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view libinterp/corefcn/qz.cc @ 23219:3ac9f9ecfae5 stable
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author | John W. Eaton <jwe@octave.org> |
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date | Wed, 22 Feb 2017 12:39:29 -0500 |
parents | e9a0469dedd9 |
children | 092078913d54 1bc0e610e293 |
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/* Copyright (C) 1998-2017 A. S. Hodel This file is part of Octave. Octave is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3 of the License, or (at your option) any later version. Octave is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with Octave; see the file COPYING. If not, see <http://www.gnu.org/licenses/>. */ // Generalized eigenvalue balancing via LAPACK // Author: A. S. Hodel <scotte@eng.auburn.edu> #undef DEBUG #undef DEBUG_SORT #undef DEBUG_EIG #if defined (HAVE_CONFIG_H) # include "config.h" #endif #include <cfloat> #include <iostream> #include <iomanip> #include "f77-fcn.h" #include "lo-lapack-proto.h" #include "lo-math.h" #include "qr.h" #include "quit.h" #include "defun.h" #include "error.h" #include "errwarn.h" #include "ovl.h" #include "oct-map.h" #include "ov.h" #include "pager.h" #if defined (DEBUG) || defined (DEBUG_SORT) # include "pr-output.h" #endif #include "symtab.h" #include "utils.h" #include "variables.h" typedef octave_idx_type (*sort_function) (const octave_idx_type& LSIZE, const double& ALPHA, const double& BETA, const double& S, const double& P); extern "C" { // Van Dooren's code (netlib.org: toms/590) for reordering // GEP. Only processes Z, not Q. F77_RET_T F77_FUNC (dsubsp, DSUBSP) (const F77_INT& NMAX, const F77_INT& N, F77_DBLE* A, F77_DBLE* B, F77_DBLE* Z, sort_function, const F77_DBLE& EPS, F77_INT& NDIM, F77_INT& FAIL, F77_INT* IND); } // fcrhp, fin, fout, folhp: // Routines for ordering of generalized eigenvalues. // Return 1 if test is passed, 0 otherwise. // fin: |lambda| < 1 // fout: |lambda| >= 1 // fcrhp: real(lambda) >= 0 // folhp: real(lambda) < 0 static octave_idx_type fcrhp (const octave_idx_type& lsize, const double& alpha, const double& beta, const double& s, const double&) { if (lsize == 1) return (alpha * beta >= 0 ? 1 : -1); else return (s >= 0 ? 1 : -1); } static octave_idx_type fin (const octave_idx_type& lsize, const double& alpha, const double& beta, const double&, const double& p) { octave_idx_type retval; if (lsize == 1) retval = (fabs (alpha) < fabs (beta) ? 1 : -1); else retval = (fabs (p) < 1 ? 1 : -1); #if defined (DEBUG) std::cout << "qz: fin: retval=" << retval << std::endl; #endif return retval; } static octave_idx_type folhp (const octave_idx_type& lsize, const double& alpha, const double& beta, const double& s, const double&) { if (lsize == 1) return (alpha * beta < 0 ? 1 : -1); else return (s < 0 ? 1 : -1); } static octave_idx_type fout (const octave_idx_type& lsize, const double& alpha, const double& beta, const double&, const double& p) { if (lsize == 1) return (fabs (alpha) >= fabs (beta) ? 1 : -1); else return (fabs (p) >= 1 ? 