SRC/zgghd3.f(3) Library Functions Manual SRC/zgghd3.f(3) NAME SRC/zgghd3.f SYNOPSIS Functions/Subroutines subroutine zgghd3 (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info) ZGGHD3 Function/Subroutine Documentation subroutine zgghd3 (character compq, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb, complex*16, dimension( ldq, * ) q, integer ldq, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, integer info) ZGGHD3 Purpose: !> !> ZGGHD3 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 ZGGHD3 reduces the original !> problem to generalized Hessenberg form. !> !> This is a blocked variant of CGGHRD, using matrix-matrix !> multiplications for parts of the computation to enhance performance. !> 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 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 !> A is 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 !> LDA is INTEGER !> The leading dimension of the array A. LDA >= max(1,N). !> B !> B is 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 !> LDB is INTEGER !> The leading dimension of the array B. LDB >= max(1,N). !> Q !> Q is 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 !> LDQ is INTEGER !> The leading dimension of the array Q. !> LDQ >= N if COMPQ='V' or 'I'; LDQ >= 1 otherwise. !> Z !> Z is 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 !> LDZ is INTEGER !> The leading dimension of the array Z. !> LDZ >= N if COMPZ='V' or 'I'; LDZ >= 1 otherwise. !> WORK !> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) !> On exit, if INFO = 0, WORK(1) returns the optimal LWORK. !> LWORK !> LWORK is INTEGER !> The length of the array WORK. LWORK >= 1. !> For optimum performance LWORK >= 6*N*NB, where NB is the !> optimal blocksize. !> !> If LWORK = -1, then a workspace query is assumed; the routine !> only calculates the optimal size of the WORK array, returns !> this value as the first entry of the WORK array, and no error !> message related to LWORK is issued by XERBLA. !> 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 Univ. of Colorado Denver NAG Ltd. Further Details: !> !> This routine reduces A to Hessenberg form and maintains B in triangular form !> using a blocked variant of Moler and Stewart's original algorithm, !> as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti !> (BIT 2008). !> Definition at line 225 of file zgghd3.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 SRC/zgghd3.f(3)