# sgghd3.f - Man Page

SRC/sgghd3.f

## Synopsis

### Functions/Subroutines

subroutine **sgghd3** (compq, compz, n, ilo, ihi, a, lda, b, ldb, q, ldq, z, ldz, work, lwork, info)**SGGHD3**

## Function/Subroutine Documentation

### subroutine sgghd3 (character compq, character compz, integer n, integer ilo, integer ihi, real, dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b, integer ldb, real, dimension( ldq, * ) q, integer ldq, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)

**SGGHD3**

**Purpose:**

SGGHD3 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 A*x = lambda*B*x, 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. This subroutine simultaneously reduces A to a Hessenberg matrix H: Q**T*A*Z = H and transforms B to another upper triangular matrix T: Q**T*B*Z = T in order to reduce the problem to its standard form H*y = lambda*T*y where y = Z**T*x. 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 Q1 * A * Z1**T = (Q1*Q) * H * (Z1*Z)**T Q1 * B * Z1**T = (Q1*Q) * T * (Z1*Z)**T If Q1 is the orthogonal matrix from the QR factorization of B in the original equation A*x = lambda*B*x, then SGGHD3 reduces the original problem to generalized Hessenberg form. This is a blocked variant of SGGHRD, 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 orthogonal matrix Q is returned; = 'V': Q must contain an orthogonal 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 orthogonal matrix Z is returned; = 'V': Z must contain an orthogonal 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 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.

*A*A is 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.

*LDA*LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).

*B*B is 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.

*LDB*LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).

*Q*Q is REAL array, dimension (LDQ, N) On entry, if COMPQ = 'V', the orthogonal matrix Q1, typically from the QR factorization of B. On exit, if COMPQ='I', the orthogonal 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 REAL array, dimension (LDZ, N) On entry, if COMPZ = 'V', the orthogonal matrix Z1. On exit, if COMPZ='I', the orthogonal 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 REAL array, dimension (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 **228** of file **sgghd3.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page sgghd3(3) is an alias of sgghd3.f(3).

Tue Nov 28 2023 12:08:42 Version 3.12.0 LAPACK