
|
572 #endif |
|
3183
|
573 |
|
3185
|
574 aa = (bqr.Q ()).transpose ()*aa; |
|
|
575 |
|
|
576 #ifdef DEBUG |
|
3538
|
577 std::cout << "qz: updated aa " << std::endl; |
|
|
578 std::cout << "bqr.Q () = " << std::endl << bqr.Q () << std::endl; |
|
3183
|
579 |
|
3185
|
580 if (compq == 'V') |
|
3538
|
581 std::cout << "QQ =" << QQ << std::endl; |
|
3185
|
582 #endif |
|
3183
|
583 |
|
3185
|
584 if (compq == 'V') |
|
|
585 QQ = QQ*bqr.Q (); |
|
3183
|
586 |
|
3185
|
587 #ifdef DEBUG |
|
3538
|
588 std::cout << "qz: precursors done..." << std::endl; |
|
3185
|
589 #endif |
|
3183
|
590 |
|
3185
|
591 #ifdef DEBUG |
|
3538
|
592 std::cout << "qz: compq = " << compq << ", compz = " << compz << std::endl; |
|
3185
|
593 #endif |
|
3183
|
594 |
|
3185
|
595 // reduce to generalized hessenberg form |
|
|
596 F77_XFCN (dgghrd, DGGHRD, |
|
4552
|
597 (F77_CONST_CHAR_ARG2 (&compq, 1), |
|
|
598 F77_CONST_CHAR_ARG2 (&compz, 1), |
|
|
599 nn, ilo, ihi, aa.fortran_vec (), |
|
|
600 nn, bb.fortran_vec (), nn, QQ.fortran_vec (), nn, |
|
|
601 ZZ.fortran_vec (), nn, info |
|
|
602 F77_CHAR_ARG_LEN (1) |
|
|
603 F77_CHAR_ARG_LEN (1))); |
|
3183
|
604 |
|
3185
|
605 if (f77_exception_encountered) |
|
|
606 { |
|
|
607 error ("unrecoverable error in qz (dgghrd)"); |
|
|
608 return retval; |
|
|
609 } |
|
3183
|
610 |
|
3185
|
611 // check if just computing generalized eigenvalues or if we're |
|
|
612 // actually computing the decomposition |
|
3183
|
613 |
|
3185
|
614 // reduce to generalized Schur form |
|
|
615 F77_XFCN (dhgeqz, DHGEQZ, |
|
4552
|
616 (F77_CONST_CHAR_ARG2 (&qz_job, 1), |
|
|
617 F77_CONST_CHAR_ARG2 (&compq, 1), |
|
|
618 F77_CONST_CHAR_ARG2 (&compz, 1), |
|
|
619 nn, ilo, ihi, aa.fortran_vec (), nn, bb.fortran_vec (), |
|
|
620 nn, alphar.fortran_vec (), alphai.fortran_vec (), |
|
|
621 betar.fortran_vec (), QQ.fortran_vec (), nn, |
|
|
622 ZZ.fortran_vec (), nn, work.fortran_vec (), nn, info |
|
|
623 F77_CHAR_ARG_LEN (1) |
|
|
624 F77_CHAR_ARG_LEN (1) |
|
|
625 F77_CHAR_ARG_LEN (1))); |
|
3183
|
626 |
|
3185
|
627 if (f77_exception_encountered) |
|
|
628 { |
|
|
629 error ("unrecoverable error in qz (dhgeqz)"); |
|
|
630 return retval; |
|
|
631 } |
|
|
632 } |
|
3183
|
633 |
|
|
634 // order the QZ decomposition? |
|
3185
|
635 if (ord_job[0] != 'N') |
|
3183
|
636 { |
|
3185
|
637 if (complex_case) |
|
|
638 { |
|
|
639 // probably not needed, but better be safe |
|
|
640 error ("qz: cannot re-order complex qz decomposition."); |
|
|
641 return retval; |
|
|
642 } |
|
|
643 else |
|
|
644 { |
|
|
645 #ifdef DEBUG_SORT |
|
3538
|
646 std::cout << "qz: ordering eigenvalues: ord_job = " << ord_job[0] << std::endl; |
|
3185
|
647 #endif |
|
3183
|
648 |
|
3185
|
649 // declared static to avoid vfork/long jump compiler complaints |
|
|
650 static sort_function sort_test; |
|
|
651 sort_test = NULL; |
|
3183
|
652 |
|
3185
|
653 switch (ord_job[0]) |
|
|
654 { |
|
|
655 case 'S': |
|
|
656 sort_test = &fin; |
|
|
657 break; |
|
3183
|