1 : -1); } //FIXME: Matlab does not produce lambda as the first output argument. // Compatibility problem? DEFUN (qz, args, nargout, doc: /* -*- texinfo -*- @deftypefn {} {@var{lambda} =} qz (@var{A}, @var{B}) @deftypefnx {} {@var{lambda} =} qz (@var{A}, @var{B}, @var{opt}) QZ@tie{}decomposition of the generalized eigenvalue problem (@math{A x = s B x}). There are three ways to call this function: @enumerate @item @code{@var{lambda} = qz (@var{A}, @var{B})} Computes the generalized eigenvalues @tex $\lambda$ @end tex @ifnottex @var{lambda} @end ifnottex of @math{(A - s B)}. @item @code{[AA, BB, Q, Z, V, W, @var{lambda}] = qz (@var{A}, @var{B})} Computes QZ@tie{}decomposition, generalized eigenvectors, and generalized eigenvalues of @math{(A - s B)} @tex $$ AV = BV{ \rm diag }(\lambda) $$ $$ W^T A = { \rm diag }(\lambda)W^T B $$ $$ AA = Q^T AZ, BB = Q^T BZ $$ @end tex @ifnottex @example @group A * V = B * V * diag (@var{lambda}) W' * A = diag (@var{lambda}) * W' * B AA = Q * A * Z, BB = Q * B * Z @end group @end example @end ifnottex with @var{Q} and @var{Z} orthogonal (unitary)= @var{I} @item @code{[AA,BB,Z@{, @var{lambda}@}] = qz (@var{A}, @var{B}, @var{opt})} As in form [2], but allows ordering of generalized eigenpairs for, e.g., solution of discrete time algebraic Riccati equations. Form 3 is not available for complex matrices, and does not compute the generalized eigenvectors @var{V}, @var{W}, nor the orthogonal matrix @var{Q}. @table @var @item opt for ordering eigenvalues of the @nospell{GEP} pencil. The leading block of the revised pencil contains all eigenvalues that satisfy: @table @asis @item @qcode{"N"} = unordered (default) @item @qcode{"S"} = small: leading block has all |lambda| @leq{} 1 @item @qcode{"B"} = big: leading block has all |lambda| @geq{} 1 @item @qcode{"-"} = negative real part: leading block has all eigenvalues in the open left half-plane @item @qcode{"+"} = non-negative real part: leading block has all eigenvalues in the closed right half-plane @end table @end table @end enumerate Note: @code{qz} performs permutation balancing, but not scaling (@pxref{XREFbalance}). The order of output arguments was selected for compatibility with @sc{matlab}. @seealso{eig, balance, lu, chol, hess, qr, qzhess, schur, svd} @end deftypefn */) { volatile int nargin = args.length (); #if defined (DEBUG) std::cout << "qz: nargin = " << nargin << ", nargout = " << nargout << std::endl; #endif if (nargin < 2 || nargin > 3 || nargout > 7) print_usage (); if (nargin == 3 && (nargout < 3 || nargout > 4)) error ("qz: invalid number of output arguments for form [3] call"); #if defined (DEBUG) std::cout << "qz: determine ordering option" << std::endl; #endif // Determine ordering option. volatile char ord_job = 0; static double safmin; if (nargin == 2) ord_job = 'N'; else { std::string tmp = args(2).xstring_value ("qz: OPT must be a string"); if (! tmp.empty ()) ord_job = tmp[0]; if (! (ord_job == 'N' || ord_job == 'n' || ord_job == 'S' || ord_job == 's' || ord_job == 'B' || ord_job == 'b' || ord_job == '+' || ord_job == '-')) error ("qz: invalid order option"); // overflow constant required by dlag2 F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG2 ("S", 1), safmin F77_CHAR_ARG_LEN (1)); #if defined (DEBUG_EIG) std::cout << "qz: initial value of safmin=" << setiosflags (std::ios::scientific) << safmin << std::endl; #endif // Some machines (e.g., DEC alpha) get safmin = 0; // for these, use eps instead to avoid problems in dlag2. if (safmin == 0) { #if defined (DEBUG_EIG) std::cout << "qz: DANGER WILL ROBINSON: safmin is 0!" << std::endl; #endif F77_FUNC (xdlamch, XDLAMCH) (F77_CONST_CHAR_ARG2 ("E", 1), safmin F77_CHAR_ARG_LEN (1)); #if defined (DEBUG_EIG) std::cout << "qz: safmin set to " << setiosflags (std::ios::scientific) << safmin << std::endl; #endif } } #if defined (DEBUG) std::cout << "qz: check argument 1" << std::endl; #endif // Argument 1: check if it's okay dimensioned. octave_idx_type nn = args(0).rows (); #if defined (DEBUG) std::cout << "argument 1 dimensions: (" << nn << "," << args(0).columns () << ")" << std::endl; #endif octave_value_list retval; if (args(0).is_empty ()) { warn_empty_arg ("qz: parameter 1; continuing"); return octave_value_list (2, Matrix ()); } else if (args(0).columns () != nn) err_square_matrix_required ("qz", "A"); // Argument 1: dimensions look good; get the value. Matrix aa; ComplexMatrix caa; if (args(0).is_complex_type ()) caa = args(0).complex_matrix_value (); else aa = args(0).matrix_value (); #if defined (DEBUG) std::cout << "qz: check argument 2" << std::endl; #endif // Extract argument 2 (bb, or cbb if complex). if ((nn != args(1).columns ()) || (nn != args(1).rows ())) err_nonconformant (); Matrix bb; ComplexMatrix cbb; if (args(1).is_complex_type ()) cbb = args(1).complex_matrix_value (); else bb = args(1).matrix_value (); // Both matrices loaded, now let's check what kind of arithmetic: // declared volatile to avoid compiler warnings about long jumps, // vforks. volatile int complex_case = (args(0).is_complex_type () || args(1).is_complex_type ()); if (nargin == 3 && complex_case) error ("qz: cannot re-order complex qz decomposition"); // First, declare variables used in both the real and complex case. Matrix QQ(nn,nn), ZZ(nn,nn), VR(nn,nn), VL(nn,nn); RowVector alphar(nn), alphai(nn), betar(nn); ComplexRowVector xalpha(nn), xbeta(nn); ComplexMatrix CQ(nn,nn), CZ(nn,nn), CVR(nn,nn), CVL(nn,nn); octave_idx_type ilo, ihi, info; char compq = (nargout >= 3 ? 'V' : 'N'); char compz = ((nargout >= 4 || nargin == 3)? 'V' : 'N'); // Initialize Q, Z to identity if we need either of them. if (compq == 'V' || compz == 'V') for (octave_idx_type ii = 0; ii < nn; ii++) for (octave_idx_type jj = 0; jj < nn; jj++) { OCTAVE_QUIT; QQ(ii,jj) = ZZ(ii,jj) = (ii == jj ? 1.0 : 0.0); } // Always perform permutation balancing. const char bal_job = 'P'; RowVector lscale (nn), rscale (nn), work (6 * nn), rwork (nn); if (complex_case) { #if defined (DEBUG) if (compq == 'V') std::cout << "qz: performing balancing; CQ=" << std::endl << CQ << std::endl; #endif if (args(0).is_real_type ()) caa = ComplexMatrix (aa); if (args(1).is_real_type ()) cbb = ComplexMatrix (bb); if (compq == 'V') CQ = ComplexMatrix (QQ); if (compz == 'V') CZ = ComplexMatrix (ZZ); F77_XFCN (zggbal, ZGGBAL, (F77_CONST_CHAR_ARG2 (&bal_job, 1), nn, F77_DBLE_CMPLX_ARG (caa.