# zgghrd.f - Man Page

SRC/zgghrd.f

## Synopsis

### Functions/Subroutines

subroutine **zgghrd** (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, info)**ZGGHRD**

## Function/Subroutine Documentation

### subroutine zgghrd (character compq, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, integer info)

**ZGGHRD**

**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.

**Parameters***COMPQ*COMPQ is 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*COMPZ is CHARACTER*1 = 'N': do not compute Z; = 'I': Z is initialized to the unit matrix, and the unitary matrix Z is returned; = 'V': Z must contain a unitary matrix Z1 on entry, and the product Z1*Z is returned.

*N*N is INTEGER The order of the matrices A and B. N >= 0.

*ILO*ILO is INTEGER

*IHI*IHI is 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*A is 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*LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*B*B is 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*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*Q*Q is 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*LDQ is INTEGER The leading dimension of the array Q. LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise.

*Z*Z is 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*LDZ is INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise.

*INFO*INFO is INTEGER = 0: successful exit. < 0: if INFO = -i, the i-th argument had an illegal value.

**Author**Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**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).

Definition at line **202** of file **zgghrd.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page zgghrd(3) is an alias of zgghrd.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK