# zgghrd (l) - Linux Manuals

## NAME

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

## 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 Z1**H (Q1*Q) (Z1*Z)**H

Q1 Z1**H (Q1*Q) (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
= aqNaq: do not compute Q;
= aqIaq: Q is initialized to the unit matrix, and the unitary matrix Q is returned; = aqVaq: Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned.
COMPZ (input) CHARACTER*1
= aqNaq: do not compute Q;
= aqIaq: Q is initialized to the unit matrix, and the unitary matrix Q is returned; = aqVaq: 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 = aqVaq, the unitary matrix Q1, typically from the QR factorization of B. On exit, if COMPQ=aqIaq, the unitary matrix Q, and if COMPQ = aqVaq, the product Q1*Q. Not referenced if COMPQ=aqNaq.
LDQ (input) INTEGER
The leading dimension of the array Q. LDQ >= N if COMPQ=aqVaq or aqIaq; LDQ >= 1 otherwise.
Z (input/output) COMPLEX*16 array, dimension (LDZ, N)
On entry, if COMPZ = aqVaq, the unitary matrix Z1. On exit, if COMPZ=aqIaq, the unitary matrix Z, and if COMPZ = aqVaq, the product Z1*Z. Not referenced if COMPZ=aqNaq.
LDZ (input) INTEGER
The leading dimension of the array Z. LDZ >= N if COMPZ=aqVaq or aqIaq; 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).