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3183
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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 2, or (at your option) any |
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10 later version. |
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11 |
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12 Octave is distributed in the hope that it will be useful, but WITHOUT |
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13 ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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14 FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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15 for more details. |
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16 |
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17 You should have received a copy of the GNU General Public License |
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18 along with Octave; see the file COPYING. If not, write to the Free |
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19 Software Foundation, 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. |
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20 |
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21 */ |
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22 |
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23 // Generalized eigenvalue balancing via LAPACK |
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24 // Written by A. S. Hodel <scotte@eng.auburn.edu> |
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25 |
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26 #undef DEBUG |
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27 #undef DEBUG_SORT |
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28 #undef DEBUG_EIG |
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29 |
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30 #include "config.h" |
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31 |
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32 #include <cfloat> |
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33 #include <iostream.h> |
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34 #include <iomanip.h> |
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35 #include <math.h> |
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36 #include <string.h> |
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37 #include <strstream.h> |
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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 "defun-dld.h" |
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42 #include "error.h" |
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43 #include "f77-fcn.h" |
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44 #include "gripes.h" |
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45 #include "help.h" |
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46 #include "oct-obj.h" |
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47 #include "oct-map.h" |
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48 #include "ov.h" |
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49 #include "pager.h" |
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50 #if defined(DEBUG) || defined(DEBUG_SORT) |
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51 #include "pr-output.h" |
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52 #endif |
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53 #include "symtab.h" |
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54 #include "utils.h" |
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55 #include "variables.h" |
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56 |
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57 typedef int (*sort_function) (const int& LSIZE, const double& ALPHA, |
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58 const double& BETA, const double& S, const double& P); |
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59 |
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60 extern "C" |
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61 { |
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62 int F77_FCN( dggbal, DGGBAL) (const char* JOB, const int& N, |
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63 double* A, const int& LDA, double* B, const int& LDB, |
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64 int& ILO, int & IHI, double* LSCALE, |
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65 double* RSCALE, double* WORK, int& INFO, long ); |
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66 |
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67 int F77_FCN( dggbak, DGGBAK) (const char* JOB, const char* SIDE, |
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68 const int& N, const int& ILO, const int& IHI, |
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69 double* LSCALE, double* RSCALE, int& M, |
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70 double* V, const int& LDV, int& INFO, long, long); |
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71 |
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72 int F77_FCN( dgghrd, DGGHRD) ( const char* COMPQ, const char* COMPZ, |
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73 const int& N, const int& ILO, const int& IHI, double* A, const int& LDA, |
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74 double* B, const int& LDB, double* Q, const int& LDQ, double* Z, |
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75 const int& LDZ, int& INFO, const long, const long); |
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76 |
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77 int F77_FCN( dhgeqz, DHGEQZ) ( const char* JOB, const char* COMPQ, |
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78 const char* COMPZ, const int& N, const int& ILO, const int& IHI, |
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79 double* A, const int& LDA, double* B, const int& LDB, |
