cgghrd-f(3)

cgghrd.f –

Synopsis

Functions/Subroutines

subroutine cgghrd (COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO)
CGGHRD

Function/Subroutine Documentation

subroutine cgghrd (characterCOMPQ, characterCOMPZ, integerN, integerILO, integerIHI, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, complex, dimension( ldq, * )Q, integerLDQ, complex, dimension( ldz, * )Z, integerLDZ, integerINFO)

CGGHRD

Purpose:

 CGGHRD 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 CGGHRD 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 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 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 CGGBAL; 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 is 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

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

 B is COMPLEX 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 is COMPLEX 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 is COMPLEX 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.

Date:

November 2011

Further Details:

 This routine reduces A to Hessenberg and B to triangular form by
 an unblocked reduction, as described in <em>Matrix_Computations</em>,
 by Golub and van Loan (Johns Hopkins Press).

Definition at line 204 of file cgghrd.f.

Author

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