SLATRZ 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.

M

          M is INTEGER


          The number of rows of the matrix A.  M >= 0.

N

          N is INTEGER


          The number of columns of the matrix A.  N >= 0.

L

          L is INTEGER


          The number of columns of the matrix A containing the


          meaningful part of the Householder vectors. N-M >= L >= 0.

A

          A is 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 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

          LDA is INTEGER


          The leading dimension of the array A.  LDA >= max(1,M).

TAU

          TAU is REAL array, dimension (M)


          The scalar factors of the elementary reflectors.

WORK

          WORK is REAL array, dimension (M)

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

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 )**T,   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 ).