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80 double* ALPHAR, double* ALPHAI, double* BETA, double* Q, |
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81 const int& LDQ, double* Z, const int& LDZ, double* WORK, |
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82 const int& LWORK, int& INFO, const long, const long, const long ); |
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83 |
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84 int F77_FCN( dlag2, DLAG2) ( double* A, const int& LDA, double* B, |
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85 const int& LDB, const double& SAFMIN, double& SCALE1, double& SCALE2, |
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86 double& WR1, double& WR2, double& WI ); |
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87 |
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88 // Van Dooren's code (netlib.org: toms/590) for reordering |
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89 // GEP. Only processes Z, not Q. |
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90 int F77_FCN( dsubsp, DSUBSP) ( const int& NMAX, const int& N, double* A, |
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91 double* B, double* Z, sort_function, const double& EPS, |
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92 int& NDIM, int& FAIL, int* IND); |
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93 |
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94 // documentation for DTGEVC incorrectly states that VR, VL are |
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95 // complex*16; they are declared in DTGEVC as double precision |
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96 // (probably a cut and paste problem fro ZTGEVC) |
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97 int F77_FCN( dtgevc, DTGEVC) ( const char* SIDE, const char* HOWMNY, |
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98 int* SELECT, const int& N, double* A, const int& LDA, double* B, |
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99 const int& LDB, double* VL, const int& LDVL, double* VR, |
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100 const int& LDVR, const int& MM, int& M, double* WORK, int& INFO, |
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101 long, long ); |
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102 |
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103 int F77_FCN ( xdlamch, XDLAMCH) (const char* cmach, double& retval, long); |
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104 int F77_FCN ( xdlange, XDLANGE) (const char*, const int&, |
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105 const int&, const double*, |
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106 const int&, double*, double&); |
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107 } |
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108 |
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109 // fcrhp, fin, fout, folhp: |
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110 // routines for ordering of generalized eigenvalues |
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111 // return 1 if test is passed, 0 otherwise |
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112 // fin: |lambda| < 1 |
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113 // fout: |lambda| >= 1 |
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114 // fcrhp: real(lambda) >= 0 |
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115 // folhp: real(lambda) < 0 |
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116 |
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117 static int fcrhp(const int& lsize, const double& alpha, |
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118 const double& beta, const double& s, const double& p) |
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119 { |
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120 if(lsize == 1) |
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121 return (alpha*beta >= 0 ? 1 : -1); |
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122 else |
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123 return (s >= 0 ? 1 : -1); |
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124 } |
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125 static int fin(const int& lsize, const double& alpha, |
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126 const double& beta, const double& s, const double& p) |
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127 { |
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128 int retval; |
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129 if(lsize == 1) |
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130 retval = (fabs(alpha) < fabs(beta) ? 1 : -1); |
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131 else |
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132 retval = (fabs(p) < 1 ? 1 : -1); |
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133 |
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134 #ifdef DEBUG |
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135 cout << "qz: fin: retval=" << retval << endl; |
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136 #endif |
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137 return retval; |
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138 } |
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139 static int folhp(const int& lsize, const double& alpha, |
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140 const double& beta, const double& s, const double& p) |
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141 { |
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142 if(lsize == 1) |
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143 return (alpha*beta < 0 ? 1 : -1); |
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144 else |
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145 return (s < 0 ? 1 : -1); |
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146 } |
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147 static int fout(const int& lsize, const double& alpha, |