658 |
|
3185
|
659 case 'B': |
|
|
660 sort_test = &fout; |
|
|
661 break; |
|
3183
|
662 |
|
3185
|
663 case '+': |
|
|
664 sort_test = &fcrhp; |
|
|
665 break; |
|
3183
|
666 |
|
3185
|
667 case '-': |
|
|
668 sort_test = &folhp; |
|
|
669 break; |
|
|
670 |
|
|
671 default: |
|
3427
|
672 // invalid order option (should never happen, since we |
|
3185
|
673 // checked the options at the top). |
|
|
674 panic_impossible (); |
|
|
675 break; |
|
3550
|
676 } |
|
3183
|
677 |
|
5275
|
678 octave_idx_type ndim, fail; |
|
3185
|
679 double inf_norm; |
|
|
680 |
|
|
681 F77_XFCN (xdlange, XDLANGE, |
|
4552
|
682 (F77_CONST_CHAR_ARG2 ("I", 1), |
|
4566
|
683 nn, nn, aa.data (), nn, work.fortran_vec (), inf_norm |
|
4552
|
684 F77_CHAR_ARG_LEN (1))); |
|
3185
|
685 |
|
|
686 double eps = DBL_EPSILON*inf_norm*nn; |
|
|
687 |
|
|
688 #ifdef DEBUG_SORT |
|
3538
|
689 std::cout << "qz: calling dsubsp: aa=" << std::endl; |
|
3531
|
690 octave_print_internal (std::cout, aa, 0); |
|
3538
|
691 std::cout << std::endl << "bb=" << std::endl; |
|
3531
|
692 octave_print_internal (std::cout, bb, 0); |
|
3185
|
693 if (compz == 'V') |
|
|
694 { |
|
3538
|
695 std::cout << std::endl << "ZZ=" << std::endl; |
|
3531
|
696 octave_print_internal (std::cout, ZZ, 0); |
|
3185
|
697 } |
|
3538
|
698 std::cout << std::endl; |
|
|
699 std::cout << "alphar = " << std::endl; |
|
3531
|
700 octave_print_internal (std::cout, (Matrix) alphar, 0); |
|
3538
|
701 std::cout << std::endl << "alphai = " << std::endl; |
|
3531
|
702 octave_print_internal (std::cout, (Matrix) alphai, 0); |
|
3538
|
703 std::cout << std::endl << "beta = " << std::endl; |
|
3531
|
704 octave_print_internal (std::cout, (Matrix) betar, 0); |
|
3538
|
705 std::cout << std::endl; |
|
3185
|
706 #endif |
|
|
707 |
|
5275
|
708 Array<octave_idx_type> ind (nn); |
|
3550
|
709 |
|
3185
|
710 F77_XFCN (dsubsp, DSUBSP, |
|
4552
|
711 (nn, nn, aa.fortran_vec (), bb.fortran_vec (), |
|
|
712 ZZ.fortran_vec (), sort_test, eps, ndim, fail, |
|
3550
|
713 ind.fortran_vec ())); |
|
3185
|
714 |
|
|
715 #ifdef DEBUG |
|
3538
|
716 std::cout << "qz: back from dsubsp: aa=" << std::endl; |
|
3531
|
717 octave_print_internal (std::cout, aa, 0); |
|
3538
|
718 std::cout << std::endl << "bb=" << std::endl; |
|
3531
|
719 octave_print_internal (std::cout, bb, 0); |
|
3185
|
720 if (compz == 'V') |
|
|
721 { |
|
3538
|
722 std::cout << std::endl << "ZZ=" << std::endl; |
|
3531
|
723 octave_print_internal (std::cout, ZZ, 0); |
|
3185
|
724 } |
|
3538
|
725 std::cout << std::endl; |
|
3185
|
726 #endif |
|
|
727 |
|
|
728 // manually update alphar, alphai, betar |
|
|
729 static int jj; |
|
|
730 |
|
|
731 jj=0; |
|
|
732 while (jj < nn) |
|
|
733 { |
|
|
734 #ifdef DEBUG_EIG |
|
3538
|
735 std::cout << "computing gen eig #" << jj << std::endl; |
|
3185
|
736 #endif |
|
|
737 |
|
|
738 static int zcnt; // number of zeros in this block |
|
|
739 |
|
|
740 if (jj == (nn-1)) |
|
|
741 zcnt = 1; |
|
|
742 else if (aa(jj+1,jj) == 0) |
