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*> \brief \b DLARFT forms the triangular factor T of a block reflector H = I - vtvH
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DLARFT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarft.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dlarft.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dlarft.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
* INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
* DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DLARFT forms the triangular factor T of a real block reflector H
*> of order n, which is defined as a product of k elementary reflectors.
*>
*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular;
*>
*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular.
*>
*> If STOREV = 'C', the vector which defines the elementary reflector
*> H(i) is stored in the i-th column of the array V, and
*>
*> H = I - V * T * V**T
*>
*> If STOREV = 'R', the vector which defines the elementary reflector
*> H(i) is stored in the i-th row of the array V, and
*>
*> H = I - V**T * T * V
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] DIRECT
*> \verbatim
*> DIRECT is CHARACTER*1
*> Specifies the order in which the elementary reflectors are
*> multiplied to form the block reflector:
*> = 'F': H = H(1) H(2) . . . H(k) (Forward)
*> = 'B': H = H(k) . . . H(2) H(1) (Backward)
*> \endverbatim
*>
*> \param[in] STOREV
*> \verbatim
*> STOREV is CHARACTER*1
*> Specifies how the vectors which define the elementary
*> reflectors are stored (see also Further Details):
*> = 'C': columnwise
*> = 'R': rowwise
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the block reflector H. N >= 0.
*> \endverbatim
*>
*> \param[in] K
*> \verbatim
*> K is INTEGER
*> The order of the triangular factor T (= the number of
*> elementary reflectors). K >= 1.
*> \endverbatim
*>
*> \param[in] V
*> \verbatim
*> V is DOUBLE PRECISION array, dimension
*> (LDV,K) if STOREV = 'C'
*> (LDV,N) if STOREV = 'R'
*> The matrix V. See further details.
*> \endverbatim
*>
*> \param[in] LDV
*> \verbatim
*> LDV is INTEGER
*> The leading dimension of the array V.
*> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.
*> \endverbatim
*>
*> \param[in] TAU
*> \verbatim
*> TAU is DOUBLE PRECISION array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is DOUBLE PRECISION array, dimension (LDT,K)
*> The k by k triangular factor T of the block reflector.
*> If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is
*> lower triangular. The rest of the array is not used.
*> \endverbatim
*>
*> \param[in] LDT
*> \verbatim
*> LDT is INTEGER
*> The leading dimension of the array T. LDT >= K.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date December 2016
*
*> \ingroup doubleOTHERauxiliary
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> The shape of the matrix V and the storage of the vectors which define
*> the H(i) is best illustrated by the following example with n = 5 and
*> k = 3. The elements equal to 1 are not stored.
*>
*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R':
*>
*> V = ( 1 ) V = ( 1 v1 v1 v1 v1 )
*> ( v1 1 ) ( 1 v2 v2 v2 )
*> ( v1 v2 1 ) ( 1 v3 v3 )
*> ( v1 v2 v3 )
*> ( v1 v2 v3 )
*>
*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R':
*>
*> V = ( v1 v2 v3 ) V = ( v1 v1 1 )
*> ( v1 v2 v3 ) ( v2 v2 v2 1 )
*> ( 1 v2 v3 ) ( v3 v3 v3 v3 1 )
*> ( 1 v3 )
*> ( 1 )
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* -- LAPACK auxiliary 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 ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
DOUBLE PRECISION T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ONE, ZERO
PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
* .. External Subroutines ..
EXTERNAL DGEMV, DTRMV
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
PREVLASTV = N
DO I = 1, K
PREVLASTV = MAX( I, PREVLASTV )
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO J = 1, I
T( J, I ) = ZERO
END DO
ELSE
*
* general case
*
IF( LSAME( STOREV, 'C' ) ) THEN
* Skip any trailing zeros.
DO LASTV = N, I+1, -1
IF( V( LASTV, I ).NE.ZERO ) EXIT
END DO
DO J = 1, I-1
T( J, I ) = -TAU( I ) * V( I , J )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**T * V(i:j,i)
*
CALL DGEMV( 'Transpose', J-I, I-1, -TAU( I ),
$ V( I+1, 1 ), LDV, V( I+1, I ), 1, ONE,
$ T( 1, I ), 1 )
ELSE
* Skip any trailing zeros.
DO LASTV = N, I+1, -1
IF( V( I, LASTV ).NE.ZERO ) EXIT
END DO
DO J = 1, I-1
T( J, I ) = -TAU( I ) * V( J , I )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**T
*
CALL DGEMV( 'No transpose', I-1, J-I, -TAU( I ),
$ V( 1, I+1 ), LDV, V( I, I+1 ), LDV, ONE,
$ T( 1, I ), 1 )
END IF
*
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
CALL DTRMV( 'Upper', 'No transpose', 'Non-unit', I-1, T,
$ LDT, T( 1, I ), 1 )
T( I, I ) = TAU( I )
IF( I.GT.1 ) THEN
PREVLASTV = MAX( PREVLASTV, LASTV )
ELSE
PREVLASTV = LASTV
END IF
END IF
END DO
ELSE
PREVLASTV = 1
DO I = K, 1, -1
IF( TAU( I ).EQ.ZERO ) THEN
*
* H(i) = I
*
DO J = I, K
T( J, I ) = ZERO
END DO
ELSE
*
* general case
*
IF( I.LT.K ) THEN
IF( LSAME( STOREV, 'C' ) ) THEN
* Skip any leading zeros.
DO LASTV = 1, I-1
IF( V( LASTV, I ).NE.ZERO ) EXIT
END DO
DO J = I+1, K
T( J, I ) = -TAU( I ) * V( N-K+I , J )
END DO
J = MAX( LASTV, PREVLASTV )
*
* T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**T * V(j:n-k+i,i)
*
CALL DGEMV( 'Transpose', N-K+I-J, K-I, -TAU( I ),
$ V( J, I+1 ), LDV, V( J, I ), 1, ONE,
$ T( I+1, I ), 1 )
ELSE
* Skip any leading zeros.
DO LASTV = 1, I-1
IF( V( I, LASTV ).NE.ZERO ) EXIT
END DO
DO J = I+1, K
T( J, I ) = -TAU( I ) * V( J, N-K+I )
END DO
J = MAX( LASTV, PREVLASTV )
*
* T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**T
*
CALL DGEMV( 'No transpose', K-I, N-K+I-J,
$ -TAU( I ), V( I+1, J ), LDV, V( I, J ), LDV,
$ ONE, T( I+1, I ), 1 )
END IF
*
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL DTRMV( 'Lower', 'No transpose', 'Non-unit', K-I,
$ T( I+1, I+1 ), LDT, T( I+1, I ), 1 )
IF( I.GT.1 ) THEN
PREVLASTV = MIN( PREVLASTV, LASTV )
ELSE
PREVLASTV = LASTV
END IF
END IF
T( I, I ) = TAU( I )
END IF
END DO
END IF
RETURN
*
* End of DLARFT
*
END
|