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148 const double& beta, const double& s, const double& p) |
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149 { |
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150 if(lsize == 1) |
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151 return (fabs(alpha) >= fabs(beta) ? 1 : -1); |
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152 else |
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153 return (fabs(p) >= 1 ? 1 : -1); |
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154 } |
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155 |
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156 DEFUN_DLD (qz, args, nargout, |
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157 "Usage: lambda = qz(A,B) form [1]\n\ |
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158 [AA,BB,Q,Z{,V,W,lambda}] = qz(A,B) form [2]\n\ |
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159 [AA,BB,Z{,lambda}] = qz(A,B,opt) form [3]\n\ |
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160 Generalized eigenvalue problem A x = s B x \n\ |
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161 |
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162 Form [1]: Computes the generalized eigenvalues lambda of (A - sB).\n\ |
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163 Form [2]: Computes qz decomposition, generalized eigenvectors, and \n\ |
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164 generalized eigenvalues of (A - sB)\n\ |
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165 A V = B V diag(lambda)\n\ |
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166 W' A = diag(lambda) W' B\n\ |
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167 AA = Q'*A*Z, BB = Q'*B*Z with Q, Z orthogonal (unitary)= I\n\ |
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168 Form [3]: As in form [2], but allows ordering of generalized eigenpairs\n\ |
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169 for (e.g.) solution of discrete time algebraic Riccati equations.\n\ |
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170 Form 3 is not available for complex matrices and does not compute\n\ |
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171 the generalized eigenvectors V, W, nor the orthogonal matrix Q.\n\ |
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172 \n\ |
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173 opt: for ordering eigenvalues of the GEP pencil. The leading block\n\ |
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174 of the revised pencil contains all eigenvalues that satisfy:\n\ |
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175 \"N\" = unordered (default) \n\ |
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176 \"S\" = small: leading block has all |lambda| <=1 \n\ |
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177 \"B\" = big: leading block has all |lambda >= 1 \n\ |
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178 \"-\" = negative real part: leading block has all eigenvalues\n\ |
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179 in the open left half-plant\n\ |
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180 \"+\" = nonnegative real part: leading block has all eigenvalues\n\ |
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181 in the closed right half-plane\n\ |
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182 \n\ |
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183 Note: Permutation balancing is performed, but not scaling (see balance)\n\ |
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184 Order of output arguments was selected for compatibility with MATLAB\n\ |
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185 \n\ |
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186 See also: balance, dare, eig, schur\n") |
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187 { |
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188 octave_value_list retval; |
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189 int nargin = args.length (); |
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190 |
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191 #ifdef DEBUG |
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192 cout << "qz: nargin = " << nargin << ", nargout = " << nargout << endl; |
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193 #endif |
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194 |
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195 if (nargin < 2 || nargin > 3 || nargout > 7 ) |
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196 { |
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197 print_usage ("qz"); |
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198 return retval; |
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199 } |
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200 else if(nargin == 3 && (nargout < 3 || nargout > 4)) |
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201 { |
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202 error("qz: Illegal number of output arguments for form [3] call"); |
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203 } |
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204 |
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205 #ifdef DEBUG |
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206 cout << "qz: determine ordering option" << endl; |
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207 #endif |
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208 |
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209 // Determine ordering option |
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210 string ord_job; |
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211 static double safmin; |
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212 if(nargin == 2) |
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213 ord_job = "N"; |
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214 else if( !args(2).is_string() ) |
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215 error("qz: argument 3 must be a string"); |