|
|
743 zcnt = 1; |
|
|
744 else zcnt = 2; |
|
|
745 |
|
|
746 if (zcnt == 1) // real zero |
|
|
747 { |
|
|
748 #ifdef DEBUG_EIG |
|
3538
|
749 std::cout << " single gen eig:" << std::endl; |
|
|
750 std::cout << " alphar(" << jj << ") = " << aa(jj,jj) << std::endl; |
|
|
751 std::cout << " betar( " << jj << ") = " << bb(jj,jj) << std::endl; |
|
|
752 std::cout << " alphai(" << jj << ") = 0" << std::endl; |
|
3185
|
753 #endif |
|
|
754 |
|
|
755 alphar(jj) = aa(jj,jj); |
|
|
756 alphai(jj) = 0; |
|
|
757 betar(jj) = bb(jj,jj); |
|
|
758 } |
|
|
759 else |
|
|
760 { |
|
|
761 // complex conjugate pair |
|
|
762 #ifdef DEBUG_EIG |
|
3538
|
763 std::cout << "qz: calling dlag2:" << std::endl; |
|
3531
|
764 std::cout << "safmin=" |
|
3538
|
765 << setiosflags (std::ios::scientific) << safmin << std::endl; |
|
3185
|
766 |
|
|
767 for (int idr = jj; idr <= jj+1; idr++) |
|
|
768 { |
|
|
769 for (int idc = jj; idc <= jj+1; idc++) |
|
|
770 { |
|
3531
|
771 std::cout << "aa(" << idr << "," << idc << ")=" |
|
3538
|
772 << aa(idr,idc) << std::endl; |
|
3531
|
773 std::cout << "bb(" << idr << "," << idc << ")=" |
|
3538
|
774 << bb(idr,idc) << std::endl; |
|
3185
|
775 } |
|
|
776 } |
|
|
777 #endif |
|
|
778 |
|
5775
|
779 // FIXME -- probably should be using |
|
4566
|
780 // fortran_vec instead of &aa(jj,jj) here. |
|
|
781 |
|
3185
|
782 double scale1, scale2, wr1, wr2, wi; |
|
4629
|
783 const double *aa_ptr = aa.data () + jj*nn+jj; |
|
|
784 const double *bb_ptr = bb.data () + jj*nn+jj; |
|
3185
|
785 F77_XFCN (dlag2, DLAG2, |
|
4629
|
786 (aa_ptr, nn, bb_ptr, nn, safmin, |
|
3185
|
787 scale1, scale2, wr1, wr2, wi)); |
|
|
788 |
|
|
789 #ifdef DEBUG_EIG |
|
3531
|
790 std::cout << "dlag2 returns: scale1=" << scale1 |
|
3538
|
791 << "\tscale2=" << scale2 << std::endl |
|
3185
|
792 << "\twr1=" << wr1 << "\twr2=" << wr2 |
|
3538
|
793 << "\twi=" << wi << std::endl; |
|
3185
|
794 #endif |
|
|
795 |
|
|
796 // just to be safe, check if it's a real pair |
|
|
797 if (wi == 0) |
|
|
798 { |
|
|
799 alphar(jj) = wr1; |
|
|
800 alphai(jj) = 0; |
|
|
801 betar(jj) = scale1; |
|
|
802 alphar(jj+1) = wr2; |
|
|
803 alphai(jj+1) = 0; |
|
|
804 betar(jj+1) = scale2; |
|
|
805 } |
|
|
806 else |
|
|
807 { |
|
|
808 alphar(jj) = alphar(jj+1)=wr1; |
|
|
809 alphai(jj) = -(alphai(jj+1) = wi); |
|
|
810 betar(jj) = betar(jj+1) = scale1; |
|
|
811 } |
|
|
812 } |
|
|
813 |
|
|
814 // advance past this block |
|
|
815 jj += zcnt; |
|
|
816 } |
|
|
817 |
|
|
818 #ifdef DEBUG_SORT |
|
3538
|
819 std::cout << "qz: back from dsubsp: aa=" << std::endl; |
|
3531
|
820 octave_print_internal (std::cout, aa, 0); |
|
3538
|
821 std::cout << std::endl << "bb=" << std::endl; |
|
3531
|
822 octave_print_internal (std::cout, bb, 0); |
|
3185
|
823 |
|
|
824 if (compz == 'V') |
|
|
825 { |
|
3538
|
826 std::cout << std::endl << "ZZ=" << std::endl; |
|
3531
|
827 octave_print_internal (std::cout, ZZ, 0); |
|
3185
|
828 } |
|
3538
|
829 std::cout << std::endl << "qz: ndim=" << ndim << std::endl |
