CTZRQF(3)      LAPACK routine of NEC Numeric Library Collection      CTZRQF(3)



NAME
       CTZRQF

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



PURPOSE
            This routine is deprecated and has been replaced by routine CTZRZF.

            CTZRQF 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)
                     M is INTEGER
                     The number of rows of the matrix A.  M >= 0.

           N         (input)
                     N is INTEGER
                     The number of columns of the matrix A.  N >= M.

           A         (input/output)
                     A is COMPLEX 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
                     unitary matrix Z as a product of M elementary reflectors.

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

           TAU       (output)
                     TAU is COMPLEX array, dimension (M)
                     The scalar factors of the elementary reflectors.

           INFO      (output)
                     INFO is 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 )**H,   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 ).



LAPACK routine                  31 October 2017                      CTZRQF(3)