• NAME
• SYNOPSIS
• F95 INTERFACE
• C INTERFACE
• PURPOSE
• ARGUMENTS
• FURTHER DETAILS

NAME

zhgeqz - implement a single-shift version of the QZ method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation det( A-w(i) B ) = 0 If JOB='S', then the pair (A,B) is simultaneously reduced to Schur form (i.e., A and B are both upper triangular) by applying one unitary tranformation (usually called Q) on the left and another (usually called Z) on the right

SYNOPSIS

  SUBROUTINE ZHGEQZ( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, 
 *      ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO)
  CHARACTER * 1 JOB, COMPQ, COMPZ
  DOUBLE COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*), WORK(*)
  INTEGER N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
  DOUBLE PRECISION RWORK(*)

  SUBROUTINE ZHGEQZ_64( JOB, COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, 
 *      LDB, ALPHA, BETA, Q, LDQ, Z, LDZ, WORK, LWORK, RWORK, INFO)
  CHARACTER * 1 JOB, COMPQ, COMPZ
  DOUBLE COMPLEX A(LDA,*), B(LDB,*), ALPHA(*), BETA(*), Q(LDQ,*), Z(LDZ,*), WORK(*)
  INTEGER*8 N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
  DOUBLE PRECISION RWORK(*)

F95 INTERFACE

  SUBROUTINE HGEQZ( JOB, COMPQ, COMPZ, [N], ILO, IHI, A, [LDA], B, 
 *       [LDB], ALPHA, BETA, Q, [LDQ], Z, [LDZ], [WORK], [LWORK], [RWORK], 
 *       [INFO])
  CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ
  COMPLEX(8), DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX(8), DIMENSION(:,:) :: A, B, Q, Z
  INTEGER :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
  REAL(8), DIMENSION(:) :: RWORK

  SUBROUTINE HGEQZ_64( JOB, COMPQ, COMPZ, [N], ILO, IHI, A, [LDA], B, 
 *       [LDB], ALPHA, BETA, Q, [LDQ], Z, [LDZ], [WORK], [LWORK], [RWORK], 
 *       [INFO])
  CHARACTER(LEN=1) :: JOB, COMPQ, COMPZ
  COMPLEX(8), DIMENSION(:) :: ALPHA, BETA, WORK
  COMPLEX(8), DIMENSION(:,:) :: A, B, Q, Z
  INTEGER(8) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, LWORK, INFO
  REAL(8), DIMENSION(:) :: RWORK

C INTERFACE

#include <sunperf.h>

void zhgeqz(char job, char compq, char compz, int n, int ilo, int ihi, doublecomplex *a, int lda, doublecomplex *b, int ldb, doublecomplex *alpha, doublecomplex *beta, doublecomplex *q, int ldq, doublecomplex *z, int ldz, int *info);

void zhgeqz_64(char job, char compq, char compz, long n, long ilo, long ihi, doublecomplex *a, long lda, doublecomplex *b, long ldb, doublecomplex *alpha, doublecomplex *beta, doublecomplex *q, long ldq, doublecomplex *z, long ldz, long *info);

PURPOSE

zhgeqz implements a single-shift version of the QZ method for finding the generalized eigenvalues w(i)=ALPHA(i)/BETA(i) of the equation A are then ALPHA(1),...,ALPHA(N), and of B are BETA(1),...,BETA(N).

If JOB='S' and COMPQ and COMPZ are 'V' or 'I', then the unitary transformations used to reduce (A,B) are accumulated into the arrays Q and Z s.t.:

(in) A(in) Z(in)* = Q(out) A(out) Z(out)* (in) B(in) Z(in)* = Q(out) B(out) Z(out)*

Ref: C.B. Moler & G.W. Stewart, ``An Algorithm for Generalized Matrixigenvalue Problems'', SIAM J. Numer. Anal., 10(1973),p. 241--256.

