stzrzf(3S)

NAME
     STZRZF - reduce the M-by-N ( M<=N ) real upper trapezoidal matrix A to
     upper triangular form by means of orthogonal transformations

SYNOPSIS
     SUBROUTINE STZRZF( M, N, A, LDA, TAU, WORK, LWORK, INFO )

         INTEGER        INFO, LDA, LWORK, M, N

         REAL           A( LDA, * ), TAU( * ), WORK( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     STZRZF 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

        A = ( R  0 ) * Z,

     where Z is an N-by-N orthogonal matrix and R is an M-by-M 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 >= 0.

     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.

     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 (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 i-th argument had an illegal value

FURTHER DETAILS
     Based on contributions by
       A. 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 ).
SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.