*> \brief \b ZTZRQF
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
*
* .. Scalar Arguments ..
* INTEGER INFO, LDA, M, N
* ..
* .. Array Arguments ..
* COMPLEX*16 A( LDA, * ), TAU( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> This routine is deprecated and has been replaced by routine ZTZRZF.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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows of the matrix A. M >= 0.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The number of columns of the matrix A. N >= M.
*> \endverbatim
*>
*> \param[in,out] A
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,M).
*> \endverbatim
*>
*> \param[out] TAU
*> \verbatim
*> TAU is COMPLEX*16 array, dimension (M)
*> The scalar factors of the elementary reflectors.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup complex16OTHERcomputational
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 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 ).
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
*
* -- 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
*
END