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LAPACK-3

man page of LAPACK-3

LAPACK-3: reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is

```NAME
LAPACK-3  -  reduces  a  pair  of complex matrices (A,B) to generalized
upper Hessenberg form using  unitary  transformations,  where  A  is  a
general matrix and B is upper triangular

SYNOPSIS
SUBROUTINE ZGGHRD( 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

COMPLEX*16     A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )

PURPOSE
ZGGHRD reduces a pair of complex matrices (A,B)  to  generalized  upper
Hessenberg  form  using  unitary  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 unitary matrix Q to the left side
of the equation.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**H*A*Z = H
and transforms B to another upper triangular matrix T:
Q**H*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**H*x.
The unitary 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**H = (Q1*Q) * H * (Z1*Z)**H
Q1 * B * Z1**H = (Q1*Q) * T * (Z1*Z)**H
If Q1 is the unitary matrix from the QR factorization of B in the
original equation A*x = lambda*B*x, then ZGGHRD 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
unitary matrix Q is returned;
= 'V': Q must contain a unitary matrix Q1 on entry,
and the product Q1*Q is returned.

COMPZ   (input) CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
unitary matrix Q is returned;
= 'V': Q must contain a unitary matrix Q1 on entry,
and the product Q1*Q 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 ZGGBAL; 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) COMPLEX*16 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) COMPLEX*16 array, dimension (LDB, N)
On entry, the N-by-N upper triangular matrix B.
On exit, the upper triangular matrix T = Q**H 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) COMPLEX*16 array, dimension (LDQ, N)
On entry, if COMPQ = 'V', the unitary matrix Q1, typically
from the QR factorization of B.
On exit, if COMPQ='I', the unitary 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) COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = 'V', the unitary matrix Z1.
On exit, if COMPZ='I', the unitary 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)     March 2011                       LAPACK-3(3)
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