.TH SGGHRD 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME SGGHRD - a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangular .SH SYNOPSIS .TP 19 SUBROUTINE SGGHRD( COMPQ, COMPZ, N, ILO, IHI, A, LDA, B, LDB, Q, LDQ, Z, LDZ, INFO ) .TP 19 .ti +4 CHARACTER COMPQ, COMPZ .TP 19 .ti +4 INTEGER IHI, ILO, INFO, LDA, LDB, LDQ, LDZ, N .TP 19 .ti +4 REAL A( LDA, * ), B( LDB, * ), Q( LDQ, * ), Z( LDZ, * ) .SH PURPOSE SGGHRD reduces a pair of real matrices (A,B) to generalized upper Hessenberg form using orthogonal transformations, where A is a general matrix and B is upper triangular. The form of the generalized eigenvalue problem is .br A*x = lambda*B*x, .br and B is typically made upper triangular by computing its QR factorization and moving the orthogonal matrix Q to the left side of the equation. .br This subroutine simultaneously reduces A to a Hessenberg matrix H: Q**T*A*Z = H .br and transforms B to another upper triangular matrix T: .br Q**T*B*Z = T .br in order to reduce the problem to its standard form .br H*y = lambda*T*y .br where y = Z**T*x. .br The orthogonal 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 .br Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T .br Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T .br If Q1 is the orthogonal matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then SGGHRD reduces the original problem to generalized Hessenberg form. .br .SH ARGUMENTS .TP 8 COMPQ (input) CHARACTER*1 = \(aqN\(aq: do not compute Q; .br = \(aqI\(aq: Q is initialized to the unit matrix, and the orthogonal matrix Q is returned; = \(aqV\(aq: Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned. .TP 8 COMPZ (input) CHARACTER*1 = \(aqN\(aq: do not compute Z; .br = \(aqI\(aq: Z is initialized to the unit matrix, and the orthogonal matrix Z is returned; = \(aqV\(aq: Z must contain an orthogonal matrix Z1 on entry, and the product Z1*Z is returned. .TP 8 N (input) INTEGER The order of the matrices A and B. N >= 0. .TP 8 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 SGGBAL; 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. .TP 8 A (input/output) REAL 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. .TP 8 LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). .TP 8 B (input/output) REAL array, dimension (LDB, N) On entry, the N-by-N upper triangular matrix B. On exit, the upper triangular matrix T = Q**T B Z. The elements below the diagonal are set to zero. .TP 8 LDB (input) INTEGER The leading dimension of the array B. LDB >= max(1,N). .TP 8 Q (input/output) REAL array, dimension (LDQ, N) On entry, if COMPQ = \(aqV\(aq, the orthogonal matrix Q1, typically from the QR factorization of B. On exit, if COMPQ=\(aqI\(aq, the orthogonal matrix Q, and if COMPQ = \(aqV\(aq, the product Q1*Q. Not referenced if COMPQ=\(aqN\(aq. .TP 8 LDQ (input) INTEGER The leading dimension of the array Q. LDQ >= N if COMPQ=\(aqV\(aq or \(aqI\(aq; LDQ >= 1 otherwise. .TP 8 Z (input/output) REAL array, dimension (LDZ, N) On entry, if COMPZ = \(aqV\(aq, the orthogonal matrix Z1. On exit, if COMPZ=\(aqI\(aq, the orthogonal matrix Z, and if COMPZ = \(aqV\(aq, the product Z1*Z. Not referenced if COMPZ=\(aqN\(aq. .TP 8 LDZ (input) INTEGER The leading dimension of the array Z. LDZ >= N if COMPZ=\(aqV\(aq or \(aqI\(aq; LDZ >= 1 otherwise. .TP 8 INFO (output) INTEGER = 0: successful exit. .br < 0: if INFO = -i, the i-th argument had an illegal value. .SH 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.) .br