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Merge pull request #5064 from martin-frbg/lapack1080 

Replace LAPACK ?LARFT with a recursive implementation (Reference-LAPACK PR 1080)
tags/v0.3.29
Martin Kroeker GitHub 1 year ago
parent
5527eda561 d035e80d33
commit
f422845b6d
No known key found for this signature in database GPG Key ID: B5690EEEBB952194
10 changed files with 3109 additions and 570 deletions
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  1. +8
    -2
      lapack-netlib/SRC/VARIANTS/Makefile
  2. +2
    -0
      lapack-netlib/SRC/VARIANTS/README
  3. +328
    -0
      lapack-netlib/SRC/VARIANTS/larft/LL_LVL2/clarft.f
  4. +326
    -0
      lapack-netlib/SRC/VARIANTS/larft/LL_LVL2/dlarft.f
  5. +326
    -0
      lapack-netlib/SRC/VARIANTS/larft/LL_LVL2/slarft.f
  6. +327
    -0
      lapack-netlib/SRC/VARIANTS/larft/LL_LVL2/zlarft.f
  7. +451
    -145
      lapack-netlib/SRC/clarft.f
  8. +445
    -140
      lapack-netlib/SRC/dlarft.f
  9. +444
    -139
      lapack-netlib/SRC/slarft.f
  10. +452
    -144
      lapack-netlib/SRC/zlarft.f

+ 8
- 2
lapack-netlib/SRC/VARIANTS/Makefile View File

@@ -30,9 +30,11 @@ LUREC = lu/REC/cgetrf.o lu/REC/dgetrf.o lu/REC/sgetrf.o lu/REC/zgetrf.o

QRLL = qr/LL/cgeqrf.o qr/LL/dgeqrf.o qr/LL/sgeqrf.o qr/LL/zgeqrf.o

LARFTL2 = larft/LL-LVL2/clarft.o larft/LL-LVL2/dlarft.o larft/LL-LVL2/slarft.o larft/LL-LVL2/zlarft.o


.PHONY: all
all: cholrl.a choltop.a lucr.a lull.a lurec.a qrll.a
all: cholrl.a choltop.a lucr.a lull.a lurec.a qrll.a larftl2.a

cholrl.a: $(CHOLRL)
$(AR) $(ARFLAGS) $@ $^
@@ -58,9 +60,13 @@ qrll.a: $(QRLL)
$(AR) $(ARFLAGS) $@ $^
$(RANLIB) $@

larftl2.a: $(LARFTL2)
$(AR) $(ARFLAGS) $@ $^
$(RANLIB) $@

.PHONY: clean cleanobj cleanlib
clean: cleanobj cleanlib
cleanobj:
rm -f $(CHOLRL) $(CHOLTOP) $(LUCR) $(LULL) $(LUREC) $(QRLL)
rm -f $(CHOLRL) $(CHOLTOP) $(LUCR) $(LULL) $(LUREC) $(QRLL) $(LARFTL2)
cleanlib:
rm -f *.a

+ 2
- 0
lapack-netlib/SRC/VARIANTS/README View File

@@ -23,6 +23,7 @@ This directory contains several variants of LAPACK routines in single/double/com
- [sdcz]geqrf with QR Left Looking Level 3 BLAS version algorithm [2]- Directory: SRC/VARIANTS/qr/LL
- [sdcz]potrf with Cholesky Right Looking Level 3 BLAS version algorithm [2]- Directory: SRC/VARIANTS/cholesky/RL
- [sdcz]potrf with Cholesky Top Level 3 BLAS version algorithm [2]- Directory: SRC/VARIANTS/cholesky/TOP
- [sdcz]larft using a Left Looking Level 2 BLAS version algorithm - Directory: SRC/VARIANTS/larft/LL-LVL2

References:For a more detailed description please refer to
- [1] Toledo, S. 1997. Locality of Reference in LU Decomposition with Partial Pivoting. SIAM J. Matrix Anal. Appl. 18, 4 (Oct. 1997),
@@ -44,6 +45,7 @@ Corresponding libraries created in SRC/VARIANTS:
- QR Left Looking : qrll.a
- Cholesky Right Looking : cholrl.a
- Cholesky Top : choltop.a
- LARFT Level 2: larftl2.a


===========


+ 328
- 0
lapack-netlib/SRC/VARIANTS/larft/LL_LVL2/clarft.f View File

@@ -0,0 +1,328 @@
*> \brief \b CLARFT VARIANT: left-looking Level 2 BLAS version of the algorithm
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download CLARFT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/clarft.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/clarft.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/clarft.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE CLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
* INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
* COMPLEX T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CLARFT forms the triangular factor T of a complex 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**H
*>
*> 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**H * 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 COMPLEX 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 COMPLEX array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is COMPLEX 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.
*
*> \ingroup larft
*
*> \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 CLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
COMPLEX T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX ONE, ZERO
PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ),
$ ZERO = ( 0.0E+0, 0.0E+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
* .. External Subroutines ..
EXTERNAL CGEMM, CGEMV, CTRMV
* ..
* .. 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( PREVLASTV, I )
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 ) * CONJG( V( I , J ) )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**H * V(i:j,i)
*
CALL CGEMV( 'Conjugate 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)**H
*
CALL CGEMM( 'N', 'C', I-1, 1, J-I, -TAU( I ),
$ V( 1, I+1 ), LDV, V( I, I+1 ), LDV,
$ ONE, T( 1, I ), LDT )
END IF
*
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
CALL CTRMV( '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 ) * CONJG( 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)**H * V(j:n-k+i,i)
*
CALL CGEMV( 'Conjugate 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)**H
*
CALL CGEMM( 'N', 'C', K-I, 1, N-K+I-J,
$ -TAU( I ),
$ V( I+1, J ), LDV, V( I, J ), LDV,
$ ONE, T( I+1, I ), LDT )
END IF
*
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL CTRMV( '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 CLARFT
*
END

+ 326
- 0
lapack-netlib/SRC/VARIANTS/larft/LL_LVL2/dlarft.f View File

@@ -0,0 +1,326 @@
*> \brief \b DLARFT VARIANT: left-looking Level 2 BLAS version of the algorithm
*
* =========== 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.
*
*> \ingroup larft
*
*> \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 --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. 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

+ 326
- 0
lapack-netlib/SRC/VARIANTS/larft/LL_LVL2/slarft.f View File

@@ -0,0 +1,326 @@
*> \brief \b SLARFT VARIANT: left-looking Level 2 BLAS version of the algorithm.
*
* =========== 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.
*
*> \ingroup larft
*
*> \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 --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. 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

