---
name: "zgghrd"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/zgghrd/3lapack"
---

# zgghrd(3lapack)

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

## NAME

ZGGHRD - 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

## SYNOPSIS

- SUBROUTINE ZGGHRD(

  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

- COMPLEX\*16

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

## PURPOSE

ZGGHRD 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 ZGGHRD 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 ZGGBAL; 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\*16 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\*16 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\*16 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\*16 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).\