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216 else |
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217 { |
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218 ord_job = args(2).string_value(); |
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219 if(ord_job[0] != 'N' && ord_job[0] != 'S' && ord_job[0] != 'B' |
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220 && ord_job[0] != '+' && ord_job[0] != '-') |
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221 error("qz: illegal order option"); |
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222 |
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223 // overflow constant required by dlag2 |
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224 F77_XFCN ( xdlamch, XDLAMCH, ("S", safmin, 1L)); |
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225 |
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226 #ifdef DEBUG_EIG |
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227 cout << "qz: initial value of safmin=" << setiosflags(ios::scientific) |
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228 << safmin << endl; |
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229 #endif |
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230 |
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231 // some machines (e.g., DEC alpha) get safmin = 0; |
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232 // for these, use eps instead to avoid problems in dlag2 |
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233 if(safmin == 0) |
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234 { |
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235 #ifdef DEBUG_EIG |
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236 cout << "qz: DANGER WILL ROBINSON: safmin is 0!" << endl; |
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237 #endif |
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238 |
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239 F77_XFCN ( xdlamch, XDLAMCH, ("E", safmin, 1L)); |
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240 |
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241 #ifdef DEBUG_EIG |
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242 cout << "qz: safmin set to " << setiosflags(ios::scientific) |
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243 << safmin << endl; |
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244 #endif |
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245 } |
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246 } |
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247 |
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248 #ifdef DEBUG |
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249 cout << "qz: check argument 1" << endl; |
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250 #endif |
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251 |
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252 // Argument 1: check if it's o.k. dimensioned |
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253 int nn = args(0).rows(); |
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254 |
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255 #ifdef DEBUG |
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256 cout << "argument 1 dimensions: (" << nn << "," << args(0).columns() << ")" |
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257 << endl; |
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258 #endif |
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259 int arg_is_empty = empty_arg ("qz", nn, args(0).columns()); |
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260 if (arg_is_empty < 0) |
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261 { |
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262 gripe_empty_arg("qz: parameter 1",0); |
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263 return retval; |
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264 } |
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265 else if (arg_is_empty > 0) |
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266 { |
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267 gripe_empty_arg("qz: parameter 1; continuing",0); |
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268 return octave_value_list (2, Matrix ()); |
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269 } |
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270 else if (args(0).columns() != nn) |
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271 { |
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272 gripe_square_matrix_required ("qz"); |
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273 return retval; |
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274 } |
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275 |
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276 // Argument 1: dimensions look good; get the value |
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277 Matrix aa; |
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278 ComplexMatrix caa; |
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279 if (args(0).is_complex_type ()) |
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280 caa = args(0).complex_matrix_value (); |
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281 else |
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282 aa = args(0).matrix_value (); |
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283 if (error_state) |
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284 return retval; |
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285 |
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286 #ifdef DEBUG |
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287 cout << "qz: check argument 2" << endl; |
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288 #endif |
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289 |
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290 // Extract argument 2 (bb, or cbb if complex) |
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291 if( (nn != args(1).columns()) || (nn != args(1).rows() )) |