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (cbb.fortran_vec ()), nn, ilo, ihi, lscale.fortran_vec (), rscale.fortran_vec (), work.fortran_vec (), info F77_CHAR_ARG_LEN (1))); } else { #if defined (DEBUG) if (compq == 'V') std::cout << "qz: performing balancing; QQ=" << std::endl << QQ << std::endl; #endif F77_XFCN (dggbal, DGGBAL, (F77_CONST_CHAR_ARG2 (&bal_job, 1), nn, aa.fortran_vec (), nn, bb.fortran_vec (), nn, ilo, ihi, lscale.fortran_vec (), rscale.fortran_vec (), work.fortran_vec (), info F77_CHAR_ARG_LEN (1))); } // Since we just want the balancing matrices, we can use dggbal // for both the real and complex cases; left first #if 0 if (compq == 'V') { F77_XFCN (dggbak, DGGBAK, (F77_CONST_CHAR_ARG2 (&bal_job, 1), F77_CONST_CHAR_ARG2 ("L", 1), nn, ilo, ihi, lscale.data (), rscale.data (), nn, QQ.fortran_vec (), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); #if defined (DEBUG) if (compq == 'V') std::cout << "qz: balancing done; QQ=" << std::endl << QQ << std::endl; #endif } // then right if (compz == 'V') { F77_XFCN (dggbak, DGGBAK, (F77_CONST_CHAR_ARG2 (&bal_job, 1), F77_CONST_CHAR_ARG2 ("R", 1), nn, ilo, ihi, lscale.data (), rscale.data (), nn, ZZ.fortran_vec (), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); #if defined (DEBUG) if (compz == 'V') std::cout << "qz: balancing done; ZZ=" << std::endl << ZZ << std::endl; #endif } #endif static char qz_job; qz_job = (nargout < 2 ? 'E' : 'S'); if (complex_case) { // Complex case. // The QR decomposition of cbb. octave::math::qr<ComplexMatrix> cbqr (cbb); // The R matrix of QR decomposition for cbb. cbb = cbqr.R (); // (Q*)caa for following work. caa = (cbqr.Q ().hermitian ()) * caa; CQ = CQ * cbqr.Q (); F77_XFCN (zgghrd, ZGGHRD, (F77_CONST_CHAR_ARG2 (&compq, 1), F77_CONST_CHAR_ARG2 (&compz, 1), nn, ilo, ihi, F77_DBLE_CMPLX_ARG (caa.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (cbb.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (CQ.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (CZ.fortran_vec ()), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); ComplexRowVector cwork (1 * nn); F77_XFCN (zhgeqz, ZHGEQZ, (F77_CONST_CHAR_ARG2 (&qz_job, 1), F77_CONST_CHAR_ARG2 (&compq, 1), F77_CONST_CHAR_ARG2 (&compz, 1), nn, ilo, ihi, F77_DBLE_CMPLX_ARG (caa.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (cbb.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (xalpha.fortran_vec ()), F77_DBLE_CMPLX_ARG (xbeta.fortran_vec ()), F77_DBLE_CMPLX_ARG (CQ.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (CZ.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (cwork.fortran_vec ()), nn, rwork.fortran_vec (), info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); if (compq == 'V') { // Left eigenvector. F77_XFCN (zggbak, ZGGBAK, (F77_CONST_CHAR_ARG2 (&bal_job, 1), F77_CONST_CHAR_ARG2 ("L", 1), nn, ilo, ihi, lscale.data (), rscale.data (), nn, F77_DBLE_CMPLX_ARG (CQ.fortran_vec ()), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); } // Right eigenvector. if (compz == 'V') { F77_XFCN (zggbak, ZGGBAK, (F77_CONST_CHAR_ARG2 (&bal_job, 1), F77_CONST_CHAR_ARG2 ("R", 1), nn, ilo, ihi, lscale.data (), rscale.data (), nn, F77_DBLE_CMPLX_ARG (CZ.fortran_vec ()), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); } } else { #if defined (DEBUG) std::cout << "qz: peforming qr decomposition of bb" << std::endl; #endif // Compute the QR factorization of bb. octave::math::qr<Matrix> bqr (bb); #if defined (DEBUG) std::cout << "qz: qr (bb) done; now peforming qz decomposition" << std::endl; #endif bb = bqr.R (); #if defined (DEBUG) std::cout << "qz: extracted bb" << std::endl; #endif aa = (bqr.Q ()).transpose () * aa; #if defined (DEBUG) std::cout << "qz: updated aa " << std::endl; std::cout << "bqr.Q () = " << std::endl << bqr.Q () << std::endl; if (compq == 'V') std::cout << "QQ =" << QQ << std::endl; #endif if (compq == 'V') QQ = QQ * bqr.Q (); #if defined (DEBUG) std::cout << "qz: precursors done..." << std::endl; #endif #if defined (DEBUG) std::cout << "qz: compq = " << compq << ", compz = " << compz << std::endl; #endif // Reduce to generalized Hessenberg form. F77_XFCN (dgghrd, DGGHRD, (F77_CONST_CHAR_ARG2 (&compq, 1), F77_CONST_CHAR_ARG2 (&compz, 1), nn, ilo, ihi, aa.fortran_vec (), nn, bb.fortran_vec (), nn, QQ.fortran_vec (), nn, ZZ.fortran_vec (), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); // Check if just computing generalized eigenvalues or if we're // actually computing the decomposition. // Reduce to generalized Schur form. F77_XFCN (dhgeqz, DHGEQZ, (F77_CONST_CHAR_ARG2 (&qz_job, 1), F77_CONST_CHAR_ARG2 (&compq, 1), F77_CONST_CHAR_ARG2 (&compz, 1), nn, ilo, ihi, aa.fortran_vec (), nn, bb.fortran_vec (), nn, alphar.fortran_vec (), alphai.fortran_vec (), betar.fortran_vec (), QQ.fortran_vec (), nn, ZZ.fortran_vec (), nn, work.fortran_vec (), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); if (compq == 'V') { F77_XFCN (dggbak, DGGBAK, (F77_CONST_CHAR_ARG2 (&bal_job, 1), F77_CONST_CHAR_ARG2 ("L", 1), nn, ilo, ihi, lscale.data (), rscale.data (), nn, QQ.fortran_vec (), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); #if defined (DEBUG) if (compq == 'V') std::cout << "qz: balancing done; QQ=" << std::endl << QQ << std::endl; #endif } // then right if (compz == 'V') { F77_XFCN (dggbak, DGGBAK, (F77_CONST_CHAR_ARG2 (&bal_job, 1), F77_CONST_CHAR_ARG2 ("R", 1), nn, ilo, ihi, lscale.data (), rscale.data (), nn, ZZ.fortran_vec (), nn, info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); #if defined (DEBUG) if (compz == 'V') std::cout << "qz: balancing done; ZZ=" << std::endl << ZZ << std::endl; #endif } } // Order the QZ decomposition? if (! (ord_job == 'N' || ord_job == 'n')) { if (complex_case) // Probably not needed, but better be safe. error ("qz: cannot re-order complex qz decomposition"); #if defined (DEBUG_SORT) std::cout << "qz: ordering eigenvalues: ord_job = " << ord_job << std::endl; #endif // Declared static to avoid vfork/long jump compiler complaints. static sort_function sort_test; sort_test = 0; switch (ord_job) { case 'S': case 's': sort_test = &fin; break; case 'B': case 'b': sort_test = &fout; break; case '+': sort_test = &fcrhp; break; case '-': sort_test = &folhp; break; default: // Invalid order option (should never happen, since we // checked the options at the top). panic_impossible (); break; } octave_idx_type ndim, fail; double inf_norm; F77_XFCN (xdlange, XDLANGE, (F77_CONST_CHAR_ARG2 ("I", 1), nn, nn, aa.data (), nn, work.fortran_vec (), inf_norm F77_CHAR_ARG_LEN (1))); double eps = std::numeric_limits<double>::epsilon () * inf_norm * nn; #if defined (DEBUG_SORT) std::cout << "qz: calling dsubsp: aa=" << std::endl; octave_print_internal (std::cout, aa, 0); std::cout << std::endl << "bb=" << std::endl; octave_print_internal (std::cout, bb, 0); if (compz == 'V') { std::cout << std::endl << "ZZ=" << std::endl; octave_print_internal (std::cout, ZZ, 0); } std::cout << std::endl; std::cout << "alphar = " << std::endl; octave_print_internal (std::cout, (Matrix) alphar, 0); std::cout << std::endl << "alphai = " << std::endl; octave_print_internal (std::cout, (Matrix) alphai, 0); std::cout << std::endl << "beta = " << std::endl; octave_print_internal (std::cout, (Matrix) betar, 0); std::cout << std::endl; #endif Array<octave_idx_type> ind (dim_vector (nn, 1)); F77_XFCN (dsubsp, DSUBSP, (nn, nn, aa.fortran_vec (), bb.fortran_vec (), ZZ.fortran_vec (), sort_test, eps, ndim, fail, ind.fortran_vec ())); #if defined (DEBUG) std::cout << "qz: back from dsubsp: aa=" << std::endl; octave_print_internal (std::cout, aa, 0); std::cout << std::endl << "bb=" << std::endl; octave_print_internal (std::cout, bb, 0); if (compz == 'V') { std::cout << std::endl << "ZZ=" << std::endl; octave_print_internal (std::cout, ZZ, 0); } std::cout << std::endl; #endif // Manually update alphar, alphai, betar. static int jj; jj = 0; while (jj < nn) { #if defined (DEBUG_EIG) std::cout << "computing gen eig #" << jj << std::endl; #endif // Number of zeros in this block. static int zcnt; if (jj == (nn-1)) zcnt = 1; else if (aa(jj+1,jj) == 0) zcnt = 1; else zcnt = 2; if (zcnt == 1) { // Real zero. #if defined (DEBUG_EIG) std::cout << " single gen eig:" << std::endl; std::cout << " alphar(" << jj << ") = " << aa(jj,jj) << std::endl; std::cout << " betar(" << jj << ") = " << bb(jj,jj) << std::endl; std::cout << " alphai(" << jj << ") = 0" << std::endl; #endif alphar(jj) = aa(jj,jj); alphai(jj) = 0; betar(jj) = bb(jj,jj); } else { // Complex conjugate pair. #if defined (DEBUG_EIG) std::cout << "qz: calling dlag2:" << std::endl; std::cout << "safmin=" << setiosflags (std::ios::scientific) << safmin << std::endl; for (int idr = jj; idr <= jj+1; idr++) { for (int idc = jj; idc <= jj+1; idc++) { std::cout << "aa(" << idr << "," << idc << ")=" << aa(idr,idc) << std::endl; std::cout << "bb(" << idr << "," << idc << ")=" << bb(idr,idc) << std::endl; } } #endif // FIXME: probably should be using // fortran_vec instead of &aa(jj,jj) here. double scale1, scale2, wr1, wr2, wi; const double *aa_ptr = aa.data () + jj * nn + jj; const double *bb_ptr = bb.data () + jj * nn + jj; F77_XFCN (dlag2, DLAG2, (aa_ptr, nn, bb_ptr, nn, safmin, scale1, scale2, wr1, wr2, wi)); #if defined (DEBUG_EIG) std::cout << "dlag2 returns: scale1=" << scale1 << "\tscale2=" << scale2 << std::endl << "\twr1=" << wr1 << "\twr2=" << wr2 << "\twi=" << wi << std::endl; #endif // Just to be safe, check if it's a real pair. if (wi == 0) { alphar(jj) = wr1; alphai(jj) = 0; betar(jj) = scale1; alphar(jj+1) = wr2; alphai(jj+1) = 0; betar(jj+1) = scale2; } else { alphar(jj) = alphar(jj+1) = wr1; alphai(jj) = -(alphai(jj+1) = wi); betar(jj) = betar(jj+1) = scale1; } } // Advance past this block. jj += zcnt; } #if defined (DEBUG_SORT) std::cout << "qz: back from dsubsp: aa=" << std::endl; octave_print_internal (std::cout, aa, 0); std::cout << std::endl << "bb=" << std::endl; octave_print_internal (std::cout, bb, 0); if (compz == 'V') { std::cout << std::endl << "ZZ=" << std::endl; octave_print_internal (std::cout, ZZ, 0); } std::cout << std::endl << "qz: ndim=" << ndim << std::endl << "fail=" << fail << std::endl; std::cout << "alphar = " << std::endl; octave_print_internal (std::cout, (Matrix) alphar, 0); std::cout << std::endl << "alphai = " << std::endl; octave_print_internal (std::cout, (Matrix) alphai, 0); std::cout << std::endl << "beta = " << std::endl; octave_print_internal (std::cout, (Matrix) betar, 0); std::cout << std::endl; #endif } // Compute generalized eigenvalues? ComplexColumnVector gev; if (nargout < 2 || nargout == 7 || (nargin == 3 && nargout == 4)) { if (complex_case) { int cnt = 0; for (int ii = 0; ii < nn; ii++) cnt++; ComplexColumnVector tmp (cnt); cnt = 0; for (int ii = 0; ii < nn; ii++) tmp(cnt++) = xalpha(ii) / xbeta(ii); gev = tmp; } else { #if defined (DEBUG) std::cout << "qz: computing generalized eigenvalues" << std::endl; #endif // Return finite generalized eigenvalues. int cnt = 0; for (int ii = 0; ii < nn; ii++) if (betar(ii) != 0) cnt++; ComplexColumnVector tmp (cnt); cnt = 0; for (int ii = 0; ii < nn; ii++) if (betar(ii) != 0) tmp(cnt++) = Complex(alphar(ii), alphai(ii))/betar(ii); gev = tmp; } } // Right, left eigenvector matrices. if (nargout >= 5) { // Which side to compute? char side = (nargout == 5 ? 'R' : 'B'); // Compute all of them and backtransform char howmny = 'B'; // Dummy pointer; select is not used. octave_idx_type *select = 0; if (complex_case) { CVL = CQ; CVR = CZ; ComplexRowVector cwork2 (2 * nn); RowVector rwork2 (8 * nn); octave_idx_type m; F77_XFCN (ztgevc, ZTGEVC, (F77_CONST_CHAR_ARG2 (&side, 1), F77_CONST_CHAR_ARG2 (&howmny, 1), select, nn, F77_DBLE_CMPLX_ARG (caa.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (cbb.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (CVL.fortran_vec ()), nn, F77_DBLE_CMPLX_ARG (CVR.fortran_vec ()), nn, nn, m, F77_DBLE_CMPLX_ARG (cwork2.fortran_vec ()), rwork2.fortran_vec (), info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); } else { #if defined (DEBUG) std::cout << "qz: computing generalized eigenvectors" << std::endl; #endif VL = QQ; VR = ZZ; octave_idx_type m; F77_XFCN (dtgevc, DTGEVC, (F77_CONST_CHAR_ARG2 (&side, 1), F77_CONST_CHAR_ARG2 (&howmny, 1), select, nn, aa.fortran_vec (), nn, bb.fortran_vec (), nn, VL.fortran_vec (), nn, VR.fortran_vec (), nn, nn, m, work.fortran_vec (), info F77_CHAR_ARG_LEN (1) F77_CHAR_ARG_LEN (1))); // Now construct the complex form of VV, WW. int jj = 0; while (jj < nn) { OCTAVE_QUIT; // See if real or complex eigenvalue. // Column increment; assume complex eigenvalue. int cinc = 2; if (jj == (nn-1)) // Single column. cinc = 1; else if (aa(jj+1,jj) == 0) cinc = 1; // Now copy the eigenvector (s) to CVR, CVL. if (cinc == 1) { for (int ii = 0; ii < nn; ii++) CVR(ii,jj) = VR(ii,jj); if (side == 'B') for (int ii = 0; ii < nn; ii++) CVL(ii,jj) = VL(ii,jj); } else { // Double column; complex vector. for (int ii = 0; ii < nn; ii++) { CVR(ii,jj) = Complex (VR(ii,jj), VR(ii,jj+1)); CVR(ii,jj+1) = Complex (VR(ii,jj), -VR(ii,jj+1)); } if (side == 'B') for (int ii = 0; ii < nn; ii++) { CVL(ii,jj) = Complex (VL(ii,jj), VL(ii,jj+1)); CVL(ii,jj+1) = Complex (VL(ii,jj), -VL(ii,jj+1)); } } // Advance to next eigenvectors (if any). jj += cinc; } } } switch (nargout) { case 7: retval(6) = gev; case 6: // Return eigenvectors. retval(5) = CVL; case 5: // Return eigenvectors. retval(4) = CVR; case 4: if (nargin == 3) { #if defined (DEBUG) std::cout << "qz: sort: retval(3) = gev = " << std::endl; octave_print_internal (std::cout, gev); std::cout << std::endl; #endif retval(3) = gev; } else { if (complex_case) retval(3) = CZ; else retval(3) = ZZ; } case 3: if (nargin == 3) { if (complex_case) retval(2) = CZ; else retval(2) = ZZ; } else { if (complex_case) retval(2) = CQ.hermitian (); else retval(2) = QQ.transpose (); } case 2: { if (complex_case) { #if defined (DEBUG) std::cout << "qz: retval(1) = cbb = " << std::endl; octave_print_internal (std::cout, cbb, 0); std::cout << std::endl << "qz: retval(0) = caa = " <<std::endl; octave_print_internal (std::cout, caa, 0); std::cout << std::endl; #endif retval(1) = cbb; retval(0) = caa; } else { #if defined (DEBUG) std::cout << "qz: retval(1) = bb = " << std::endl; octave_print_internal (std::cout, bb, 0); std::cout << std::endl << "qz: retval(0) = aa = " <<std::endl; octave_print_internal (std::cout, aa, 0); std::cout << std::endl; #endif retval(1) = bb; retval(0) = aa; } } break; case 1: case 0: #if defined (DEBUG) std::cout << "qz: retval(0) = gev = " << gev << std::endl; #endif retval(0) = gev; break; default: error ("qz: too many return arguments"); break; } #if defined (DEBUG) std::cout << "qz: exiting (at long last)" << std::endl; #endif return retval; } /* %!shared a, b, c %! a = [1 2; 0 3]; %! b = [1 0; 0 0]; %! c = [0 1; 0 0]; %!assert (qz (a,b), 1) %!assert (isempty (qz (a,c))) ## Exaple 7.7.3 in Golub & Van Loan %!test %! a = [ 10 1 2; %! 1 2 -1; %! 1 1 2]; %! b = reshape (1:9,3,3); %! [aa, bb, q, z, v, w, lambda] = qz (a, b); %! sz = length (lambda); %! observed = (b * v * diag ([lambda;0])) (:, 1:sz); %! assert ((a*v)(:, 1:sz), observed, norm (observed) * 1e-14); %! observed = (diag ([lambda;0]) * w' * b) (1:sz, :); %! assert ((w'*a)(1:sz, :) , observed, norm (observed) * 1e-13); %! assert (q * a * z, aa, norm (aa) * 1e-14); %! assert (q * b * z, bb, norm (bb) * 1e-14); %!test %! A = [0, 0, -1, 0; 1, 0, 0, 0; -1, 0, -2, -1; 0, -1, 1, 0]; %! B = [0, 0, 0, 0; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]; %! [AA, BB, Q, Z1] = qz (A, B); %! [AA, BB, Z2] = qz (A, B, '-'); %! assert (Z1, Z2); */