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

NAME

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

SYNOPSIS

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

  SUBROUTINE SGGHRD_64( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, 
 *      LDQ, Z, LDZ, INFO)
  CHARACTER * 1 COMPQ, COMPZ
  INTEGER*8 N, ILO, IHI, LDA, LDB, LDQ, LDZ, INFO
  REAL A(LDA,*), B(LDB,*), Q(LDQ,*), Z(LDZ,*)

F95 INTERFACE

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

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

C INTERFACE

#include <sunperf.h>

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

void sgghrd_64(char compq, char compz, long n, long ilo, long ihi, float *a, long lda, float *b, long ldb, float *q, long ldq, float *z, long ldz, long *info);

PURPOSE

sgghrd reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal 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 orthogonal, and ' means transpose.

The orthogonal 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
  orthogonal matrix Q is returned;
   = 'V': Q must contain an orthogonal matrix Q1 on entry,
  and the product Q1*Q is returned.

• COMPZ (input)
  

   = 'N': do not compute Z;

   = 'I': Z is initialized to the unit matrix, and the
  orthogonal matrix Z is returned;
   = 'V': Z must contain an orthogonal matrix Z1 on entry,
  and the product Z1*Z 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 SGGBAL; 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 the 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 orthogonal 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 an orthogonal 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 orthogonal 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 an orthogonal 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.)