+ 327
- 0
lapack-netlib/SRC/VARIANTS/larft/LL_LVL2/zlarft.f View File

@@ -0,0 +1,327 @@
*> \brief \b ZLARFT VARIANT: left-looking Level 2 BLAS version of the algorithm.
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download ZLARFT + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlarft.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlarft.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlarft.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
* INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
* COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZLARFT forms the triangular factor T of a complex 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**H
*>
*> 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**H * 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 COMPLEX*16 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 COMPLEX*16 array, dimension (K)
*> TAU(i) must contain the scalar factor of the elementary
*> reflector H(i).
*> \endverbatim
*>
*> \param[out] T
*> \verbatim
*> T is COMPLEX*16 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.
*
*> \ingroup larft
*
*> \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 ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
COMPLEX*16 ONE, ZERO
PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
$ ZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
* .. External Subroutines ..
EXTERNAL ZGEMV, ZTRMV, ZGEMM
* ..
* .. 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( PREVLASTV, I )
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 ) * CONJG( V( I , J ) )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**H * V(i:j,i)
*
CALL ZGEMV( 'Conjugate 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)**H
*
CALL ZGEMM( 'N', 'C', I-1, 1, J-I, -TAU( I ),
$ V( 1, I+1 ), LDV, V( I, I+1 ), LDV,
$ ONE, T( 1, I ), LDT )
END IF
*
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
CALL ZTRMV( '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 ) * CONJG( 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)**H * V(j:n-k+i,i)
*
CALL ZGEMV( 'Conjugate 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)**H
*
CALL ZGEMM( 'N', 'C', K-I, 1, N-K+I-J,
$ -TAU( I ),
$ V( I+1, J ), LDV, V( I, J ), LDV,
$ ONE, T( I+1, I ), LDT )
END IF
*
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL ZTRMV( '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 ZLARFT
*
END

+ 451
- 145
lapack-netlib/SRC/clarft.f View File

@@ -18,7 +18,7 @@
* Definition:
* ===========
*
* SUBROUTINE CLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
* RECURSIVE SUBROUTINE CLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
@@ -130,7 +130,7 @@
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexOTHERauxiliary
*> \ingroup larft
*
*> \par Further Details:
* =====================
@@ -159,167 +159,473 @@
*> \endverbatim
*>
* =====================================================================
SUBROUTINE CLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
RECURSIVE SUBROUTINE CLARFT( DIRECT, STOREV, N, K, V, LDV,
$ TAU, T, LDT )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* .. Scalar Arguments
*
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
COMPLEX T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* =====================================================================
COMPLEX T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* .. Parameters ..
COMPLEX ONE, ZERO
PARAMETER ( ONE = ( 1.0E+0, 0.0E+0 ),
$ ZERO = ( 0.0E+0, 0.0E+0 ) )
* ..
*
COMPLEX ONE, NEG_ONE, ZERO
PARAMETER(ONE=1.0E+0, ZERO = 0.0E+0, NEG_ONE=-1.0E+0)
*
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
*
INTEGER I,J,L
LOGICAL QR,LQ,QL,DIRF,COLV
*
* .. External Subroutines ..
EXTERNAL CGEMM, CGEMV, CTRMV
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
EXTERNAL CTRMM,CGEMM,CLACPY
*
* .. External Functions..
*
LOGICAL LSAME
EXTERNAL LSAME
*
* .. Intrinsic Functions..
*
INTRINSIC CONJG
*
* The general scheme used is inspired by the approach inside DGEQRT3
* which was (at the time of writing this code):
* Based on the algorithm of Elmroth and Gustavson,
* IBM J. Res. Develop. Vol 44 No. 4 July 2000.
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
PREVLASTV = N
DO I = 1, K
PREVLASTV = MAX( PREVLASTV, I )
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 ) * CONJG( V( I , J ) )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**H * V(i:j,i)
*
CALL CGEMV( 'Conjugate 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)**H
*
CALL CGEMM( 'N', 'C', I-1, 1, J-I, -TAU( I ),
$ V( 1, I+1 ), LDV, V( I, I+1 ), LDV,
$ ONE, T( 1, I ), LDT )
END IF
*
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
CALL CTRMV( '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
IF(N.EQ.0.OR.K.EQ.0) THEN
RETURN
END IF
*
* Base case
*
IF(N.EQ.1.OR.K.EQ.1) THEN
T(1,1) = TAU(1)
RETURN
END IF
*
* Beginning of executable statements
*
L = K / 2
*
* Determine what kind of Q we need to compute
* We assume that if the user doesn't provide 'F' for DIRECT,
* then they meant to provide 'B' and if they don't provide
* 'C' for STOREV, then they meant to provide 'R'
*
DIRF = LSAME(DIRECT,'F')
COLV = LSAME(STOREV,'C')
*
* QR happens when we have forward direction in column storage
*
QR = DIRF.AND.COLV
*
* LQ happens when we have forward direction in row storage
*
LQ = DIRF.AND.(.NOT.COLV)
*
* QL happens when we have backward direction in column storage
*
QL = (.NOT.DIRF).AND.COLV
*
* The last case is RQ. Due to how we structured this, if the
* above 3 are false, then RQ must be true, so we never store
* this
* RQ happens when we have backward direction in row storage
* RQ = (.NOT.DIRF).AND.(.NOT.COLV)
*
IF(QR) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} 0 |
* |V_{2,1} V_{2,2}|
* |V_{3,1} V_{3,2}|
* |---------------|
*
* V_{1,1}\in\C^{l,l} unit lower triangular
* V_{2,1}\in\C^{k-l,l} rectangular
* V_{3,1}\in\C^{n-k,l} rectangular
*
* V_{2,2}\in\C^{k-l,k-l} unit lower triangular
* V_{3,2}\in\C^{n-k,k-l} rectangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\C^{l, l} upper triangular
* T_{2,2}\in\C^{k-l, k-l} upper triangular
* T_{1,2}\in\C^{l, k-l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_1*T_{1,1}*V_1')*(I - V_2*T_{2,2}*V_2')
* = I - V_1*T_{1,1}*V_1' - V_2*T_{2,2}*V_2' + V_1*T_{1,1}*V_1'*V_2*T_{2,2}*V_2'
*
* Define T{1,2} = -T_{1,1}*V_1'*V_2*T_{2,2}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL CLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL CLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)
*
* Compute T_{1,2}
* T_{1,2} = V_{2,1}'
*
DO J = 1, L
DO I = 1, K-L
T(J, L+I) = CONJG(V(L+I, J))
END DO
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 ) * CONJG( 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)**H * V(j:n-k+i,i)
*
CALL CGEMV( 'Conjugate 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)**H
*
CALL CGEMM( 'N', 'C', K-I, 1, N-K+I-J, -TAU( I ),
$ V( I+1, J ), LDV, V( I, J ), LDV,
$ ONE, T( I+1, I ), LDT )
END IF
*
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL CTRMV( '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
*
* T_{1,2} = T_{1,2}*V_{2,2}
*
CALL CTRMM('Right', 'Lower', 'No transpose', 'Unit', L,
$ K-L, ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
* T_{1,2} = V_{3,1}'*V_{3,2} + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL CGEMM('Conjugate', 'No transpose', L, K-L, N-K, ONE,
$ V(K+1, 1), LDV, V(K+1, L+1), LDV, ONE, T(1, L+1),
$ LDT)
*
* At this point, we have that T_{1,2} = V_1'*V_2
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL CTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL CTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)

