NAME

zgghrd - reduce a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is a general matrix and B is upper triangular

SYNOPSIS

  SUBROUTINE ZGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, 
 *      LDQ, Z, LDZ, INFO)
  CHARACTER * 1 COMPQ, COMPZ
  DOUBLE COMPLEX A(LDA,*), B(LDB,*), Q(LDQ,*), Z(LDZ,*)
  INTEGER N, ILO, IHI, LDA, LDB, LDQ, LDZ, INFO

  SUBROUTINE ZGGHRD_64( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, 
 *      LDQ, Z, LDZ, INFO)
  CHARACTER * 1 COMPQ, COMPZ
  DOUBLE COMPLEX A(LDA,*), B(LDB,*), Q(LDQ,*), Z(LDZ,*)
  INTEGER*8 N, ILO, IHI, LDA, LDB, LDQ, LDZ, INFO

F95 INTERFACE

  SUBROUTINE GGHRD( COMPQ, COMPZ, [N], ILO, IHI, A, [LDA], B, [LDB], 
 *       Q, [LDQ], Z, [LDZ], [INFO])
  CHARACTER(LEN=1) :: COMPQ, COMPZ
  COMPLEX(8), DIMENSION(:,:) :: A, B, Q, Z
  INTEGER :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, INFO

  SUBROUTINE GGHRD_64( COMPQ, COMPZ, [N], ILO, IHI, A, [LDA], B, [LDB], 
 *       Q, [LDQ], Z, [LDZ], [INFO])
  CHARACTER(LEN=1) :: COMPQ, COMPZ
  COMPLEX(8), DIMENSION(:,:) :: A, B, Q, Z
  INTEGER(8) :: N, ILO, IHI, LDA, LDB, LDQ, LDZ, INFO

C INTERFACE

#include <sunperf.h>

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

void zgghrd_64(char compq, char compz, long n, long ilo, long ihi, doublecomplex *a, long lda, doublecomplex *b, long ldb, doublecomplex *q, long ldq, doublecomplex *z, long ldz, long *info);

PURPOSE

zgghrd reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is a general matrix and B is upper triangular: Q' * A * Z = H and Q' * B * Z = T, where H is upper Hessenberg, T is upper triangular, and Q and Z are unitary, and ' means conjugate transpose.

The unitary matrices Q and Z are determined as products of Givens rotations. They may either be formed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that

1 * A * Z1' = (Q1*Q) * H * (Z1*Z)' 1 * B * Z1' = (Q1*Q) * T * (Z1*Z)'

ARGUMENTS

COMPQ (input)

 = 'N': do not compute Q;

 = 'I': Q is initialized to the unit matrix, and the
unitary matrix Q is returned;
 = 'V': Q must contain a unitary matrix Q1 on entry,
and the product Q1*Q is returned.

COMPZ (input)

 = 'N': do not compute Q;

 = 'I': Q is initialized to the unit matrix, and the
unitary matrix Q is returned;
 = 'V': Q must contain a unitary matrix Q1 on entry,
and the product Q1*Q is returned.

N (input)
The order of the matrices A and B. N > = 0.
ILO (input)
It is assumed that A is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to CGGBAL; otherwise they should be set to 1 and N respectively. 1 < = ILO < = IHI < = N, if N > 0; ILO =1 and IHI =0, if N =0.
IHI (input)
See description of ILO.
A (input/output)
On entry, the N-by-N general matrix to be reduced. On exit, the upper triangle and the first subdiagonal of A are overwritten with the upper Hessenberg matrix H, and the rest is set to zero.
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. On exit, the upper triangular matrix T = Q' B Z. The elements below the diagonal are set to zero.
LDB (input)
The leading dimension of the array B. LDB > = max(1,N).
Q (input/output)
If COMPQ ='N': Q is not referenced.
If COMPQ ='I': on entry, Q need not be set, and on exit it contains the unitary matrix Q, where Q' is the product of the Givens transformations which are applied to A and B on the left. If COMPQ ='V': on entry, Q must contain a unitary matrix Q1, and on exit this is overwritten by Q1*Q.
LDQ (input)
The leading dimension of the array Q. LDQ > = N if COMPQ ='V' or 'I'; LDQ > = 1 otherwise.
Z (input/output)
If COMPZ ='N': Z is not referenced.
If COMPZ ='I': on entry, Z need not be set, and on exit it contains the unitary matrix Z, which is the product of the Givens transformations which are applied to A and B on the right. If COMPZ ='V': on entry, Z must contain a unitary matrix Z1, and on exit this is overwritten by Z1*Z.
LDZ (input)
The leading dimension of the array Z. LDZ > = N if COMPZ ='V' or 'I'; LDZ > = 1 otherwise.

INFO (output)

 = 0:  successful exit.

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

FURTHER DETAILS

This routine reduces A to Hessenberg and B to triangular form by an unblocked reduction, as described in _Matrix_Computations_, by Golub and van Loan (Johns Hopkins Press).