|
|
830 << "fail=" << fail << std::endl; |
|
|
831 std::cout << "alphar = " << std::endl; |
|
3531
|
832 octave_print_internal (std::cout, (Matrix) alphar, 0); |
|
3538
|
833 std::cout << std::endl << "alphai = " << std::endl; |
|
3531
|
834 octave_print_internal (std::cout, (Matrix) alphai, 0); |
|
3538
|
835 std::cout << std::endl << "beta = " << std::endl; |
|
3531
|
836 octave_print_internal (std::cout, (Matrix) betar, 0); |
|
3538
|
837 std::cout << std::endl; |
|
3185
|
838 #endif |
|
|
839 } |
|
3183
|
840 } |
|
3185
|
841 |
|
3183
|
842 // compute generalized eigenvalues? |
|
|
843 ComplexColumnVector gev; |
|
3185
|
844 |
|
|
845 if (nargout < 2 || nargout == 7 || (nargin == 3 && nargout == 4)) |
|
3183
|
846 { |
|
3185
|
847 if (complex_case) |
|
|
848 { |
|
|
849 error ("complex case not yet implemented"); |
|
|
850 return retval; |
|
|
851 } |
|
|
852 else |
|
|
853 { |
|
|
854 #ifdef DEBUG |
|
3538
|
855 std::cout << "qz: computing generalized eigenvalues" << std::endl; |
|
3185
|
856 #endif |
|
3183
|
857 |
|
3185
|
858 // return finite generalized eigenvalues |
|
|
859 int cnt = 0; |
|
|
860 |
|
|
861 for (int ii = 0; ii < nn; ii++) |
|
|
862 if (betar(ii) != 0) |
|
|
863 cnt++; |
|
|
864 |
|
|
865 ComplexColumnVector tmp(cnt); |
|
|
866 |
|
3671
|
867 cnt = 0; |
|
3185
|
868 for (int ii = 0; ii < nn; ii++) |
|
|
869 if (betar(ii) != 0) |
|
3671
|
870 tmp(cnt++) = Complex(alphar(ii), alphai(ii))/betar(ii); |
|
3185
|
871 gev = tmp; |
|
|
872 } |
|
3183
|
873 } |
|
|
874 |
|
|
875 // right, left eigenvector matrices |
|
3185
|
876 if (nargout >= 5) |
|
3183
|
877 { |
|
3185
|
878 char side = (nargout == 5 ? 'R' : 'B'); // which side to compute? |
|
|
879 char howmny = 'B'; // compute all of them and backtransform |
|
5275
|
880 octave_idx_type *select = NULL; // dummy pointer; select is not used. |
|
|
881 octave_idx_type m; |
|
3185
|
882 |
|
|
883 if (complex_case) |
|
|
884 { |
|
|
885 error ("complex type not yet implemented"); |
|
|
886 return retval; |
|
|
887 } |
|
|
888 else |
|
|
889 { |
|
|
890 #ifdef DEBUG |
|
3538
|
891 std::cout << "qz: computing generalized eigenvectors" << std::endl; |
|
3185
|
892 #endif |
|
|
893 |
|
|
894 VL = QQ; |
|
|
895 VR = ZZ; |
|
|
896 |
|
|
897 F77_XFCN (dtgevc, DTGEVC, |
|
4552
|
898 (F77_CONST_CHAR_ARG2 (&side, 1), |
|
|
899 F77_CONST_CHAR_ARG2 (&howmny, 1), |
|
|
900 select, nn, aa.fortran_vec (), nn, bb.fortran_vec (), |
|
|
901 nn, VL.fortran_vec (), nn, VR.fortran_vec (), nn, nn, |
|
|
902 m, work.fortran_vec (), info |
|
|
903 F77_CHAR_ARG_LEN (1) |
|
|
904 F77_CHAR_ARG_LEN (1))); |
|
3185
|
905 |
|
|
906 if (f77_exception_encountered) |
|
|
907 { |
|
|
908 error ("unrecoverable error in qz (dtgevc)"); |
|
|
909 return retval; |
|
|
910 } |
|
3183
|
911 |
|
3185
|
912 // now construct the complex form of VV, WW |
|
|
913 int jj = 0; |
|
|
914 |
|
|
915 while (jj < nn) |
|
|
916 { |
|
4153
|
917 OCTAVE_QUIT; |
|
|
918 |
|
3185
|
