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## man page of LAPACK-3

### LAPACK-3: reduces a pair of complex matrices (A,B) to generalized upper Hessenberg form using unitary transformations, where A is

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## NAME

LAPACK-3 - 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). LAPACK routine (version 3.2) March 2011 LAPACK-3(3)

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