zgghrd()

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

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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

[in]	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. !>
[in]	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. !>
[in]	N	!> N is INTEGER !> The order of the matrices A and B. N >= 0. !>
[in]	ILO	!> ILO is INTEGER !>
[in]	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. !>
[in,out]	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. !>
[in]	LDA	!> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !>
[in,out]	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. !>
[in]	LDB	!> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !>
[in,out]	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'. !>
[in]	LDQ	!> LDQ is INTEGER !> The leading dimension of the array Q. !> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. !>
[in,out]	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'. !>
[in]	LDZ	!> LDZ is INTEGER !> The leading dimension of the array Z. !> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. !>
[out]	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).
!>