dlatrz(3S)

NAME
     DLATRZ - factor the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] =
     [ A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means of orthogonal
     transformations

SYNOPSIS
     SUBROUTINE DLATRZ( M, N, L, A, LDA, TAU, WORK )

         INTEGER        L, LDA, M, N

         DOUBLE         PRECISION 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
     DLATRZ factors the M-by-(M+L) real upper trapezoidal matrix [ A1 A2 ] = [
     A(1:M,1:M) A(1:M,N-L+1:N) ] as ( R 0 ) * Z, by means of orthogonal
     transformations. Z is an (M+L)-by-(M+L) orthogonal matrix and, R and A1
     are M-by-M upper triangular matrices.


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.

     L       (input) INTEGER
             The number of columns of the matrix A containing the meaningful
             part of the Householder vectors. N-M >= L >= 0.

     A       (input/output) DOUBLE PRECISION 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 N-L+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) DOUBLE PRECISION array, dimension (M)
             The scalar factors of the elementary reflectors.

     WORK    (workspace) DOUBLE PRECISION array, dimension (M)

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 l element vector. tau and z( k ) are
     chosen to annihilate the elements of the kth row of A2.

     The scalar tau is returned in the kth element of TAU and the vector u( k
     ) in the kth row of A2, such that the elements of z( k ) are in  a( k, l
     + 1 ), ..., a( k, n ). The elements of R are returned in the upper
     triangular part of A1.

     Z is given by

        Z =  Z( 1 ) * Z( 2 ) * ... * Z( m ).


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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