
DGGHRD(l) ) DGGHRD(l)
NAME
DGGHRD  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 DGGHRD( 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
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )
PURPOSE
DGGHRD 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
Q1 * A * Z1' = (Q1*Q) * H * (Z1*Z)'
Q1 * B * Z1' = (Q1*Q) * T * (Z1*Z)'
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 It is assumed that A is already upper triangular in rows
and columns 1:ILO1 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.
A (input/output) DOUBLE PRECISION array, dimension (LDA, N)
On entry, the NbyN 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) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
B (input/output) DOUBLE PRECISION array, dimension (LDB, N)
On entry, the NbyN 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) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ, N)
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) INTEGER
The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 oth
erwise.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)
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) INTEGER
The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 oth
erwise.
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = i, the ith 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 version 3.0 15 June 2000 DGGHRD(l) 
