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239 lines
6.8 KiB
239 lines
6.8 KiB
2 years ago
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*> \brief \b ZTZRQF
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*
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* =========== DOCUMENTATION ===========
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*
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* Online html documentation available at
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* http://www.netlib.org/lapack/explore-html/
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*
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*> \htmlonly
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*> Download ZTZRQF + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/ztzrqf.f">
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*> [TGZ]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/ztzrqf.f">
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*> [ZIP]</a>
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/ztzrqf.f">
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*> [TXT]</a>
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*> \endhtmlonly
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
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*
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* .. Scalar Arguments ..
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* INTEGER INFO, LDA, M, N
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* ..
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* .. Array Arguments ..
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* COMPLEX*16 A( LDA, * ), TAU( * )
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* ..
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*
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*
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*> \par Purpose:
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* =============
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*>
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*> \verbatim
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*>
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*> This routine is deprecated and has been replaced by routine ZTZRZF.
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*>
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*> ZTZRQF reduces the M-by-N ( M<=N ) complex upper trapezoidal matrix A
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*> to upper triangular form by means of unitary transformations.
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*>
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*> The upper trapezoidal matrix A is factored as
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*>
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*> A = ( R 0 ) * Z,
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*>
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*> where Z is an N-by-N unitary matrix and R is an M-by-M upper
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*> triangular matrix.
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*> \endverbatim
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*
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* Arguments:
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* ==========
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*
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*> \param[in] M
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*> \verbatim
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*> M is INTEGER
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*> The number of rows of the matrix A. M >= 0.
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*> \endverbatim
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*>
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*> \param[in] N
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*> \verbatim
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*> N is INTEGER
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*> The number of columns of the matrix A. N >= M.
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*> \endverbatim
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*>
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*> \param[in,out] A
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*> \verbatim
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*> A is COMPLEX*16 array, dimension (LDA,N)
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*> On entry, the leading M-by-N upper trapezoidal part of the
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*> array A must contain the matrix to be factorized.
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*> On exit, the leading M-by-M upper triangular part of A
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*> contains the upper triangular matrix R, and elements M+1 to
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*> N of the first M rows of A, with the array TAU, represent the
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*> unitary matrix Z as a product of M elementary reflectors.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,M).
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*> \endverbatim
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*>
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*> \param[out] TAU
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*> \verbatim
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*> TAU is COMPLEX*16 array, dimension (M)
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*> The scalar factors of the elementary reflectors.
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*> \endverbatim
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*>
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*> \param[out] INFO
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*> \verbatim
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*> INFO is INTEGER
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*> = 0: successful exit
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*> < 0: if INFO = -i, the i-th argument had an illegal value
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*> \endverbatim
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*
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* Authors:
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* ========
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*
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*> \author Univ. of Tennessee
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*> \author Univ. of California Berkeley
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*> \author Univ. of Colorado Denver
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*> \author NAG Ltd.
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*
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*> \ingroup complex16OTHERcomputational
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*
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*> \par Further Details:
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* =====================
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*>
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*> \verbatim
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*>
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*> The factorization is obtained by Householder's method. The kth
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*> transformation matrix, Z( k ), whose conjugate transpose is used to
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*> introduce zeros into the (m - k + 1)th row of A, is given in the form
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*>
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*> Z( k ) = ( I 0 ),
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*> ( 0 T( k ) )
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*>
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*> where
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*>
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*> T( k ) = I - tau*u( k )*u( k )**H, u( k ) = ( 1 ),
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*> ( 0 )
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*> ( z( k ) )
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*>
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*> tau is a scalar and z( k ) is an ( n - m ) element vector.
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*> tau and z( k ) are chosen to annihilate the elements of the kth row
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*> of X.
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*>
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*> The scalar tau is returned in the kth element of TAU and the vector
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*> u( k ) in the kth row of A, such that the elements of z( k ) are
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*> in a( k, m + 1 ), ..., a( k, n ). The elements of R are returned in
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*> the upper triangular part of A.
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*>
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*> Z is given by
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*>
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*> Z = Z( 1 ) * Z( 2 ) * ... * Z( m ).
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*> \endverbatim
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*>
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* =====================================================================
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SUBROUTINE ZTZRQF( M, N, A, LDA, TAU, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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*
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* .. Scalar Arguments ..
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INTEGER INFO, LDA, M, N
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* ..
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* .. Array Arguments ..
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COMPLEX*16 A( LDA, * ), TAU( * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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COMPLEX*16 CONE, CZERO
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PARAMETER ( CONE = ( 1.0D+0, 0.0D+0 ),
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$ CZERO = ( 0.0D+0, 0.0D+0 ) )
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* ..
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* .. Local Scalars ..
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INTEGER I, K, M1
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COMPLEX*16 ALPHA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC DCONJG, MAX, MIN
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* ..
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* .. External Subroutines ..
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EXTERNAL XERBLA, ZAXPY, ZCOPY, ZGEMV, ZGERC, ZLACGV,
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$ ZLARFG
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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IF( M.LT.0 ) THEN
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INFO = -1
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ELSE IF( N.LT.M ) THEN
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INFO = -2
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ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
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INFO = -4
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'ZTZRQF', -INFO )
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RETURN
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END IF
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*
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* Perform the factorization.
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*
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IF( M.EQ.0 )
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$ RETURN
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IF( M.EQ.N ) THEN
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DO 10 I = 1, N
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TAU( I ) = CZERO
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10 CONTINUE
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ELSE
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M1 = MIN( M+1, N )
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DO 20 K = M, 1, -1
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*
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* Use a Householder reflection to zero the kth row of A.
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* First set up the reflection.
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*
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A( K, K ) = DCONJG( A( K, K ) )
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CALL ZLACGV( N-M, A( K, M1 ), LDA )
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ALPHA = A( K, K )
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CALL ZLARFG( N-M+1, ALPHA, A( K, M1 ), LDA, TAU( K ) )
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A( K, K ) = ALPHA
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TAU( K ) = DCONJG( TAU( K ) )
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*
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IF( TAU( K ).NE.CZERO .AND. K.GT.1 ) THEN
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*
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* We now perform the operation A := A*P( k )**H.
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*
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* Use the first ( k - 1 ) elements of TAU to store a( k ),
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* where a( k ) consists of the first ( k - 1 ) elements of
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* the kth column of A. Also let B denote the first
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* ( k - 1 ) rows of the last ( n - m ) columns of A.
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*
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CALL ZCOPY( K-1, A( 1, K ), 1, TAU, 1 )
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*
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* Form w = a( k ) + B*z( k ) in TAU.
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*
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CALL ZGEMV( 'No transpose', K-1, N-M, CONE, A( 1, M1 ),
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$ LDA, A( K, M1 ), LDA, CONE, TAU, 1 )
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*
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* Now form a( k ) := a( k ) - conjg(tau)*w
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* and B := B - conjg(tau)*w*z( k )**H.
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*
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CALL ZAXPY( K-1, -DCONJG( TAU( K ) ), TAU, 1, A( 1, K ),
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$ 1 )
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CALL ZGERC( K-1, N-M, -DCONJG( TAU( K ) ), TAU, 1,
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$ A( K, M1 ), LDA, A( 1, M1 ), LDA )
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END IF
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20 CONTINUE
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END IF
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*
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RETURN
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*
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* End of ZTZRQF
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*
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END
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