ELSE IF(LQ) THEN
*
* Break V apart into 6 components
*
* V = |----------------------|
* |V_{1,1} V_{1,2} V{1,3}|
* |0 V_{2,2} V{2,3}|
* |----------------------|
*
* V_{1,1}\in\C^{l,l} unit upper triangular
* V_{1,2}\in\C^{l,k-l} rectangular
* V_{1,3}\in\C^{l,n-k} rectangular
*
* V_{2,2}\in\C^{k-l,k-l} unit upper triangular
* V_{2,3}\in\C^{k-l,n-k} rectangular
*
* Where l = floor(k/2)
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\C^{l, l} upper triangular
* T_{2,2}\in\C^{k-l, k-l} upper triangular
* T_{1,2}\in\C^{l, k-l} rectangular
*
* Then, consider the product:
*
* (I - V_1'*T_{1,1}*V_1)*(I - V_2'*T_{2,2}*V_2)
* = I - V_1'*T_{1,1}*V_1 - V_2'*T_{2,2}*V_2 + V_1'*T_{1,1}*V_1*V_2'*T_{2,2}*V_2
*
* Define T_{1,2} = -T_{1,1}*V_1*V_2'*T_{2,2}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL CLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL CLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)

*
* Compute T_{1,2}
* T_{1,2} = V_{1,2}
*
CALL CLACPY('All', L, K-L, V(1, L+1), LDV, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*V_{2,2}'
*
CALL CTRMM('Right', 'Upper', 'Conjugate', 'Unit', L, K-L,
$ ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
* T_{1,2} = V_{1,3}*V_{2,3}' + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL CGEMM('No transpose', 'Conjugate', L, K-L, N-K, ONE,
$ V(1, K+1), LDV, V(L+1, K+1), LDV, ONE, T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1*V_2'
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL CTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)

*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL CTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1,L+1), LDT, T(1, L+1), LDT)
ELSE IF(QL) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} V_{1,2}|
* |V_{2,1} V_{2,2}|
* |0 V_{3,2}|
* |---------------|
*
* V_{1,1}\in\C^{n-k,k-l} rectangular
* V_{2,1}\in\C^{k-l,k-l} unit upper triangular
*
* V_{1,2}\in\C^{n-k,l} rectangular
* V_{2,2}\in\C^{k-l,l} rectangular
* V_{3,2}\in\C^{l,l} unit upper triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\C^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\C^{l, l} non-unit lower triangular
* T_{2,1}\in\C^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2*T_{2,2}*V_2')*(I - V_1*T_{1,1}*V_1')
* = I - V_2*T_{2,2}*V_2' - V_1*T_{1,1}*V_1' + V_2*T_{2,2}*V_2'*V_1*T_{1,1}*V_1'
*
* Define T_{2,1} = -T_{2,2}*V_2'*V_1*T_{1,1}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL CLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL CLARFT(DIRECT, STOREV, N, L, V(1, K-L+1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}'
*
DO J = 1, K-L
DO I = 1, L
T(K-L+I, J) = CONJG(V(N-K+J, K-L+I))
END DO
END DO
END IF
RETURN
*
* End of CLARFT
* T_{2,1} = T_{2,1}*V_{2,1}
*
CALL CTRMM('Right', 'Upper', 'No transpose', 'Unit', L,
$ K-L, ONE, V(N-K+1, 1), LDV, T(K-L+1, 1), LDT)

*
* T_{2,1} = V_{2,2}'*V_{2,1} + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL CGEMM('Conjugate', 'No transpose', L, K-L, N-K, ONE,
$ V(1, K-L+1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)
*
* At this point, we have that T_{2,1} = V_2'*V_1
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL CTRMM('Left', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)
*
END
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL CTRMM('Right', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
ELSE
*
* Else means RQ case
*
* Break V apart into 6 components
*
* V = |-----------------------|
* |V_{1,1} V_{1,2} 0 |
* |V_{2,1} V_{2,2} V_{2,3}|
* |-----------------------|
*
* V_{1,1}\in\C^{k-l,n-k} rectangular
* V_{1,2}\in\C^{k-l,k-l} unit lower triangular
*
* V_{2,1}\in\C^{l,n-k} rectangular
* V_{2,2}\in\C^{l,k-l} rectangular
* V_{2,3}\in\C^{l,l} unit lower triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\C^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\C^{l, l} non-unit lower triangular
* T_{2,1}\in\C^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2'*T_{2,2}*V_2)*(I - V_1'*T_{1,1}*V_1)
* = I - V_2'*T_{2,2}*V_2 - V_1'*T_{1,1}*V_1 + V_2'*T_{2,2}*V_2*V_1'*T_{1,1}*V_1
*
* Define T_{2,1} = -T_{2,2}*V_2*V_1'*T_{1,1}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL CLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL CLARFT(DIRECT, STOREV, N, L, V(K-L+1,1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}
*
CALL CLACPY('All', L, K-L, V(K-L+1, N-K+1), LDV,
$ T(K-L+1, 1), LDT)

*
* T_{2,1} = T_{2,1}*V_{1,2}'
*
CALL CTRMM('Right', 'Lower', 'Conjugate', 'Unit', L, K-L,
$ ONE, V(1, N-K+1), LDV, T(K-L+1,1), LDT)

*
* T_{2,1} = V_{2,1}*V_{1,1}' + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL CGEMM('No transpose', 'Conjugate', L, K-L, N-K, ONE,
$ V(K-L+1, 1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)

*
* At this point, we have that T_{2,1} = V_2*V_1'
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL CTRMM('Left', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)

*
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL CTRMM('Right', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
END IF
END SUBROUTINE

