CGGHRD
November 2006
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.
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.
Arguments
| COMPQ |
(input) 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 |
(input) 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 |
(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 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 |
(input/output) 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 |
(input) INTEGER
The leading dimension of the array A. LDA >= max(1,N).
|
| B |
(input/output) 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 |
(input) INTEGER
The leading dimension of the array B. LDB >= max(1,N).
|
| Q |
(input/output) 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 |
(input) INTEGER
The leading dimension of the array Q.
LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. |
| Z |
(input/output) 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 |
(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).
an unblocked reduction, as described in _Matrix_Computations_,
by Golub and van Loan (Johns Hopkins Press).