SYNOPSIS
 SUBROUTINE STZRZF(
 M, N, A, LDA, TAU, WORK, LWORK, INFO )
 INTEGER INFO, LDA, LWORK, M, N
 REAL A( LDA, * ), TAU( * ), WORK( * )
PURPOSE
STZRZF reduces the MbyN ( M<=N ) real upper trapezoidal matrix A to upper triangular form by means of orthogonal transformations. The upper trapezoidal matrix A is factored asA = ( R 0 ) * Z,
where Z is an NbyN orthogonal matrix and R is an MbyM upper triangular matrix.
ARGUMENTS
 M (input) INTEGER
 The number of rows of the matrix A. M >= 0.
 N (input) INTEGER
 The number of columns of the matrix A. N >= M.
 A (input/output) REAL array, dimension (LDA,N)
 On entry, the leading MbyN upper trapezoidal part of the array A must contain the matrix to be factorized. On exit, the leading MbyM 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.
 LDA (input) INTEGER
 The leading dimension of the array A. LDA >= max(1,M).
 TAU (output) REAL array, dimension (M)
 The scalar factors of the elementary reflectors.
 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
 LWORK (input) INTEGER
 The dimension of the array WORK. LWORK >= max(1,M). For optimum performance LWORK >= M*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 (output) INTEGER

= 0: successful exit
< 0: if INFO = i, the ith argument had an illegal value
FURTHER DETAILS
Based on contributions byA. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA The factorization is obtained by Householder's 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
Z( k ) = ( I 0 ),
( 0 T( k ) )
where
T( k ) = I  tau*u( k )*u( k )', 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.
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.
Z is given by
Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).