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ztzrqf

 NAME
      ZTZRQF - reduce the M-by-N ( M<=N ) complex upper tra-
      pezoidal matrix A to upper triangular form by means of uni-
      tary transformations

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

          INTEGER        INFO, LDA, M, N

          COMPLEX*16     A( LDA, * ), TAU( * )

 PURPOSE
      ZTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal
      matrix A to upper triangular form by means of unitary
      transformations.

      The upper trapezoidal matrix A is factored as

         A = ( R  0 ) * Z,

      where Z is an N-by-N unitary 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) COMPLEX*16 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 factor-
              ized.  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 unitary 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) COMPLEX*16 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 ), whose conjugate transpose
      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 ).