919 // see if real or complex eigenvalue |
|
|
920 int cinc = 2; // column increment; assume complex eigenvalue |
|
|
921 |
|
|
922 if (jj == (nn-1)) |
|
|
923 cinc = 1; // single column |
|
|
924 else if (aa(jj+1,jj) == 0) |
|
|
925 cinc = 1; |
|
|
926 |
|
|
927 // now copy the eigenvector (s) to CVR, CVL |
|
|
928 if (cinc == 1) |
|
|
929 { |
|
|
930 for (int ii = 0; ii < nn; ii++) |
|
|
931 CVR(ii,jj) = VR(ii,jj); |
|
|
932 |
|
|
933 if (side == 'B') |
|
|
934 for (int ii = 0; ii < nn; ii++) |
|
|
935 CVL(ii,jj) = VL(ii,jj); |
|
|
936 } |
|
|
937 else |
|
|
938 { |
|
|
939 // double column; complex vector |
|
|
940 |
|
|
941 for (int ii = 0; ii < nn; ii++) |
|
|
942 { |
|
|
943 CVR(ii,jj) = Complex (VR(ii,jj), VR(ii,jj+1)); |
|
|
944 CVR(ii,jj+1) = Complex (VR(ii,jj), -VR(ii,jj+1)); |
|
|
945 } |
|
3183
|
946 |
|
3185
|
947 if (side == 'B') |
|
|
948 for (int ii = 0; ii < nn; ii++) |
|
|
949 { |
|
|
950 CVL(ii,jj) = Complex (VL(ii,jj), VL(ii,jj+1)); |
|
|
951 CVL(ii,jj+1) = Complex (VL(ii,jj), -VL(ii,jj+1)); |
|
|
952 } |
|
|
953 } |
|
|
954 |
|
|
955 // advance to next eigenvectors (if any) |
|
|
956 jj += cinc; |
|
|
957 } |
|
|
958 } |
|
3183
|
959 } |
|
3185
|
960 |
|
|
961 switch (nargout) |
|
|
962 { |
|
|
963 case 7: |
|
|
964 retval(6) = gev; |
|
|
965 |
|
|
966 case 6: // return eigenvectors |
|
|
967 retval(5) = CVL; |
|
|
968 |
|
|
969 case 5: // return eigenvectors |
|
|
970 retval(4) = CVR; |
|
|
971 |
|
|
972 case 4: |
|
|
973 if (nargin == 3) |
|
|
974 { |
|
|
975 #ifdef DEBUG |
|
3538
|
976 std::cout << "qz: sort: retval(3) = gev = " << std::endl; |
|
3531
|
977 octave_print_internal (std::cout, gev); |
|
3538
|
978 std::cout << std::endl; |
|
3185
|
979 #endif |
|
|
980 retval(3) = gev; |
|
|
981 } |
|
|
982 else |
|
|
983 retval(3) = ZZ; |
|
|
984 |
|
|
985 case 3: |
|
|
986 if (nargin == 3) |
|
|
987 retval(2) = ZZ; |
|
|
988 else |
|
|
989 retval(2) = QQ; |
|
|
990 |
|
|
991 case 2: |
|
|
992 #ifdef DEBUG |
|
3538
|
993 std::cout << "qz: retval (1) = bb = " << std::endl; |
|
3531
|
994 octave_print_internal (std::cout, bb, 0); |
|
3538
|
995 std::cout << std::endl << "qz: retval(0) = aa = " <<std::endl; |
|
3531
|
996 octave_print_internal (std::cout, aa, 0); |
|
3538
|
997 std::cout << std::endl; |
|
3185
|
998 #endif |
|
|
999 retval(1) = bb; |
|
|
1000 retval(0) = aa; |
|
|
1001 break; |
|
|
1002 |
|
|
1003 case 1: |
|
|
1004 case 0: |
|
|
1005 #ifdef DEBUG |
|
3538
|
1006 std::cout << "qz: retval(0) = gev = " << gev << std::endl; |
|
3185
|
1007 #endif |
|
|
1008 retval(0) = gev; |
|
|
1009 break; |
|
|
1010 |
|
|
1011 default: |
|
|
1012 error ("qz: too many return arguments."); |
|
|
1013 break; |
|
3183
|
1014 } |
|
|
1015 |
|
3185
|
1016 #ifdef DEBUG |
|
3538
|
1017 std::cout << "qz: exiting (at long last)" << std::endl; |
|
3185
|
1018 #endif |
|
3183
|
1019 |
|
|
1020 return retval; |
|
|
1021 } |
|
|
1022 |
|
|
1023 /* |
|
|
1024 ;;; Local Variables: *** |
|
|
1025 ;;; mode: C++ *** |
|
|
1026 ;;; End: *** |
|
|
1027 */ |