+ 445
- 140
lapack-netlib/SRC/dlarft.f View File

@@ -18,7 +18,7 @@
* Definition:
* ===========
*
* SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
* RECURSIVE SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
@@ -130,7 +130,7 @@
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup doubleOTHERauxiliary
*> \ingroup larft
*
*> \par Further Details:
* =====================
@@ -159,165 +159,470 @@
*> \endverbatim
*>
* =====================================================================
SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
RECURSIVE SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV,
$ TAU, T, LDT )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
* .. 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 )
* ..
*
DOUBLE PRECISION ONE, NEG_ONE, ZERO
PARAMETER(ONE=1.0D+0, ZERO = 0.0D+0, NEG_ONE=-1.0D+0)
*
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
*
INTEGER I,J,L
LOGICAL QR,LQ,QL,DIRF,COLV
*
* .. External Subroutines ..
EXTERNAL DGEMV, DTRMV
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
EXTERNAL DTRMM,DGEMM,DLACPY
*
* .. External Functions..
*
LOGICAL LSAME
EXTERNAL LSAME
*
* The general scheme used is inspired by the approach inside DGEQRT3
* which was (at the time of writing this code):
* Based on the algorithm of Elmroth and Gustavson,
* IBM J. Res. Develop. Vol 44 No. 4 July 2000.
* ..
* .. 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
IF(N.EQ.0.OR.K.EQ.0) THEN
RETURN
END IF
*
* Base case
*
IF(N.EQ.1.OR.K.EQ.1) THEN
T(1,1) = TAU(1)
RETURN
END IF
*
* Beginning of executable statements
*
L = K / 2
*
* Determine what kind of Q we need to compute
* We assume that if the user doesn't provide 'F' for DIRECT,
* then they meant to provide 'B' and if they don't provide
* 'C' for STOREV, then they meant to provide 'R'
*
DIRF = LSAME(DIRECT,'F')
COLV = LSAME(STOREV,'C')
*
* QR happens when we have forward direction in column storage
*
QR = DIRF.AND.COLV
*
* LQ happens when we have forward direction in row storage
*
LQ = DIRF.AND.(.NOT.COLV)
*
* QL happens when we have backward direction in column storage
*
QL = (.NOT.DIRF).AND.COLV
*
* The last case is RQ. Due to how we structured this, if the
* above 3 are false, then RQ must be true, so we never store
* this
* RQ happens when we have backward direction in row storage
* RQ = (.NOT.DIRF).AND.(.NOT.COLV)
*
IF(QR) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} 0 |
* |V_{2,1} V_{2,2}|
* |V_{3,1} V_{3,2}|
* |---------------|
*
* V_{1,1}\in\R^{l,l} unit lower triangular
* V_{2,1}\in\R^{k-l,l} rectangular
* V_{3,1}\in\R^{n-k,l} rectangular
*
* V_{2,2}\in\R^{k-l,k-l} unit lower triangular
* V_{3,2}\in\R^{n-k,k-l} rectangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{l, l} upper triangular
* T_{2,2}\in\R^{k-l, k-l} upper triangular
* T_{1,2}\in\R^{l, k-l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_1*T_{1,1}*V_1')*(I - V_2*T_{2,2}*V_2')
* = I - V_1*T_{1,1}*V_1' - V_2*T_{2,2}*V_2' + V_1*T_{1,1}*V_1'*V_2*T_{2,2}*V_2'
*
* Define T_{1,2} = -T_{1,1}*V_1'*V_2*T_{2,2}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL DLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL DLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)
*
* Compute T_{1,2}
* T_{1,2} = V_{2,1}'
*
DO J = 1, L
DO I = 1, K-L
T(J, L+I) = V(L+I, J)
END DO
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
*
* T_{1,2} = T_{1,2}*V_{2,2}
*
CALL DTRMM('Right', 'Lower', 'No transpose', 'Unit', L,
$ K-L, ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
* T_{1,2} = V_{3,1}'*V_{3,2} + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL DGEMM('Transpose', 'No transpose', L, K-L, N-K, ONE,
$ V(K+1, 1), LDV, V(K+1, L+1), LDV, ONE,
$ T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1'*V_2
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL DTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL DTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)

ELSE IF(LQ) THEN
*
* Break V apart into 6 components
*
* V = |----------------------|
* |V_{1,1} V_{1,2} V{1,3}|
* |0 V_{2,2} V{2,3}|
* |----------------------|
*
* V_{1,1}\in\R^{l,l} unit upper triangular
* V_{1,2}\in\R^{l,k-l} rectangular
* V_{1,3}\in\R^{l,n-k} rectangular
*
* V_{2,2}\in\R^{k-l,k-l} unit upper triangular
* V_{2,3}\in\R^{k-l,n-k} rectangular
*
* Where l = floor(k/2)
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{l, l} upper triangular
* T_{2,2}\in\R^{k-l, k-l} upper triangular
* T_{1,2}\in\R^{l, k-l} rectangular
*
* Then, consider the product:
*
* (I - V_1'*T_{1,1}*V_1)*(I - V_2'*T_{2,2}*V_2)
* = I - V_1'*T_{1,1}*V_1 - V_2'*T_{2,2}*V_2 + V_1'*T_{1,1}*V_1*V_2'*T_{2,2}*V_2
*
* Define T_{1,2} = -T_{1,1}*V_1*V_2'*T_{2,2}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL DLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL DLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)

*
* Compute T_{1,2}
* T_{1,2} = V_{1,2}
*
CALL DLACPY('All', L, K-L, V(1, L+1), LDV, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*V_{2,2}'
*
CALL DTRMM('Right', 'Upper', 'Transpose', 'Unit', L, K-L,
$ ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
* T_{1,2} = V_{1,3}*V_{2,3}' + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL DGEMM('No transpose', 'Transpose', L, K-L, N-K, ONE,
$ V(1, K+1), LDV, V(L+1, K+1), LDV, ONE,
$ T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1*V_2'
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL DTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)

*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL DTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)
ELSE IF(QL) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} V_{1,2}|
* |V_{2,1} V_{2,2}|
* |0 V_{3,2}|
* |---------------|
*
* V_{1,1}\in\R^{n-k,k-l} rectangular
* V_{2,1}\in\R^{k-l,k-l} unit upper triangular
*
* V_{1,2}\in\R^{n-k,l} rectangular
* V_{2,2}\in\R^{k-l,l} rectangular
* V_{3,2}\in\R^{l,l} unit upper triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\R^{l, l} non-unit lower triangular
* T_{2,1}\in\R^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2*T_{2,2}*V_2')*(I - V_1*T_{1,1}*V_1')
* = I - V_2*T_{2,2}*V_2' - V_1*T_{1,1}*V_1' + V_2*T_{2,2}*V_2'*V_1*T_{1,1}*V_1'
*
* Define T_{2,1} = -T_{2,2}*V_2'*V_1*T_{1,1}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL DLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL DLARFT(DIRECT, STOREV, N, L, V(1, K-L+1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}'
*
DO J = 1, K-L
DO I = 1, L
T(K-L+I, J) = V(N-K+J, K-L+I)
END DO
END DO
END IF
RETURN
*
* End of DLARFT
* T_{2,1} = T_{2,1}*V_{2,1}
*
CALL DTRMM('Right', 'Upper', 'No transpose', 'Unit', L,
$ K-L, ONE, V(N-K+1, 1), LDV, T(K-L+1, 1), LDT)

