DGGHRD - reduce a pair of real matrices (A,B) to general
ized upper Hessenberg form using orthogonal transforma
tions, where A is a general matrix and B is upper triangu
lar
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 general
ized upper Hessenberg form using orthogonal transforma
tions, where A is a general matrix and B is upper triangu
lar: Q' * A * Z = H and Q' * B * Z = T, where H is upper
Hessenberg, T is upper triangular, and Q and Z are orthog
onal, and ' means transpose.
The orthogonal matrices Q and Z are determined as products
of Givens rotations. They may either be formed explic
itly, 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
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.
A (input/output) DOUBLE PRECISION array, dimension
(LDA, N)
On entry, the N-by-N general matrix to be reduced.
On exit, the upper triangle and the first subdiag
onal of A are overwritten with the upper Hessen
berg 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 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) 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 otherwise.
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
INFO (output) INTEGER
= 0: successful exit.
< 0: if INFO = -i, the i-th argument had an ille
gal 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 Hop
kins Press.)
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