.TH SGGHRD 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) "
.SH NAME
SGGHRD - 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
.SH SYNOPSIS
.TP 19
SUBROUTINE SGGHRD(
COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q,
LDQ, Z, LDZ, INFO )
.TP 19
.ti +4
CHARACTER
COMPQ, COMPZ
.TP 19
.ti +4
INTEGER
IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N
.TP 19
.ti +4
REAL
A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
Z( LDZ, * )
.SH 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. The form of the
generalized eigenvalue problem is
.br
A*x = lambda*B*x,
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and B is typically made upper triangular by computing its QR
factorization and moving the orthogonal matrix Q to the left side
of the equation.
.br
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
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and transforms B to another upper triangular matrix T:
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Q**T*B*Z = T
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in order to reduce the problem to its standard form
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H*y = lambda*T*y
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where y = Z**T*x.
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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
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Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T
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Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
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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.
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.SH ARGUMENTS
.TP 8
COMPQ (input) CHARACTER*1
= \(aqN\(aq: do not compute Q;
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= \(aqI\(aq: Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= \(aqV\(aq: Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
.TP 8
COMPZ (input) CHARACTER*1
= \(aqN\(aq: do not compute Z;
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= \(aqI\(aq: Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= \(aqV\(aq: Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
.TP 8
N (input) INTEGER
The order of the matrices A and B. N >= 0.
.TP 8
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; 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.
.TP 8
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 overwritten with the upper Hessenberg matrix H, and the
rest is set to zero.
.TP 8
LDA (input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
.TP 8
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.
.TP 8
LDB (input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
.TP 8
Q (input/output) REAL array, dimension (LDQ, N)
On entry, if COMPQ = \(aqV\(aq, the orthogonal matrix Q1,
typically from the QR factorization of B.
On exit, if COMPQ=\(aqI\(aq, the orthogonal matrix Q, and if
COMPQ = \(aqV\(aq, the product Q1*Q.
Not referenced if COMPQ=\(aqN\(aq.
.TP 8
LDQ (input) INTEGER
The leading dimension of the array Q.
LDQ >= N if COMPQ=\(aqV\(aq or \(aqI\(aq; LDQ >= 1 otherwise.
.TP 8
Z (input/output) REAL array, dimension (LDZ, N)
On entry, if COMPZ = \(aqV\(aq, the orthogonal matrix Z1.
On exit, if COMPZ=\(aqI\(aq, the orthogonal matrix Z, and if
COMPZ = \(aqV\(aq, the product Z1*Z.
Not referenced if COMPZ=\(aqN\(aq.
.TP 8
LDZ (input) INTEGER
The leading dimension of the array Z.
LDZ >= N if COMPZ=\(aqV\(aq or \(aqI\(aq; LDZ >= 1 otherwise.
.TP 8
INFO (output) INTEGER
= 0: successful exit.
.br
< 0: if INFO = -i, the i-th argument had an illegal value.
.SH 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.)
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