*
* T_{2,1} = V_{2,2}'*V_{2,1} + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL DGEMM('Transpose', 'No transpose', L, K-L, N-K, ONE,
$ V(1, K-L+1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)
*
* At this point, we have that T_{2,1} = V_2'*V_1
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL DTRMM('Left', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)
*
END
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL DTRMM('Right', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
ELSE
*
* Else means RQ case
*
* Break V apart into 6 components
*
* V = |-----------------------|
* |V_{1,1} V_{1,2} 0 |
* |V_{2,1} V_{2,2} V_{2,3}|
* |-----------------------|
*
* V_{1,1}\in\R^{k-l,n-k} rectangular
* V_{1,2}\in\R^{k-l,k-l} unit lower triangular
*
* V_{2,1}\in\R^{l,n-k} rectangular
* V_{2,2}\in\R^{l,k-l} rectangular
* V_{2,3}\in\R^{l,l} unit lower triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\R^{l, l} non-unit lower triangular
* T_{2,1}\in\R^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2'*T_{2,2}*V_2)*(I - V_1'*T_{1,1}*V_1)
* = I - V_2'*T_{2,2}*V_2 - V_1'*T_{1,1}*V_1 + V_2'*T_{2,2}*V_2*V_1'*T_{1,1}*V_1
*
* Define T_{2,1} = -T_{2,2}*V_2*V_1'*T_{1,1}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL DLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL DLARFT(DIRECT, STOREV, N, L, V(K-L+1, 1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}
*
CALL DLACPY('All', L, K-L, V(K-L+1, N-K+1), LDV,
$ T(K-L+1, 1), LDT)

*
* T_{2,1} = T_{2,1}*V_{1,2}'
*
CALL DTRMM('Right', 'Lower', 'Transpose', 'Unit', L, K-L,
$ ONE, V(1, N-K+1), LDV, T(K-L+1, 1), LDT)

*
* T_{2,1} = V_{2,1}*V_{1,1}' + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL DGEMM('No transpose', 'Transpose', L, K-L, N-K, ONE,
$ V(K-L+1, 1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)

*
* At this point, we have that T_{2,1} = V_2*V_1'
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL DTRMM('Left', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)

*
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL DTRMM('Right', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
END IF
END SUBROUTINE

+ 444
- 139
lapack-netlib/SRC/slarft.f View File

@@ -18,7 +18,7 @@
* Definition:
* ===========
*
* SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
* RECURSIVE SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
@@ -127,10 +127,10 @@
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author Johnathan Rhyne, Univ. of Colorado Denver (original author, 2024)
*> \author NAG Ltd.
*
*> \ingroup realOTHERauxiliary
*> \ingroup larft
*
*> \par Further Details:
* =====================
@@ -159,165 +159,470 @@
*> \endverbatim
*>
* =====================================================================
SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
RECURSIVE SUBROUTINE SLARFT( DIRECT, STOREV, N, K, V, LDV,
$ TAU, T, LDT )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
* .. 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 )
* ..
*
REAL ONE, NEG_ONE, ZERO
PARAMETER(ONE=1.0E+0, ZERO = 0.0E+0, NEG_ONE=-1.0E+0)
*
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
*
INTEGER I,J,L
LOGICAL QR,LQ,QL,DIRF,COLV
*
* .. External Subroutines ..
EXTERNAL SGEMV, STRMV
* ..
* .. External Functions ..
*
EXTERNAL STRMM,SGEMM,SLACPY
*
* .. External Functions..
*
LOGICAL LSAME
EXTERNAL LSAME
*
* The general scheme used is inspired by the approach inside DGEQRT3
* which was (at the time of writing this code):
* Based on the algorithm of Elmroth and Gustavson,
* IBM J. Res. Develop. Vol 44 No. 4 July 2000.
* ..
* .. 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
IF(N.EQ.0.OR.K.EQ.0) THEN
RETURN
END IF
*
* Base case
*
IF(N.EQ.1.OR.K.EQ.1) THEN
T(1,1) = TAU(1)
RETURN
END IF
*
* Beginning of executable statements
*
L = K / 2
*
* Determine what kind of Q we need to compute
* We assume that if the user doesn't provide 'F' for DIRECT,
* then they meant to provide 'B' and if they don't provide
* 'C' for STOREV, then they meant to provide 'R'
*
DIRF = LSAME(DIRECT,'F')
COLV = LSAME(STOREV,'C')
*
* QR happens when we have forward direction in column storage
*
QR = DIRF.AND.COLV
*
* LQ happens when we have forward direction in row storage
*
LQ = DIRF.AND.(.NOT.COLV)
*
* QL happens when we have backward direction in column storage
*
QL = (.NOT.DIRF).AND.COLV
*
* The last case is RQ. Due to how we structured this, if the
* above 3 are false, then RQ must be true, so we never store
* this
* RQ happens when we have backward direction in row storage
* RQ = (.NOT.DIRF).AND.(.NOT.COLV)
*
IF(QR) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} 0 |
* |V_{2,1} V_{2,2}|
* |V_{3,1} V_{3,2}|
* |---------------|
*
* V_{1,1}\in\R^{l,l} unit lower triangular
* V_{2,1}\in\R^{k-l,l} rectangular
* V_{3,1}\in\R^{n-k,l} rectangular
*
* V_{2,2}\in\R^{k-l,k-l} unit lower triangular
* V_{3,2}\in\R^{n-k,k-l} rectangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{l, l} upper triangular
* T_{2,2}\in\R^{k-l, k-l} upper triangular
* T_{1,2}\in\R^{l, k-l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_1*T_{1,1}*V_1')*(I - V_2*T_{2,2}*V_2')
* = I - V_1*T_{1,1}*V_1' - V_2*T_{2,2}*V_2' + V_1*T_{1,1}*V_1'*V_2*T_{2,2}*V_2'
*
* Define T_{1,2} = -T_{1,1}*V_1'*V_2*T_{2,2}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL SLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL SLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)
*
* Compute T_{1,2}
* T_{1,2} = V_{2,1}'
*
DO J = 1, L
DO I = 1, K-L
T(J, L+I) = V(L+I, J)
END DO
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
*
* T_{1,2} = T_{1,2}*V_{2,2}
*
CALL STRMM('Right', 'Lower', 'No transpose', 'Unit', L,
$ K-L, ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
* T_{1,2} = V_{3,1}'*V_{3,2} + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL SGEMM('Transpose', 'No transpose', L, K-L, N-K, ONE,
$ V(K+1, 1), LDV, V(K+1, L+1), LDV, ONE,
$ T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1'*V_2
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL STRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL STRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)