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292 { |
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293 gripe_nonconformant (); |
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294 return retval; |
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295 } |
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296 Matrix bb; |
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297 ComplexMatrix cbb; |
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298 if (args(1).is_complex_type ()) |
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299 cbb = args(1).complex_matrix_value (); |
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300 else |
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301 bb = args(1).matrix_value (); |
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302 if (error_state) |
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303 return retval; |
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304 |
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305 // Both matrices loaded, now let's check what kind of arithmetic: |
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306 //declared static to avoid compiler warnings about long jumps, vforks. |
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307 static int complex_case |
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308 = (args(0).is_complex_type() || args(1).is_complex_type()); |
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309 |
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310 if(nargin == 3 && complex_case) |
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311 error("qz: cannot re-order complex qz decomposition."); |
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312 |
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313 // first, declare variables used in both the real and complex case |
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314 Matrix QQ(nn,nn), ZZ(nn,nn), VR(nn,nn), VL(nn,nn); |
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315 RowVector alphar(nn), alphai(nn), betar(nn); |
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316 |
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317 ComplexMatrix CQ(nn,nn), CZ(nn,nn),CVR(nn,nn),CVL(nn,nn); |
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318 int ilo, ihi, info; |
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319 char compq = (nargout >= 3 ? 'V' : 'N'), |
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320 compz = (nargout >= 4 ? 'V' : 'N'); |
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321 |
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322 // initialize Q,Z to identity if we need either of them |
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323 if(compq == 'V' || compz == 'V') |
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324 for(int ii=0; ii < nn ; ii++) |
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325 for( int jj=0; jj < nn ; jj++) |
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326 QQ(ii,jj) = ZZ(ii,jj) = (ii == jj ? 1.0 : 0.0); |
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327 |
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328 // always perform permutation balancing |
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329 char bal_job = 'P'; |
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330 RowVector lscale(nn), rscale(nn), work(6*nn); |
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331 |
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332 if(complex_case) |
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333 error("Complex case not implemented yet"); |
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334 else |
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335 { |
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336 #ifdef DEBUG |
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337 if(compq == 'V') |
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338 cout << "qz: performing balancing; QQ=" << endl << QQ << endl; |
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339 #endif |
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340 |
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341 F77_XFCN( dggbal, DGGBAL, (&bal_job, nn, aa.fortran_vec(), |
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342 nn, bb.fortran_vec() , nn, ilo, ihi, lscale.fortran_vec(), |
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343 rscale.fortran_vec(), work.fortran_vec(), info , 1L)); |
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344 if(f77_exception_encountered) |
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345 (*current_liboctave_error_handler) ("unrecoverable error in qz(bal)"); |
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346 } |
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347 |
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348 // Since we just want the balancing matrices, we can use dggbal |
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349 // for both the real and complex cases; |
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350 // left first |
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351 if(compq == 'V') |
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352 { |
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353 F77_XFCN( dggbak, DGGBAK, (&bal_job, "L", |
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354 nn, ilo, ihi, lscale.fortran_vec(), |
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355 rscale.fortran_vec(), nn, QQ.fortran_vec(), |
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356 nn, info, 1L, 1L)); |
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357 |
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358 #ifdef DEBUG |
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359 if(compq == 'V') cout << "qz: balancing done; QQ=" << endl << QQ << endl; |
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360 #endif |
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361 |
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362 if(f77_exception_encountered) |