ARGUMENTS

• JOB (input)
  

   = 'E': compute only ALPHA and BETA.  A and B will not
  necessarily be put into generalized Schur form.
   = 'S': put A and B into generalized Schur form, as well
  as computing ALPHA and BETA.

• COMPQ (input)
  

   = 'N': do not modify Q.

   = 'V': multiply the array Q on the right by the conjugate
  transpose of the unitary tranformation that is
  applied to the left side of A and B to reduce them
  to Schur form.
   = 'I': like COMPQ ='V', except that Q will be initialized to
  the identity first.

• COMPZ (input)
  

   = 'N': do not modify Z.

   = 'V': multiply the array Z on the right by the unitary
  tranformation that is applied to the right side of
  A and B to reduce them to Schur form.
   = 'I': like COMPZ ='V', except that Z will be initialized to
  the identity first.

• N (input)
  The order of the matrices A, B, Q, and Z. N > = 0.

• ILO (input)
  It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. 1 < = ILO < = IHI < = N, if N > 0; ILO =1 and IHI =0, if N =0.

• IHI (input)
  It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. 1 < = ILO < = IHI < = N, if N > 0; ILO =1 and IHI =0, if N =0.

• A (input/output)
  On entry, the N-by-N upper Hessenberg matrix A. Elements below the subdiagonal must be zero. If JOB ='S', then on exit A and B will have been simultaneously reduced to upper triangular form. If JOB ='E', then on exit A will have been destroyed.

• LDA (input)
  The leading dimension of the array A. LDA > = max( 1, N ).

• B (input/output)
  On entry, the N-by-N upper triangular matrix B. Elements below the diagonal must be zero. If JOB ='S', then on exit A and B will have been simultaneously reduced to upper triangular form. If JOB ='E', then on exit B will have been destroyed.

• LDB (input)
  The leading dimension of the array B. LDB > = max( 1, N ).

• ALPHA (output)
  The diagonal elements of A when the pair (A,B) has been reduced to Schur form. ALPHA(i)/BETA(i) i =1,...,N are the generalized eigenvalues.

• BETA (output)
  The diagonal elements of B when the pair (A,B) has been reduced to Schur form. ALPHA(i)/BETA(i) i =1,...,N are the generalized eigenvalues. A and B are normalized so that BETA(1),...,BETA(N) are non-negative real numbers.

• Q (input/output)
  If COMPQ ='N', then Q will not be referenced. If COMPQ ='V' or 'I', then the conjugate transpose of the unitary transformations which are applied to A and B on the left will be applied to the array Q on the right.

• LDQ (input)
  The leading dimension of the array Q. LDQ > = 1. If COMPQ ='V' or 'I', then LDQ > = N.

• Z (input/output)
  If COMPZ ='N', then Z will not be referenced. If COMPZ ='V' or 'I', then the unitary transformations which are applied to A and B on the right will be applied to the array Z on the right.

• LDZ (input)
  The leading dimension of the array Z. LDZ > = 1. If COMPZ ='V' or 'I', then LDZ > = N.

• WORK (workspace)
  On exit, if INFO > = 0, WORK(1) returns the optimal LWORK.

• LWORK (input)
  The dimension of the array WORK. LWORK > = max(1,N).

  If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

• RWORK (workspace)
  dimension(N)

• INFO (output)
  

   = 0: successful exit

   < 0: if INFO  = -i, the i-th argument had an illegal value

   = 1,...,N: the QZ iteration did not converge.  (A,B) is not
  in Schur form, but ALPHA(i) and BETA(i),
  i =INFO+1,...,N should be correct.
   = N+1,...,2*N: the shift calculation failed.  (A,B) is not
  in Schur form, but ALPHA(i) and BETA(i),
  i =INFO-N+1,...,N should be correct.
   > 2*N:     various "impossible" errors.

FURTHER DETAILS

We assume that complex ABS works as long as its value is less than overflow.