Provided by: liblapack-doc_3.3.1-1_all NAME

LAPACK-3  -  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  Hessenberg  form  using
orthogonal  transformations,  where  A is a general matrix and B is upper triangular.  The
form of the
generalized eigenvalue problem is
A*x = lambda*B*x,
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.
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
postmultiplied 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; 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) 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.

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.

FURTHERDETAILS

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)               April 2011                            SGGHRD(3lapack)