ELSE IF(LQ) THEN
*
* Break V apart into 6 components
*
* V = |----------------------|
* |V_{1,1} V_{1,2} V{1,3}|
* |0 V_{2,2} V{2,3}|
* |----------------------|
*
* V_{1,1}\in\R^{l,l} unit upper triangular
* V_{1,2}\in\R^{l,k-l} rectangular
* V_{1,3}\in\R^{l,n-k} rectangular
*
* V_{2,2}\in\R^{k-l,k-l} unit upper triangular
* V_{2,3}\in\R^{k-l,n-k} rectangular
*
* Where l = floor(k/2)
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{l, l} upper triangular
* T_{2,2}\in\R^{k-l, k-l} upper triangular
* T_{1,2}\in\R^{l, k-l} rectangular
*
* Then, consider the product:
*
* (I - V_1'*T_{1,1}*V_1)*(I - V_2'*T_{2,2}*V_2)
* = I - V_1'*T_{1,1}*V_1 - V_2'*T_{2,2}*V_2 + V_1'*T_{1,1}*V_1*V_2'*T_{2,2}*V_2
*
* Define T_{1,2} = -T_{1,1}*V_1*V_2'*T_{2,2}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL SLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL SLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)

*
* Compute T_{1,2}
* T_{1,2} = V_{1,2}
*
CALL SLACPY('All', L, K-L, V(1, L+1), LDV, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*V_{2,2}'
*
CALL STRMM('Right', 'Upper', 'Transpose', 'Unit', L, K-L,
$ ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
* T_{1,2} = V_{1,3}*V_{2,3}' + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL SGEMM('No transpose', 'Transpose', L, K-L, N-K, ONE,
$ V(1, K+1), LDV, V(L+1, K+1), LDV, ONE,
$ T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1*V_2'
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL STRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)

*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL STRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)
ELSE IF(QL) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} V_{1,2}|
* |V_{2,1} V_{2,2}|
* |0 V_{3,2}|
* |---------------|
*
* V_{1,1}\in\R^{n-k,k-l} rectangular
* V_{2,1}\in\R^{k-l,k-l} unit upper triangular
*
* V_{1,2}\in\R^{n-k,l} rectangular
* V_{2,2}\in\R^{k-l,l} rectangular
* V_{3,2}\in\R^{l,l} unit upper triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\R^{l, l} non-unit lower triangular
* T_{2,1}\in\R^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2*T_{2,2}*V_2')*(I - V_1*T_{1,1}*V_1')
* = I - V_2*T_{2,2}*V_2' - V_1*T_{1,1}*V_1' + V_2*T_{2,2}*V_2'*V_1*T_{1,1}*V_1'
*
* Define T_{2,1} = -T_{2,2}*V_2'*V_1*T_{1,1}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL SLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL SLARFT(DIRECT, STOREV, N, L, V(1, K-L+1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}'
*
DO J = 1, K-L
DO I = 1, L
T(K-L+I, J) = V(N-K+J, K-L+I)
END DO
END DO
END IF
RETURN
*
* End of SLARFT
* T_{2,1} = T_{2,1}*V_{2,1}
*
CALL STRMM('Right', 'Upper', 'No transpose', 'Unit', L,
$ K-L, ONE, V(N-K+1, 1), LDV, T(K-L+1, 1), LDT)

*
* T_{2,1} = V_{2,2}'*V_{2,1} + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL SGEMM('Transpose', 'No transpose', L, K-L, N-K, ONE,
$ V(1, K-L+1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)
*
* At this point, we have that T_{2,1} = V_2'*V_1
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL STRMM('Left', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)
*
END
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL STRMM('Right', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
ELSE
*
* Else means RQ case
*
* Break V apart into 6 components
*
* V = |-----------------------|
* |V_{1,1} V_{1,2} 0 |
* |V_{2,1} V_{2,2} V_{2,3}|
* |-----------------------|
*
* V_{1,1}\in\R^{k-l,n-k} rectangular
* V_{1,2}\in\R^{k-l,k-l} unit lower triangular
*
* V_{2,1}\in\R^{l,n-k} rectangular
* V_{2,2}\in\R^{l,k-l} rectangular
* V_{2,3}\in\R^{l,l} unit lower triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\R^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\R^{l, l} non-unit lower triangular
* T_{2,1}\in\R^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2'*T_{2,2}*V_2)*(I - V_1'*T_{1,1}*V_1)
* = I - V_2'*T_{2,2}*V_2 - V_1'*T_{1,1}*V_1 + V_2'*T_{2,2}*V_2*V_1'*T_{1,1}*V_1
*
* Define T_{2,1} = -T_{2,2}*V_2*V_1'*T_{1,1}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'TV
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL SLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL SLARFT(DIRECT, STOREV, N, L, V(K-L+1, 1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}
*
CALL SLACPY('All', L, K-L, V(K-L+1, N-K+1), LDV,
$ T(K-L+1, 1), LDT)

*
* T_{2,1} = T_{2,1}*V_{1,2}'
*
CALL STRMM('Right', 'Lower', 'Transpose', 'Unit', L, K-L,
$ ONE, V(1, N-K+1), LDV, T(K-L+1, 1), LDT)

*
* T_{2,1} = V_{2,1}*V_{1,1}' + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL SGEMM('No transpose', 'Transpose', L, K-L, N-K, ONE,
$ V(K-L+1, 1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)

*
* At this point, we have that T_{2,1} = V_2*V_1'
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL STRMM('Left', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)

*
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL STRMM('Right', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
END IF
END SUBROUTINE

