dgghrd(3P) Sun Performance Library dgghrd(3P)NAMEdgghrd - reduce a pair of real matrices (A,B) to generalized upper Hes‐
senberg form using orthogonal transformations, where A is a general
matrix and B is upper triangular
SYNOPSIS
SUBROUTINE DGGHRD(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
DOUBLE PRECISION A(LDA,*), B(LDB,*), Q(LDQ,*), Z(LDZ,*)
SUBROUTINE DGGHRD_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
DOUBLE PRECISION 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(8), 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(8), DIMENSION(:,:) :: A, B, Q, Z
C INTERFACE
#include <sunperf.h>
void dgghrd(char compq, char compz, int n, int ilo, int ihi, double *a,
int lda, double *b, int ldb, double *q, int ldq, double *z,
int ldz, int *info);
void dgghrd_64(char compq, char compz, long n, long ilo, long ihi, dou‐
ble *a, long lda, double *b, long ldb, double *q, long ldq,
double *z, long ldz, long *info);
PURPOSEdgghrd reduces a pair of real matrices (A,B) to generalized upper Hes‐
senberg 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 post‐
multiplied into input matrices Q1 and Z1, so that
1 * A * Z1' = (Q1*Q) * H * (Z1*Z)'
ARGUMENTS
COMPQ (input)
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the orthogo‐
nal 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 orthogo‐
nal 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 DGGBAL; 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 over‐
written 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 orthog‐
onal 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.)
6 Mar 2009 dgghrd(3P)