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363 (*current_liboctave_error_handler) ("unrecoverable error in qz(bal-L)"); |
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364 } |
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365 |
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366 // then right |
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367 if(compz == 'V') |
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368 { |
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369 F77_XFCN(dggbak, DGGBAK, (&bal_job, "R", |
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370 nn, ilo, ihi, lscale.fortran_vec(), |
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371 rscale.fortran_vec(), nn, ZZ.fortran_vec(), |
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372 nn, info, 1L, 1L)); |
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373 |
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374 #ifdef DEBUG |
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375 if(compz == 'V') cout << "qz: balancing done; ZZ=" << endl << ZZ << endl; |
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376 #endif |
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377 |
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378 if(f77_exception_encountered) |
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379 (*current_liboctave_error_handler) ("unrecoverable error in qz(bal-R)"); |
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380 } |
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381 |
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382 static char qz_job; |
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383 qz_job = (nargout < 2 ? 'E' : 'S'); |
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384 if (complex_case) |
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385 { |
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386 // complex case |
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387 if (args(0).is_real_type ()) caa = aa; |
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388 if (args(1).is_real_type ()) cbb = bb; |
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389 if(compq == 'V') CQ = QQ; |
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390 if(compz == 'V') CZ = ZZ; |
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391 error("complex case not done yet"); |
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392 } |
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393 else // real matrices case |
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394 { |
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395 #ifdef DEBUG |
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396 cout << "qz: peforming qr decomposition of bb" << endl; |
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397 #endif |
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398 |
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399 // compute the QR factorization of bb |
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400 QR bqr(bb); |
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401 |
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402 #ifdef DEBUG |
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403 cout << "qz: qr(bb) done; now peforming qz decomposition" << endl; |
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404 #endif |
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405 |
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406 bb = bqr.R(); |
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407 #ifdef DEBUG |
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408 cout << "qz: extracted bb" << endl; |
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409 #endif |
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410 |
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411 aa = (bqr.Q()).transpose()*aa; |
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412 #ifdef DEBUG |
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413 cout << "qz: updated aa " << endl; |
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414 cout << "bqr.Q () = " << endl << bqr.Q () << endl; |
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415 if(compq == 'V') cout << "QQ =" << QQ << endl; |
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416 #endif |
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417 |
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418 if(compq == 'V') QQ = QQ*bqr.Q(); |
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419 |
|
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420 #ifdef DEBUG |
|
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421 cout << "qz: precursors done..." << endl; |
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422 #endif |
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423 |
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424 |
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425 #ifdef DEBUG |
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426 cout << "qz: compq = " << compq << ", compz = " << compz << endl; |
|
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427 #endif |
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428 |
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429 // reduce to generalized hessenberg form |
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430 F77_XFCN( dgghrd, DGGHRD, (&compq, &compz, nn, ilo, ihi, aa.fortran_vec(), |
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431 nn, bb.fortran_vec(), nn, QQ.fortran_vec(), nn, ZZ.fortran_vec(), |
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432 nn, info,1L,1L)); |
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433 if(f77_exception_encountered) |
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434 (*current_liboctave_error_handler) ("unrecoverable error in qz(dgghrd)"); |
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435 |
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436 // check if just computing generalized eigenvalues or if we're |
|
|
