mirror of
https://gitlab.com/libeigen/eigen.git
synced 2024-12-21 07:19:46 +08:00
327 lines
9.9 KiB
FortranFixed
327 lines
9.9 KiB
FortranFixed
|
*> \brief \b SLARFT
|
||
|
|
*
|
||
|
|
* =========== DOCUMENTATION ===========
|
||
|
|
*
|
||
|
|
* Online html documentation available at
|
||
|
|
* http://www.netlib.org/lapack/explore-html/
|
||
|
|
*
|
||
|
|
*> \htmlonly
|
||
|
|
*> Download SLARFT + dependencies
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/slarft.f">
|
||
|
|
*> [TGZ]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/slarft.f">
|
||
|
|
*> [ZIP]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/slarft.f">
|
||
|
|
*> [TXT]</a>
|
||
|
|
*> \endhtmlonly
|
||
|
|
*
|
||
|
|
* Definition:
|
||
|
|
* ===========
|
||
|
|
*
|
||
|
|
* SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
* CHARACTER DIRECT, STOREV
|
||
|
|
* INTEGER K, LDT, LDV, N
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
* REAL T( LDT, * ), TAU( * ), V( LDV, * )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
*
|
||
|
|
*> \par Purpose:
|
||
|
|
* =============
|
||
|
|
*>
|
||
|
|
*> \verbatim
|
||
|
|
*>
|
||
|
|
*> SLARFT 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 REAL 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 REAL array, dimension (K)
|
||
|
|
*> TAU(i) must contain the scalar factor of the elementary
|
||
|
|
*> reflector H(i).
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] T
|
||
|
|
*> \verbatim
|
||
|
|
*> T is REAL 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 April 2012
|
||
|
|
*
|
||
|
|
*> \ingroup realOTHERauxiliary
|
||
|
|
*
|
||
|
|
*> \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 SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
|
||
|
|
*
|
||
|
|
* -- LAPACK auxiliary routine (version 3.4.1) --
|
||
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||
|
|
* April 2012
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
CHARACTER DIRECT, STOREV
|
||
|
|
INTEGER K, LDT, LDV, N
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
REAL T( LDT, * ), TAU( * ), V( LDV, * )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
*
|
||
|
|
* .. Parameters ..
|
||
|
|
REAL ONE, ZERO
|
||
|
|
PARAMETER ( ONE = 1.0E+0, ZERO = 0.0E+0 )
|
||
|
|
* ..
|
||
|
|
* .. Local Scalars ..
|
||
|
|
INTEGER I, J, PREVLASTV, LASTV
|
||
|
|
* ..
|
||
|
|
* .. External Subroutines ..
|
||
|
|
EXTERNAL SGEMV, STRMV
|
||
|
|
* ..
|
||
|
|
* .. 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 SGEMV( '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 SGEMV( '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 STRMV( '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 SGEMV( '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 SGEMV( '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 STRMV( '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 SLARFT
|
||
|
|
*
|
||
|
|
END
|