# cgghrd.f - Man Page

SRC/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 (character compq, character compz, integer n, integer ilo, integer ihi, complex, dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ldq, * ) q, integer ldq, complex, dimension( ldz, * ) z, integer ldz, integer info)

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 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.```

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

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).```

Definition at line 202 of file cgghrd.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page cgghrd(3) is an alias of cgghrd.f(3).

Tue Nov 28 2023 12:08:41 Version 3.12.0 LAPACK