SGGHRD(1) LAPACK routine (version 3.2) SGGHRD(1)NAME
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
SYNOPSIS
SUBROUTINE SGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ,
Z, LDZ, INFO )
CHARACTER COMPQ, COMPZ
INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
REAL A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )
PURPOSE
SGGHRD 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. The form of the generalized eigen‐
value problem is
A*x = lambda*B*x,
and B is typically made upper triangular by computing its QR factoriza‐
tion and moving the orthogonal matrix Q to the left side of the equa‐
tion.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.
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
Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
If Q1 is the orthogonal matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then SGGHRD reduces the original
problem to generalized Hessenberg form.
ARGUMENTS
COMPQ (input) CHARACTER*1
= '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) CHARACTER*1
= '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) INTEGER
The order of the matrices A and B. N >= 0.
ILO (input) INTEGER
IHI (input) INTEGER ILO and IHI mark the rows and columns
of A which are to be reduced. 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; other‐
wise they should be set to 1 and N respectively. 1 <= ILO <=
IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.
A (input/output) REAL array, dimension (LDA, N)
On entry, the N-by-N general matrix to be reduced. On exit,
the upper triangle and the first subdiagonal of A are overwrit‐
ten with the upper Hessenberg matrix H, and the rest is set to
zero.
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) REAL array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B. On exit, the
upper triangular matrix T = Q**T B Z. The elements below the
diagonal are set to zero.
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q (input/output) REAL array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the orthogonal matrix Q1, typically
from the QR factorization of B. On exit, if COMPQ='I', the
orthogonal matrix Q, and if COMPQ = 'V', the product Q1*Q. Not
referenced if COMPQ='N'.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= N if COMPQ='V' or
'I'; LDQ >= 1 otherwise.
Z (input/output) REAL array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the orthogonal matrix Z1. On exit,
if COMPZ='I', the orthogonal matrix Z, and if COMPZ = 'V', the
product Z1*Z. Not referenced if COMPZ='N'.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= N if COMPZ='V' or
'I'; LDZ >= 1 otherwise.
INFO (output) INTEGER
= 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.)
LAPACK routine (version 3.2) November 2008 SGGHRD(1)
