NAME¶

DGGHRD - reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangular

SYNOPSIS¶

SUBROUTINE DGGHRD(

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 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * )

PURPOSE¶

DGGHRD reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal 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 orthogonal matrix Q to the left side of the equation.
This subroutine simultaneously reduces A to a Hessenberg matrix H:
Q**T*A*Z = H
and transforms B to another upper triangular matrix T:
Q**T*B*Z = T
in order to reduce the problem to its standard form
H*y = lambda*T*y
where y = Z**T*x.
The orthogonal 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**T = (Q1*Q) * H * (Z1*Z)**T
Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T
If Q1 is the orthogonal matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then DGGHRD reduces the original problem to generalized Hessenberg form.

ARGUMENTS¶

COMPQ (input) CHARACTER*1

= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = 'V': Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned.

COMPZ (input) CHARACTER*1

= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the orthogonal matrix Z is returned; = 'V': Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z 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 SGGBAL; 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) DOUBLE PRECISION 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) DOUBLE PRECISION array, dimension (LDB, N)

On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**T 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) DOUBLE PRECISION array, dimension (LDQ, N)

On entry, if COMPQ = 'V', the orthogonal matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the orthogonal matrix Q, and if COMPQ = 'V', the product Q1*Q. Not referenced if COMPQ='N'.

LDQ (input) INTEGER

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

Z (input/output) DOUBLE PRECISION array, dimension (LDZ, N)

On entry, if COMPZ = 'V', the orthogonal matrix Z1. On exit, if COMPZ='I', the orthogonal matrix Z, and if COMPZ = 'V', the product Z1*Z. Not referenced if COMPZ='N'.

LDZ (input) INTEGER

The leading dimension of the array Z. LDZ >= N if COMPZ='V' or 'I'; 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.)