This routine is deprecated and has been replaced by routine DTZRZF.


 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.

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 >= M.

A

          A is 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

          LDA is INTEGER


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

TAU

          TAU is DOUBLE PRECISION array, dimension (M)


          The scalar factors of the elementary reflectors.

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

November 2011

  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 ( 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 ).