292 lines
8.7 KiB
FortranFixed
292 lines
8.7 KiB
FortranFixed
*> \brief \b DLARF1F applies an elementary reflector to a general rectangular
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* matrix assuming v(1) = 1.
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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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*> Download DLARF1F + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dlarf1f.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/dlarf1f.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/dlarf1f.f">
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*> [TXT]</a>
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*
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* Definition:
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* ===========
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*
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* SUBROUTINE DLARF1F( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
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*
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* .. Scalar Arguments ..
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* CHARACTER SIDE
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* INTEGER INCV, LDC, M, N
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* DOUBLE PRECISION TAU
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* ..
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* .. Array Arguments ..
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* DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
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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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*> DLARF1F applies a real elementary reflector H to a real m by n matrix
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*> C, from either the left or the right. H is represented in the form
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*>
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*> H = I - tau * v * v**T
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*>
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*> where tau is a real scalar and v is a real vector.
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*>
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*> If tau = 0, then H is taken to be the unit 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] SIDE
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*> \verbatim
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*> SIDE is CHARACTER*1
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*> = 'L': form H * C
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*> = 'R': form C * H
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*> \endverbatim
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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 C.
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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 C.
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*> \endverbatim
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*>
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*> \param[in] V
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*> \verbatim
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*> V is DOUBLE PRECISION array, dimension
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*> (1 + (M-1)*abs(INCV)) if SIDE = 'L'
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*> or (1 + (N-1)*abs(INCV)) if SIDE = 'R'
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*> The vector v in the representation of H. V is not used if
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*> TAU = 0. V(1) is not referenced or modified.
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*> \endverbatim
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*>
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*> \param[in] INCV
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*> \verbatim
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*> INCV is INTEGER
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*> The increment between elements of v. INCV <> 0.
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*> \endverbatim
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*>
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*> \param[in] TAU
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*> \verbatim
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*> TAU is DOUBLE PRECISION
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*> The value tau in the representation of H.
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*> \endverbatim
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*>
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*> \param[in,out] C
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*> \verbatim
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*> C is DOUBLE PRECISION array, dimension (LDC,N)
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*> On entry, the m by n matrix C.
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*> On exit, C is overwritten by the matrix H * C if SIDE = 'L',
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*> or C * H if SIDE = 'R'.
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*> \endverbatim
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*>
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*> \param[in] LDC
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*> \verbatim
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*> LDC is INTEGER
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*> The leading dimension of the array C. LDC >= max(1,M).
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*> \endverbatim
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*>
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*> \param[out] WORK
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*> \verbatim
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*> WORK is DOUBLE PRECISION array, dimension
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*> (N) if SIDE = 'L'
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*> or (M) if SIDE = 'R'
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*> \endverbatim
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*
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* To take advantage of the fact that v(1) = 1, we do the following
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* v = [ 1 v_2 ]**T
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* If SIDE='L'
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* |-----|
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* | C_1 |
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* C =| C_2 |
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* |-----|
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* C_1\in\mathbb{R}^{1\times n}, C_2\in\mathbb{R}^{m-1\times n}
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* So we compute:
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* C = HC = (I - \tau vv**T)C
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* = C - \tau vv**T C
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* w = C**T v = [ C_1**T C_2**T ] [ 1 v_2 ]**T
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* = C_1**T + C_2**T v ( DGEMM then DAXPY )
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* C = C - \tau vv**T C
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* = C - \tau vw**T
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* Giving us C_1 = C_1 - \tau w**T ( DAXPY )
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* and
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* C_2 = C_2 - \tau v_2w**T ( DGER )
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* If SIDE='R'
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*
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* C = [ C_1 C_2 ]
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* C_1\in\mathbb{R}^{m\times 1}, C_2\in\mathbb{R}^{m\times n-1}
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* So we compute:
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* C = CH = C(I - \tau vv**T)
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* = C - \tau Cvv**T
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*
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* w = Cv = [ C_1 C_2 ] [ 1 v_2 ]**T
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* = C_1 + C_2v_2 ( DGEMM then DAXPY )
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* C = C - \tau Cvv**T
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* = C - \tau wv**T
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* Giving us C_1 = C_1 - \tau w ( DAXPY )
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* and
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* C_2 = C_2 - \tau wv_2**T ( DGER )
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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 larf
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*
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* =====================================================================
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SUBROUTINE DLARF1F( SIDE, M, N, V, INCV, TAU, C, LDC, WORK )
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IMPLICIT NONE
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*
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* -- LAPACK auxiliary 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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CHARACTER SIDE
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INTEGER INCV, LDC, M, N
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DOUBLE PRECISION TAU
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* ..
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* .. Array Arguments ..
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DOUBLE PRECISION C( LDC, * ), V( * ), WORK( * )
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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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DOUBLE PRECISION ONE, ZERO
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PARAMETER ( ONE = 1.0D+0, ZERO = 0.0D+0 )
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* ..
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* .. Local Scalars ..
