cgghrd(3)

NAME

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

          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

          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

          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

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

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

 

Definition at line 204 of file cgghrd.f.

Author

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