+ 452
- 144
lapack-netlib/SRC/zlarft.f View File

@@ -18,7 +18,7 @@
* Definition:
* ===========
*
* SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
* RECURSIVE SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
*
* .. Scalar Arguments ..
* CHARACTER DIRECT, STOREV
@@ -130,7 +130,7 @@
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16OTHERauxiliary
*> \ingroup larft
*
*> \par Further Details:
* =====================
@@ -159,166 +159,474 @@
*> \endverbatim
*>
* =====================================================================
SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )
RECURSIVE SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV,
$ TAU, T, LDT )
*
* -- LAPACK auxiliary routine --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
* .. Scalar Arguments ..
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* .. Scalar Arguments
*
CHARACTER DIRECT, STOREV
INTEGER K, LDT, LDV, N
* ..
* .. Array Arguments ..
COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* =====================================================================
COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * )
* ..
*
* .. Parameters ..
COMPLEX*16 ONE, ZERO
PARAMETER ( ONE = ( 1.0D+0, 0.0D+0 ),
$ ZERO = ( 0.0D+0, 0.0D+0 ) )
* ..
*
COMPLEX*16 ONE, NEG_ONE, ZERO
PARAMETER(ONE=1.0D+0, ZERO = 0.0D+0, NEG_ONE=-1.0D+0)
*
* .. Local Scalars ..
INTEGER I, J, PREVLASTV, LASTV
* ..
*
INTEGER I,J,L
LOGICAL QR,LQ,QL,DIRF,COLV
*
* .. External Subroutines ..
EXTERNAL ZGEMV, ZTRMV, ZGEMM
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
*
EXTERNAL ZTRMM,ZGEMM,ZLACPY
*
* .. External Functions..
*
LOGICAL LSAME
EXTERNAL LSAME
*
* .. Intrinsic Functions..
*
INTRINSIC CONJG
*
* The general scheme used is inspired by the approach inside DGEQRT3
* which was (at the time of writing this code):
* Based on the algorithm of Elmroth and Gustavson,
* IBM J. Res. Develop. Vol 44 No. 4 July 2000.
* ..
* .. Executable Statements ..
*
* Quick return if possible
*
IF( N.EQ.0 )
$ RETURN
*
IF( LSAME( DIRECT, 'F' ) ) THEN
PREVLASTV = N
DO I = 1, K
PREVLASTV = MAX( PREVLASTV, I )
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 ) * CONJG( V( I , J ) )
END DO
J = MIN( LASTV, PREVLASTV )
*
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**H * V(i:j,i)
*
CALL ZGEMV( 'Conjugate 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)**H
*
CALL ZGEMM( 'N', 'C', I-1, 1, J-I, -TAU( I ),
$ V( 1, I+1 ), LDV, V( I, I+1 ), LDV,
$ ONE, T( 1, I ), LDT )
END IF
*
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i)
*
CALL ZTRMV( '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
IF(N.EQ.0.OR.K.EQ.0) THEN
RETURN
END IF
*
* Base case
*
IF(N.EQ.1.OR.K.EQ.1) THEN
T(1,1) = TAU(1)
RETURN
END IF
*
* Beginning of executable statements
*
L = K / 2
*
* Determine what kind of Q we need to compute
* We assume that if the user doesn't provide 'F' for DIRECT,
* then they meant to provide 'B' and if they don't provide
* 'C' for STOREV, then they meant to provide 'R'
*
DIRF = LSAME(DIRECT,'F')
COLV = LSAME(STOREV,'C')
*
* QR happens when we have forward direction in column storage
*
QR = DIRF.AND.COLV
*
* LQ happens when we have forward direction in row storage
*
LQ = DIRF.AND.(.NOT.COLV)
*
* QL happens when we have backward direction in column storage
*
QL = (.NOT.DIRF).AND.COLV
*
* The last case is RQ. Due to how we structured this, if the
* above 3 are false, then RQ must be true, so we never store
* this
* RQ happens when we have backward direction in row storage
* RQ = (.NOT.DIRF).AND.(.NOT.COLV)
*
IF(QR) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} 0 |
* |V_{2,1} V_{2,2}|
* |V_{3,1} V_{3,2}|
* |---------------|
*
* V_{1,1}\in\C^{l,l} unit lower triangular
* V_{2,1}\in\C^{k-l,l} rectangular
* V_{3,1}\in\C^{n-k,l} rectangular
*
* V_{2,2}\in\C^{k-l,k-l} unit lower triangular
* V_{3,2}\in\C^{n-k,k-l} rectangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\C^{l, l} upper triangular
* T_{2,2}\in\C^{k-l, k-l} upper triangular
* T_{1,2}\in\C^{l, k-l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_1*T_{1,1}*V_1')*(I - V_2*T_{2,2}*V_2')
* = I - V_1*T_{1,1}*V_1' - V_2*T_{2,2}*V_2' + V_1*T_{1,1}*V_1'*V_2*T_{2,2}*V_2'
*
* Define T_{1,2} = -T_{1,1}*V_1'*V_2*T_{2,2}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL ZLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL ZLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)
*
* Compute T_{1,2}
* T_{1,2} = V_{2,1}'
*
DO J = 1, L
DO I = 1, K-L
T(J, L+I) = CONJG(V(L+I, J))
END DO
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 ) * CONJG( 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)**H * V(j:n-k+i,i)
*
CALL ZGEMV( 'Conjugate 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)**H
*
CALL ZGEMM( 'N', 'C', K-I, 1, N-K+I-J, -TAU( I ),
$ V( I+1, J ), LDV, V( I, J ), LDV,
$ ONE, T( I+1, I ), LDT )
END IF
*
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i)
*
CALL ZTRMV( '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
*
* T_{1,2} = T_{1,2}*V_{2,2}
*
CALL ZTRMM('Right', 'Lower', 'No transpose', 'Unit', L,
$ K-L, ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
* T_{1,2} = V_{3,1}'*V_{3,2} + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL ZGEMM('Conjugate', 'No transpose', L, K-L, N-K, ONE,
$ V(K+1, 1), LDV, V(K+1, L+1), LDV, ONE,
$ T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1'*V_2
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL ZTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL ZTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)

ELSE IF(LQ) THEN
*
* Break V apart into 6 components
*
* V = |----------------------|
* |V_{1,1} V_{1,2} V{1,3}|
* |0 V_{2,2} V{2,3}|
* |----------------------|
*
* V_{1,1}\in\C^{l,l} unit upper triangular
* V_{1,2}\in\C^{l,k-l} rectangular
* V_{1,3}\in\C^{l,n-k} rectangular
*
* V_{2,2}\in\C^{k-l,k-l} unit upper triangular
* V_{2,3}\in\C^{k-l,n-k} rectangular
*
* Where l = floor(k/2)
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} T_{1,2}|
* |0 T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\C^{l, l} upper triangular
* T_{2,2}\in\C^{k-l, k-l} upper triangular
* T_{1,2}\in\C^{l, k-l} rectangular
*
* Then, consider the product:
*
* (I - V_1'*T_{1,1}*V_1)*(I - V_2'*T_{2,2}*V_2)
* = I - V_1'*T_{1,1}*V_1 - V_2'*T_{2,2}*V_2 + V_1'*T_{1,1}*V_1*V_2'*T_{2,2}*V_2
*
* Define T_{1,2} = -T_{1,1}*V_1*V_2'*T_{2,2}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{1,2}
*
* Compute T_{1,1} recursively
*
CALL ZLARFT(DIRECT, STOREV, N, L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL ZLARFT(DIRECT, STOREV, N-L, K-L, V(L+1, L+1), LDV,
$ TAU(L+1), T(L+1, L+1), LDT)

*
* Compute T_{1,2}
* T_{1,2} = V_{1,2}
*
CALL ZLACPY('All', L, K-L, V(1, L+1), LDV, T(1, L+1), LDT)
*
* T_{1,2} = T_{1,2}*V_{2,2}'
*
CALL ZTRMM('Right', 'Upper', 'Conjugate', 'Unit', L, K-L,
$ ONE, V(L+1, L+1), LDV, T(1, L+1), LDT)