437 // actually computing the decomposition |
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438 |
|
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439 // reduce to generalized Schur form |
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440 F77_XFCN( dhgeqz, DHGEQZ, ( &qz_job, &compq, &compz, nn, ilo, ihi, |
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441 aa.fortran_vec(), nn, bb.fortran_vec(), nn, alphar.fortran_vec(), |
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442 alphai.fortran_vec(), betar.fortran_vec(), QQ.fortran_vec(), |
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443 nn, ZZ.fortran_vec(), nn, work.fortran_vec(), nn, info, 1L, 1L, 1L)); |
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444 if(f77_exception_encountered) |
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445 (*current_liboctave_error_handler) ("unrecoverable error in qz(dhgeqz)"); |
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446 |
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447 } |
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448 |
|
|
449 // order the QZ decomposition? |
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450 if(ord_job[0] != 'N') |
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|
451 { |
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452 if(complex_case) // probably not needed, but better be safe |
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|
453 error("qz: cannot re-order complex qz decomposition."); |
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454 |
|
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455 else |
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|
456 { |
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|
457 #ifdef DEBUG_SORT |
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458 cout << "qz: ordering eigenvalues: ord_job = " << ord_job[0] << endl; |
|
|
459 #endif |
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460 |
|
|
461 // declared static to avoid vfork/long jump compiler complaints |
|
|
462 static sort_function sort_test; |
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|
463 sort_test = NULL; |
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464 |
|
|
465 switch(ord_job[0]) |
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|
466 { |
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467 case 'S': |
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|
468 sort_test = &fin; |
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|
469 break; |
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|
470 case 'B': |
|
|
471 sort_test = &fout; |
|
|
472 break; |
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|
473 case '+': |
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|
474 sort_test = &fcrhp; |
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|
475 break; |
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476 case '-': |
|
|
477 sort_test = &folhp; |
|
|
478 break; |
|
|
479 default: // this should never happen |
|
|
480 error("qz: illegal order option"); |
|
|
481 } |
|
|
482 |
|
|
483 int ndim, fail, ind[nn]; |
|
|
484 double inf_norm; |
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|
485 F77_XFCN (xdlange, XDLANGE, ("I", nn, nn, aa.fortran_vec (), nn, |
|
|
486 work.fortran_vec (), inf_norm)); |
|
|
487 |
|
|
488 double eps = DBL_EPSILON*inf_norm*nn; |
|
|
489 |
|
|
490 #ifdef DEBUG_SORT |
|
|
491 cout << "qz: calling dsubsp: aa=" << endl; |
|
|
492 octave_print_internal(cout,aa,0); |
|
|
493 cout << endl << "bb=" << endl; |
|
|
494 octave_print_internal(cout,bb,0); |
|
|
495 if(compz == 'V') |
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|
496 { |
|
|
497 cout << endl << "ZZ=" << endl; |
|
|
498 octave_print_internal(cout,ZZ,0); |
|
|
499 } |
|
|
500 cout << endl; |
|
|
501 cout << "alphar = " << endl; |
|
|
502 octave_print_internal(cout,(Matrix) alphar,0); |
|
|
503 cout << endl << "alphai = " << endl; |
|
|
504 octave_print_internal(cout,(Matrix) alphai,0); |
|
|
505 cout << endl << "beta = " << endl; |
|
|
506 octave_print_internal(cout,(Matrix) betar,0); |
|
|
507 cout << endl; |
|
|
508 #endif |
|
|
509 |
|
|
510 F77_XFCN( dsubsp, DSUBSP, (nn,nn,aa.fortran_vec(), bb.fortran_vec(), |
|
|
511 ZZ.fortran_vec(), sort_test, eps, ndim, fail, ind)); |
|
|
512 |
|
|
513 #ifdef DEBUG |
|
|
514 cout << "qz: back from dsubsp: aa=" << endl; |
|
|
515 octave_print_internal(cout,aa,0); |
|
|
516 cout << endl << "bb=" << endl; |
|
|
517 octave_print_internal(cout,bb,0); |
|
|
518 if(compz == 'V') |
|
|
519 { |
|
|
520 cout << endl << "ZZ=" << endl; |
|
|
521 octave_print_internal(cout,ZZ,0); |
|
|
522 } |
|
|
523 cout << endl; |
|
|
524 #endif |
|
|
525 |
|
|
526 // manually update alphar, alphai, betar |
|
|
527 static int jj; |
|
|
528 jj=0; |
|
|
529 while(jj < nn) |
|
|
530 { |
|
|
531 #ifdef DEBUG_EIG |
|
|
532 cout << "computing gen eig #" << jj << endl; |
|
|
533 #endif |
|
|
534 |
|
|
535 static int zcnt; // number of zeros in this block |
|
|
536 if(jj == (nn-1)) |
|
|
537 zcnt = 1; |
|
|
538 else if(aa(jj+1,jj) == 0) |
|
|
539 zcnt = 1; |
|
|
540 else zcnt = 2; |
|
|
541 |
|
|
542 if(zcnt == 1) // real zero |
|
|
543 { |
|
|
544 #ifdef DEBUG_EIG |
|
|
545 cout << " single gen eig:" << endl; |
|
|
546 cout << " alphar(" << jj << ") = " << aa(jj,jj) << endl; |
|
|
547 cout << " betar( " << jj << ") = " << bb(jj,jj) << endl; |
|
|
548 cout << " alphai(" << jj << ") = 0" << endl; |
|
|
549 #endif |
|
|
550 |
|
|
551 alphar(jj) = aa(jj,jj); |
|
|
552 alphai(jj) = 0; |
|
|
553 betar(jj) = bb(jj,jj); |
|
|
554 } |
|
|
555 else // complex conjugate pair |
|
|
556 { |
|
|
557 #ifdef DEBUG_EIG |
|
|
558 cout << "qz: calling dlag2:" << endl; |
|
|
559 cout << "safmin=" << setiosflags(ios::scientific) << safmin << endl; |
|
|
560 for(int idr = jj ; idr <= jj+1 ; idr++) |
|
|
561 { |
|
|
562 for(int idc = jj ; idc <= jj+1 ; idc++) |
|
|
563 { |
|
|
564 cout << "aa(" << idr << "," << idc << ")=" |
|
|
565 << aa(idr,idc) << endl; |
|
|
566 cout << "bb(" << idr << "," << idc << ")=" |
|
|
567 << bb(idr,idc) << endl; |
|
|
568 } |
|
|
569 } |
|
|
570 #endif |
|
|
571 double scale1, scale2, wr1, wr2, wi; |
