ztzrqf()

subroutine ztzrqf	(	integer	M,
		integer	N,
		complex*16, dimension( lda, * )	A,
		integer	LDA,
		complex*16, dimension( * )	TAU,
		integer	INFO
	)

ZTZRQF

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Purpose:

 This routine is deprecated and has been replaced by routine ZTZRZF.

 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.

Parameters

[in]	M	M is INTEGER The number of rows of the matrix A. M >= 0.
[in]	N	N is INTEGER The number of columns of the matrix A. N >= M.
[in,out]	A	A is COMPLEX*16 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.
[in]	LDA	LDA is INTEGER The leading dimension of the array A. LDA >= max(1,M).
[out]	TAU	TAU is COMPLEX*16 array, dimension (M) The scalar factors of the elementary reflectors.
[out]	INFO	INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date

December 2016

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 ).

Definition at line 140 of file ztzrqf.f.

*
*  -- LAPACK computational routine (version 3.7.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     December 2016
*
*     .. Scalar Arguments ..
      INTEGER            INFO, LDA, M, N
*     ..
*     .. Array Arguments ..
      COMPLEX*16         A( LDA, * ), TAU( * )
*     ..
*
* =====================================================================
*
*     .. Parameters ..
      COMPLEX*16         CONE, CZERO
      parameter( cone = ( 1.0d+0, 0.0d+0 ),
     $                   czero = ( 0.0d+0, 0.0d+0 ) )
*     ..
*     .. Local Scalars ..
      INTEGER            I, K, M1
      COMPLEX*16         ALPHA
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          dconjg, max, min
*     ..
*     .. External Subroutines ..
      EXTERNAL           xerbla, zaxpy, zcopy, zgemv, zgerc, zlacgv,
     $                   zlarfg
*     ..
*     .. Executable Statements ..
*
*     Test the input parameters.
*
      info = 0
      IF( m.LT.0 ) THEN
         info = -1
      ELSE IF( n.LT.m ) THEN
         info = -2
      ELSE IF( lda.LT.max( 1, m ) ) THEN
         info = -4
      END IF
      IF( info.NE.0 ) THEN
         CALL xerbla( 'ZTZRQF', -info )
         RETURN
      END IF
*
*     Perform the factorization.
*
      IF( m.EQ.0 )
     $   RETURN
      IF( m.EQ.n ) THEN
         DO 10 i = 1, n
            tau( i ) = czero
   10    CONTINUE
      ELSE
         m1 = min( m+1, n )
         DO 20 k = m, 1, -1
*
*           Use a Householder reflection to zero the kth row of A.
*           First set up the reflection.
*
            a( k, k ) = dconjg( a( k, k ) )
            CALL zlacgv( n-m, a( k, m1 ), lda )
            alpha = a( k, k )
            CALL zlarfg( n-m+1, alpha, a( k, m1 ), lda, tau( k ) )
            a( k, k ) = alpha
            tau( k ) = dconjg( tau( k ) )
*
            IF( tau( k ).NE.czero .AND. k.GT.1 ) THEN
*
*              We now perform the operation  A := A*P( k )**H.
*
*              Use the first ( k - 1 ) elements of TAU to store  a( k ),
*              where  a( k ) consists of the first ( k - 1 ) elements of
*              the  kth column  of  A.  Also  let  B  denote  the  first
*              ( k - 1 ) rows of the last ( n - m ) columns of A.
*
               CALL zcopy( k-1, a( 1, k ), 1, tau, 1 )
*
*              Form   w = a( k ) + B*z( k )  in TAU.
*
               CALL zgemv( 'No transpose', k-1, n-m, cone, a( 1, m1 ),
     $                     lda, a( k, m1 ), lda, cone, tau, 1 )
*
*              Now form  a( k ) := a( k ) - conjg(tau)*w
*              and       B      := B      - conjg(tau)*w*z( k )**H.
*
               CALL zaxpy( k-1, -dconjg( tau( k ) ), tau, 1, a( 1, k ),
     $                     1 )
               CALL zgerc( k-1, n-m, -dconjg( tau( k ) ), tau, 1,
     $                     a( k, m1 ), lda, a( 1, m1 ), lda )
            END IF
   20    CONTINUE
      END IF
*
      RETURN
*
*     End of ZTZRQF
*

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