cgghrd(3)

NAME

CGGHRD - reduce a pair of complex matrices (A,B) to gener

alized upper Hessenberg form using unitary transformations, where

A is a general matrix and B is upper triangular

SYNOPSIS

SUBROUTINE CGGHRD( 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         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
Z( LDZ, * )

PURPOSE

CGGHRD reduces a pair of complex matrices (A,B) to gener

alized upper Hessenberg form using unitary 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 tri

angular, and Q and Z are unitary, and ' means conjugate trans

pose.

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' = (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 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 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 CGGBAL; other

wise 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 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 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) COMPLEX 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 unitary 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 a unitary 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) COMPLEX 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 unitary 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 a unitary 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 otherwise.

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_Computa

tions_, by Golub and van Loan (Johns Hopkins Press).

LAPACK version 3.0 15 June 2000

CGGHRD(3)