*
* T_{1,2} = V_{1,3}*V_{2,3}' + T_{1,2}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL ZGEMM('No transpose', 'Conjugate', L, K-L, N-K, ONE,
$ V(1, K+1), LDV, V(L+1, K+1), LDV, ONE,
$ T(1, L+1), LDT)
*
* At this point, we have that T_{1,2} = V_1*V_2'
* All that is left is to pre and post multiply by -T_{1,1} and T_{2,2}
* respectively.
*
* T_{1,2} = -T_{1,1}*T_{1,2}
*
CALL ZTRMM('Left', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T, LDT, T(1, L+1), LDT)

*
* T_{1,2} = T_{1,2}*T_{2,2}
*
CALL ZTRMM('Right', 'Upper', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T(L+1, L+1), LDT, T(1, L+1), LDT)
ELSE IF(QL) THEN
*
* Break V apart into 6 components
*
* V = |---------------|
* |V_{1,1} V_{1,2}|
* |V_{2,1} V_{2,2}|
* |0 V_{3,2}|
* |---------------|
*
* V_{1,1}\in\C^{n-k,k-l} rectangular
* V_{2,1}\in\C^{k-l,k-l} unit upper triangular
*
* V_{1,2}\in\C^{n-k,l} rectangular
* V_{2,2}\in\C^{k-l,l} rectangular
* V_{3,2}\in\C^{l,l} unit upper triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\C^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\C^{l, l} non-unit lower triangular
* T_{2,1}\in\C^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2*T_{2,2}*V_2')*(I - V_1*T_{1,1}*V_1')
* = I - V_2*T_{2,2}*V_2' - V_1*T_{1,1}*V_1' + V_2*T_{2,2}*V_2'*V_1*T_{1,1}*V_1'
*
* Define T_{2,1} = -T_{2,2}*V_2'*V_1*T_{1,1}
*
* Then, we can define the matrix V as
* V = |-------|
* |V_1 V_2|
* |-------|
*
* So, our product is equivalent to the matrix product
* I - V*T*V'
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL ZLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL ZLARFT(DIRECT, STOREV, N, L, V(1, K-L+1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}'
*
DO J = 1, K-L
DO I = 1, L
T(K-L+I, J) = CONJG(V(N-K+J, K-L+I))
END DO
END DO
END IF
RETURN
*
* End of ZLARFT
* T_{2,1} = T_{2,1}*V_{2,1}
*
CALL ZTRMM('Right', 'Upper', 'No transpose', 'Unit', L,
$ K-L, ONE, V(N-K+1, 1), LDV, T(K-L+1, 1), LDT)

*
* T_{2,1} = V_{2,2}'*V_{2,1} + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL ZGEMM('Conjugate', 'No transpose', L, K-L, N-K, ONE,
$ V(1, K-L+1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)
*
* At this point, we have that T_{2,1} = V_2'*V_1
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL ZTRMM('Left', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)
*
END
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL ZTRMM('Right', 'Lower', 'No transpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
ELSE
*
* Else means RQ case
*
* Break V apart into 6 components
*
* V = |-----------------------|
* |V_{1,1} V_{1,2} 0 |
* |V_{2,1} V_{2,2} V_{2,3}|
* |-----------------------|
*
* V_{1,1}\in\C^{k-l,n-k} rectangular
* V_{1,2}\in\C^{k-l,k-l} unit lower triangular
*
* V_{2,1}\in\C^{l,n-k} rectangular
* V_{2,2}\in\C^{l,k-l} rectangular
* V_{2,3}\in\C^{l,l} unit lower triangular
*
* We will construct the T matrix
* T = |---------------|
* |T_{1,1} 0 |
* |T_{2,1} T_{2,2}|
* |---------------|
*
* T is the triangular factor obtained from block reflectors.
* To motivate the structure, assume we have already computed T_{1,1}
* and T_{2,2}. Then collect the associated reflectors in V_1 and V_2
*
* T_{1,1}\in\C^{k-l, k-l} non-unit lower triangular
* T_{2,2}\in\C^{l, l} non-unit lower triangular
* T_{2,1}\in\C^{k-l, l} rectangular
*
* Where l = floor(k/2)
*
* Then, consider the product:
*
* (I - V_2'*T_{2,2}*V_2)*(I - V_1'*T_{1,1}*V_1)
* = I - V_2'*T_{2,2}*V_2 - V_1'*T_{1,1}*V_1 + V_2'*T_{2,2}*V_2*V_1'*T_{1,1}*V_1
*
* Define T_{2,1} = -T_{2,2}*V_2*V_1'*T_{1,1}
*
* Then, we can define the matrix V as
* V = |---|
* |V_1|
* |V_2|
* |---|
*
* So, our product is equivalent to the matrix product
* I - V'*T*V
* This means, we can compute T_{1,1} and T_{2,2}, then use this information
* to compute T_{2,1}
*
* Compute T_{1,1} recursively
*
CALL ZLARFT(DIRECT, STOREV, N-L, K-L, V, LDV, TAU, T, LDT)
*
* Compute T_{2,2} recursively
*
CALL ZLARFT(DIRECT, STOREV, N, L, V(K-L+1, 1), LDV,
$ TAU(K-L+1), T(K-L+1, K-L+1), LDT)
*
* Compute T_{2,1}
* T_{2,1} = V_{2,2}
*
CALL ZLACPY('All', L, K-L, V(K-L+1, N-K+1), LDV,
$ T(K-L+1, 1), LDT)

*
* T_{2,1} = T_{2,1}*V_{1,2}'
*
CALL ZTRMM('Right', 'Lower', 'Conjugate', 'Unit', L, K-L,
$ ONE, V(1, N-K+1), LDV, T(K-L+1, 1), LDT)

*
* T_{2,1} = V_{2,1}*V_{1,1}' + T_{2,1}
* Note: We assume K <= N, and GEMM will do nothing if N=K
*
CALL ZGEMM('No transpose', 'Conjugate', L, K-L, N-K, ONE,
$ V(K-L+1, 1), LDV, V, LDV, ONE, T(K-L+1, 1),
$ LDT)

*
* At this point, we have that T_{2,1} = V_2*V_1'
* All that is left is to pre and post multiply by -T_{2,2} and T_{1,1}
* respectively.
*
* T_{2,1} = -T_{2,2}*T_{2,1}
*
CALL ZTRMM('Left', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, NEG_ONE, T(K-L+1, K-L+1), LDT,
$ T(K-L+1, 1), LDT)

*
* T_{2,1} = T_{2,1}*T_{1,1}
*
CALL ZTRMM('Right', 'Lower', 'No tranpose', 'Non-unit', L,
$ K-L, ONE, T, LDT, T(K-L+1, 1), LDT)
END IF
END SUBROUTINE

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