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dtzrqf

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

 SYNOPSIS
      SUBROUTINE DTZRQF( M, N, A, LDA, TAU, INFO )

          INTEGER        INFO, LDA, M, N

          DOUBLE         PRECISION A( LDA, * ), TAU( * )

 PURPOSE
      DTZRQF 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 >= M.

      A       (input/output) DOUBLE PRECISION array, dimension
              (LDA,max(1,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) DOUBLE PRECISION array, dimension (max(1,M))
              The scalar factors of the elementary reflectors.

      INFO    (output) INTEGER
              = 0:  successful exit
              < 0:  if INFO = -i, the i-th argument had an illegal

              value

 FURTHER DETAILS
      The factorization is obtained by Householder's method.  The
      kth transformation matrix, Z( k ), which is used to intro-
      duce 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 ).