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replace ?larft with a recursive implementation (Reference-LAPACK PR 1080)

tags/v0.3.29
Martin Kroeker GitHub 8 months ago
parent
5527eda561
commit
ed516994d6
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4 changed files with 1792 additions and 568 deletions
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  1. +451
    -145
      lapack-netlib/SRC/clarft.f
  2. +445
    -140
      lapack-netlib/SRC/dlarft.f
  3. +444
    -139
      lapack-netlib/SRC/slarft.f
  4. +452
    -144
      lapack-netlib/SRC/zlarft.f

+ 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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