|
|
572 F77_XFCN( dlag2, DLAG2, ( &aa(jj,jj), nn, &bb(jj,jj), nn, safmin, |
|
|
573 scale1, scale2, wr1, wr2, wi)); |
|
|
574 |
|
|
575 #ifdef DEBUG_EIG |
|
|
576 cout << "dlag2 returns: scale1=" << scale1 |
|
|
577 << "\tscale2=" << scale2 << endl |
|
|
578 << "\twr1=" << wr1 << "\twr2=" << wr2 |
|
|
579 << "\twi=" << wi << endl; |
|
|
580 #endif |
|
|
581 // just to be safe, check if it's a real pair |
|
|
582 if(wi == 0) |
|
|
583 { |
|
|
584 alphar(jj) = wr1; |
|
|
585 alphai(jj) = 0; |
|
|
586 betar(jj) = scale1; |
|
|
587 alphar(jj+1) = wr2; |
|
|
588 alphai(jj+1) = 0; |
|
|
589 betar(jj+1) = scale2; |
|
|
590 } |
|
|
591 else |
|
|
592 { |
|
|
593 alphar(jj) = alphar(jj+1)=wr1; |
|
|
594 alphai(jj) = -(alphai(jj+1) = wi); |
|
|
595 betar(jj) = betar(jj+1) = scale1; |
|
|
596 } |
|
|
597 } |
|
|
598 |
|
|
599 jj += zcnt; // advance past this block |
|
|
600 } |
|
|
601 |
|
|
602 #ifdef DEBUG_SORT |
|
|
603 cout << "qz: back from dsubsp: aa=" << endl; |
|
|
604 octave_print_internal(cout,aa,0); |
|
|
605 cout << endl << "bb=" << endl; |
|
|
606 octave_print_internal(cout,bb,0); |
|
|
607 if(compz == 'V') |
|
|
608 { |
|
|
609 cout << endl << "ZZ=" << endl; |
|
|
610 octave_print_internal(cout,ZZ,0); |
|
|
611 } |
|
|
612 cout << endl << "qz: ndim=" << ndim << endl << "fail=" << fail << endl; |
|
|
613 cout << "alphar = " << endl; |
|
|
614 octave_print_internal(cout,(Matrix) alphar,0); |
|
|
615 cout << endl << "alphai = " << endl; |
|
|
616 octave_print_internal(cout,(Matrix) alphai,0); |
|
|
617 cout << endl << "beta = " << endl; |
|
|
618 octave_print_internal(cout,(Matrix) betar,0); |
|
|
619 cout << endl; |
|
|
620 #endif |
|
|
621 } |
|
|
622 } |
|
|
623 |
|
|
624 // compute generalized eigenvalues? |
|
|
625 ComplexColumnVector gev; |
|
|
626 if(nargout < 2 || nargout == 7 || (nargin == 3 && nargout == 4)) |
|
|
627 { |
|
|
628 if(complex_case) |
|
|
629 error("complex case not yet implemented"); |
|
|
630 else |
|
|
631 { |
|
|
632 #ifdef DEBUG |
|
|
633 cout << "qz: computing generalized eigenvalues" << endl; |
|
|
634 #endif |
|
|
635 |
|
|
636 // return finite generalized eigenvalues |
|
|
637 int ii, cnt = 0; |
|
|
638 for( ii=0 ; ii < nn ; ii++) |
|
|
639 if(betar(ii) != 0) |
|
|
640 cnt++; |
|
|
641 ComplexColumnVector tmp(cnt); |
|
|
642 for( ii=0 ; ii < nn ; ii++) |
|
|
643 if(betar(ii) != 0) |
|
|
644 tmp(ii) = Complex(alphar(ii), alphai(ii))/betar(ii); |
|
|
645 gev = tmp; |
|
|
646 } |
|
|
647 } |
|
|
648 |
|
|
649 // right, left eigenvector matrices |
|
|
650 if(nargout >= 5) |
|
|
651 { |
|
|
652 char side = (nargout == 5 ? 'R' : 'B'), // which side to compute? |
|
|
653 howmny = 'B'; // compute all of them and backtransform |
|
|
654 int *select = NULL; // dummy pointer; select is not used. |
|
|
655 int m; |
|
|
656 |
|
|
657 if(complex_case) |
|
|
658 error("complex type not yet implemented"); |
|
|
659 else |
|
|
660 { |
|
|
661 #ifdef DEBUG |
|
|
662 cout << "qz: computing generalized eigenvectors" << endl; |
|
|
663 #endif |
|
|
664 |
|
|
665 VL = QQ; |
|
|
666 VR = ZZ; |
|
|
667 |
|
|
668 F77_XFCN( dtgevc, DTGEVC, ( &side, &howmny, select, nn, aa.fortran_vec(), |
|
|
669 nn, bb.fortran_vec(), nn, VL.fortran_vec(), nn, VR.fortran_vec(), |
|
|
670 nn, nn, m, work.fortran_vec(), info, 1L, 1L )); |
|
|
671 if(f77_exception_encountered) |
|
|
672 (*current_liboctave_error_handler) |
|
|
673 ("unrecoverable error in qz(dtgevc)"); |
|
|
674 |
|
|
675 // now construct the complex form of VV, WW |
|
|
676 int jj = 0; |
|
|
677 while(jj < nn) |
|
|
678 { |
|
|
679 // see if real or complex eigenvalue |
|
|
680 int cinc = 2; // column increment; assume complex eigenvalue |
|
|
681 if(jj == (nn-1)) |
|
|
682 cinc = 1; // single column |
|
|
683 else if(aa(jj+1,jj) == 0) |
|
|
684 cinc = 1; |
|
|
685 |
|
|
686 // now copy the eigenvector(s) to CVR, CVL |
|
|
687 if(cinc == 1) |
|
|
688 { |
|
|
689 int ii; |
|
|
690 for(ii = 0; ii < nn ; ii++) |
|
|
691 CVR(ii,jj) = VR(ii,jj); |
|
|
692 if(side == 'B') |
|
|
693 for(ii = 0; ii < nn ; ii++) |
|
|
694 CVL(ii,jj) = VL(ii,jj); |
|
|
695 } |
|
|
696 else // double column; complex vector |
|
|
697 { |
|
|
698 int ii; |
|
|
699 for(ii = 0; ii < nn ; ii++) |
|
|
700 { |
|
|
701 CVR(ii,jj) = Complex(VR(ii,jj),VR(ii,jj+1)); |
|
|
702 CVR(ii,jj+1) = Complex(VR(ii,jj),-VR(ii,jj+1)); |
|
|
703 } |
|
|
704 if(side == 'B') |
|
|
705 for(ii = 0; ii < nn ; ii++) |
|
|
706 { |
|
|
707 CVL(ii,jj) = Complex(VL(ii,jj),VL(ii,jj+1)); |
|
|
708 CVL(ii,jj+1) = Complex(VL(ii,jj),-VL(ii,jj+1)); |
|
|
709 } |
|
|
710 } |
|
|
711 jj += cinc; // advance to next eigenvectors (if any) |
|
|
712 } |
|
|
713 } |
|
|
714 } |
|
|
715 |
|
|
716 switch(nargout) |
|
|
717 { |
|
|
718 case 7: |
|
|
719 retval(6) = gev; |
|
|
720 case 6: // return eigenvectors |
|
|
721 retval(5) = CVL; |
|
|
722 case 5: // return eigenvectors |
|
|
723 retval(4) = CVR; |
|
|
724 case 4: |
|
|
725 if(nargin == 3) |
|
|
726 { |
|
|
727 #ifdef DEBUG |
|
|
728 cout << "qz: sort: retval(3) = gev = " << endl; |
|
|
729 octave_print_internal(cout,gev); |
|
|
730 cout << endl; |
|
|
731 #endif |
|
|
732 retval(3) = gev; |
|
|
733 } |
|
|
734 else retval(3) = ZZ; |
|
|
735 case 3: |
|
|
736 if(nargin == 3) |
|
|
737 retval(2) = ZZ; |
|
|
738 else |
|
|
739 retval(2) = QQ; |
|
|
740 case 2: |
|
|
741 #ifdef DEBUG |
|
|
742 cout << "qz: retval(1) = bb = " << endl; |
|
|
743 octave_print_internal(cout,bb,0); |
|
|
744 cout << endl << "qz: retval(0) = aa = " <<endl; |
|
|
745 octave_print_internal(cout,aa,0); |
|
|
746 cout << endl; |
|
|
747 #endif |
|
|
748 retval(1) = bb; |
|
|
749 retval(0) = aa; |
|
|
750 break; |
|
|
751 case 1: |
|
|
752 case 0: |
|
|
753 #ifdef DEBUG |
|
|
754 cout << "qz: retval(0) = gev = " << gev << endl; |
|
|
755 #endif |
|
|
756 retval(0) = gev; |
|
|
757 break; |
|
|
758 default: |
|
|
759 error("qz: too many return arguments. Sorry. "); |
|
|
760 } |
|
|
761 |
|
|
762 #ifdef DEBUG |
|
|
763 cout << "qz: exiting (at long last)" << endl; |
|
|
764 #endif |
|
|
765 |
|
|
766 return retval; |
|
|
767 } |
|
|
768 |
|
|
769 /* |
|
|
770 ;;; Local Variables: *** |
|
|
771 ;;; mode: C++ *** |
|
|
772 ;;; End: *** |
|
|
773 */ |
