zlarft.f

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*> \brief \b ZLARFT forms the triangular factor T of a block reflector H = I - vtvH
*
*  =========== DOCUMENTATION ===========
*
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*
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*
*  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 complex16OTHERauxiliary
*
*> \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 )
.EQ.            IF( TAU( I )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
.NE.                     IF( V( LASTV, I )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
.NE.                     IF( V( I, LASTV )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 )
.GT.               IF( I1 ) THEN
                  PREVLASTV = MAX( PREVLASTV, LASTV )
               ELSE
                  PREVLASTV = LASTV
               END IF
             END IF
         END DO
      ELSE
         PREVLASTV = 1
         DO I = K, 1, -1
.EQ.            IF( TAU( I )ZERO ) THEN
*
*              H(i)  =  I
*
               DO J = I, K
                  T( J, I ) = ZERO
               END DO
            ELSE
*
*              general case
*
.LT.               IF( IK ) THEN
                  IF( LSAME( STOREV, 'c' ) ) THEN
*                    Skip any leading zeros.
                     DO LASTV = 1, I-1
.NE.                        IF( V( LASTV, I )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
.NE.                        IF( V( I, LASTV )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 )
.GT.                  IF( I1 ) 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