.TH STZRQF 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME STZRQF - i deprecated and has been replaced by routine STZRZF .SH SYNOPSIS .TP 19 SUBROUTINE STZRQF( M, N, A, LDA, TAU, INFO ) .TP 19 .ti +4 INTEGER INFO, LDA, M, N .TP 19 .ti +4 REAL A( LDA, * ), TAU( * ) .SH PURPOSE This routine is deprecated and has been replaced by routine STZRZF. STZRQF reduces the M-by-N ( M<=N ) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations. The upper trapezoidal matrix A is factored as .br A = ( R 0 ) * Z, .br where Z is an N-by-N orthogonal matrix and R is an M-by-M upper triangular matrix. .br .SH ARGUMENTS .TP 8 M (input) INTEGER The number of rows of the matrix A. M >= 0. .TP 8 N (input) INTEGER The number of columns of the matrix A. N >= M. .TP 8 A (input/output) REAL array, dimension (LDA,N) On entry, the leading M-by-N upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading M-by-M upper triangular part of A contains the upper triangular matrix R, and elements M+1 to N of the first M rows of A, with the array TAU, represent the orthogonal matrix Z as a product of M elementary reflectors. .TP 8 LDA (input) INTEGER The leading dimension of the array A. LDA >= max(1,M). .TP 8 TAU (output) REAL array, dimension (M) The scalar factors of the elementary reflectors. .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .SH FURTHER DETAILS The factorization is obtained by Householder\(aqs method. The kth transformation matrix, Z( k ), which is used to introduce zeros into the ( m - k + 1 )th row of A, is given in the form .br Z( k ) = ( I 0 ), .br ( 0 T( k ) ) .br where .br T( k ) = I - tau*u( k )*u( k )\(aq, u( k ) = ( 1 ), ( 0 ) ( z( k ) ) tau is a scalar and z( k ) is an ( n - m ) element vector. tau and z( k ) are chosen to annihilate the elements of the kth row of X. .br The scalar tau is returned in the kth element of TAU and the vector u( k ) in the kth row of A, such that the elements of z( k ) are in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in the upper triangular part of A. .br Z is given by .br Z = Z( 1 ) * Z( 2 ) * ... * Z( m ). .br