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LOGICAL APPLYLEFT
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INTEGER I, LASTV, LASTC
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* ..
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* .. External Subroutines ..
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EXTERNAL DGEMV, DGER, DAXPY, DSCAL
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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INTEGER ILADLR, ILADLC
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EXTERNAL LSAME, ILADLR, ILADLC
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* ..
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* .. Executable Statements ..
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*
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APPLYLEFT = LSAME( SIDE, 'L' )
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LASTV = 1
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LASTC = 0
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IF( TAU.NE.ZERO ) THEN
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! Set up variables for scanning V. LASTV begins pointing to the end
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! of V.
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IF( APPLYLEFT ) THEN
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LASTV = M
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ELSE
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LASTV = N
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END IF
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IF( INCV.GT.0 ) THEN
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I = 1 + (LASTV-1) * INCV
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ELSE
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I = 1
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END IF
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! Look for the last non-zero row in V.
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! Since we are assuming that V(1) = 1, and it is not stored, so we
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! shouldn't access it.
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DO WHILE( LASTV.GT.1 .AND. V( I ).EQ.ZERO )
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LASTV = LASTV - 1
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I = I - INCV
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END DO
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IF( APPLYLEFT ) THEN
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! Scan for the last non-zero column in C(1:lastv,:).
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LASTC = ILADLC(LASTV, N, C, LDC)
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ELSE
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! Scan for the last non-zero row in C(:,1:lastv).
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LASTC = ILADLR(M, LASTV, C, LDC)
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END IF
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END IF
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IF( LASTC.EQ.0 ) THEN
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RETURN
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END IF
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IF( APPLYLEFT ) THEN
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*
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* Form H * C
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*
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! Check if lastv = 1. This means v = 1, So we just need to compute
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! C := HC = (1-\tau)C.
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IF( LASTV.EQ.1 ) THEN
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*
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* C(1,1:lastc) := ( 1 - tau ) * C(1,1:lastc)
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*
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CALL DSCAL(LASTC, ONE - TAU, C, LDC)
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ELSE
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*
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* w(1:lastc,1) := C(1:lastv,1:lastc)**T * v(1:lastv,1)
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*
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! w(1:lastc,1) := C(2:lastv,1:lastc)**T * v(2:lastv,1)
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CALL DGEMV( 'Transpose', LASTV-1, LASTC, ONE, C(1+1,1),
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$ LDC, V(1+INCV), INCV, ZERO, WORK, 1)
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! w(1:lastc,1) += C(1,1:lastc)**T * v(1,1) = C(1,1:lastc)**T
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CALL DAXPY(LASTC, ONE, C, LDC, WORK, 1)
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*
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* C(1:lastv,1:lastc) := C(...) - tau * v(1:lastv,1) * w(1:lastc,1)**T
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*
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! C(1, 1:lastc) := C(...) - tau * v(1,1) * w(1:lastc,1)**T
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! = C(...) - tau * w(1:lastc,1)**T
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CALL DAXPY(LASTC, -TAU, WORK, 1, C, LDC)
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! C(2:lastv,1:lastc) := C(...) - tau * v(2:lastv,1)*w(1:lastc,1)**T
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CALL DGER(LASTV-1, LASTC, -TAU, V(1+INCV), INCV, WORK, 1,
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$ C(1+1,1), LDC)
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END IF
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ELSE
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*
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* Form C * H
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*
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! Check if n = 1. This means v = 1, so we just need to compute
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! C := CH = C(1-\tau).
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IF( LASTV.EQ.1 ) THEN
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*
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* C(1:lastc,1) := ( 1 - tau ) * C(1:lastc,1)
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*
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CALL DSCAL(LASTC, ONE - TAU, C, 1)
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ELSE
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*
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* w(1:lastc,1) := C(1:lastc,1:lastv) * v(1:lastv,1)
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*
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! w(1:lastc,1) := C(1:lastc,2:lastv) * v(2:lastv,1)
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CALL DGEMV( 'No transpose', LASTC, LASTV-1, ONE,
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$ C(1,1+1), LDC, V(1+INCV), INCV, ZERO, WORK, 1 )
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! w(1:lastc,1) += C(1:lastc,1) v(1,1) = C(1:lastc,1)
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CALL DAXPY(LASTC, ONE, C, 1, WORK, 1)
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*
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* C(1:lastc,1:lastv) := C(...) - tau * w(1:lastc,1) * v(1:lastv,1)**T
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*
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! C(1:lastc,1) := C(...) - tau * w(1:lastc,1) * v(1,1)**T
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! = C(...) - tau * w(1:lastc,1)
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CALL DAXPY(LASTC, -TAU, WORK, 1, C, 1)
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! C(1:lastc,2:lastv) := C(...) - tau * w(1:lastc,1) * v(2:lastv)**T
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CALL DGER( LASTC, LASTV-1, -TAU, WORK, 1, V(1+INCV),
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$ INCV, C(1,1+1), LDC )
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END IF
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END IF
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RETURN
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*
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* End of DLARF1F
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*
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END
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