DTZRQF (3lapack)

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,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 (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 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 ).