Chimie Théorique » Opt'n Path

      SUBROUTINE DLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT )

      IMPLICIT NONE

*

*  -- LAPACK auxiliary routine (version 3.2) --

*  -- LAPACK is a software package provided by Univ. of Tennessee,    --

*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--

*     November 2006

*

*     .. Scalar Arguments ..

      CHARACTER          DIRECT, STOREV

      INTEGER            K, LDT, LDV, N

*     ..

*     .. Array Arguments ..

      DOUBLE PRECISION   T( LDT, * ), TAU( * ), V( LDV, * )

*     ..

*

*  Purpose

*  =======

*

*  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'

*

*  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 * V

*

*  Arguments

*  =========

*

*  DIRECT  (input) 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)

*

*  STOREV  (input) CHARACTER*1

*          Specifies how the vectors which define the elementary

*          reflectors are stored (see also Further Details):

*          = 'C': columnwise

*          = 'R': rowwise

*

*  N       (input) INTEGER

*          The order of the block reflector H. N >= 0.

*

*  K       (input) INTEGER

*          The order of the triangular factor T (= the number of

*          elementary reflectors). K >= 1.

*

*  V       (input/output) DOUBLE PRECISION array, dimension

*                               (LDV,K) if STOREV = 'C'

*                               (LDV,N) if STOREV = 'R'

*          The matrix V. See further details.

*

*  LDV     (input) INTEGER

*          The leading dimension of the array V.

*          If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K.

*

*  TAU     (input) DOUBLE PRECISION array, dimension (K)

*          TAU(i) must contain the scalar factor of the elementary

*          reflector H(i).

*

*  T       (output) 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.

*

*  LDT     (input) INTEGER

*          The leading dimension of the array T. LDT >= K.

*

*  Further Details

*  ===============

*

*  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; the corresponding

*  array elements are modified but restored on exit. The rest of the

*  array is not used.

*

*  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 )

*

*  =====================================================================

*

*     .. Parameters ..

      DOUBLE PRECISION   ONE, ZERO

      PARAMETER          ( ONE = 1.0D+0, ZERO = 0.0D+0 )

*     ..

*     .. Local Scalars ..

      INTEGER            I, J, PREVLASTV, LASTV

      DOUBLE PRECISION   VII

*     ..

*     .. 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 20 I = 1, K

            PREVLASTV = MAX( I, PREVLASTV )

            IF( TAU( I ).EQ.ZERO ) THEN

*

*              H(i)  =  I

*

               DO 10 J = 1, I

                  T( J, I ) = ZERO

   10          CONTINUE

            ELSE

*

*              general case

*

               VII = V( I, I )

               V( I, I ) = ONE

               IF( LSAME( STOREV, 'C' ) ) THEN

!                 Skip any trailing zeros.

                  DO LASTV = N, I+1, -1

                     IF( V( LASTV, I ).NE.ZERO ) EXIT

                  END DO

                  J = MIN( LASTV, PREVLASTV )

*

*                 T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)' * V(i:j,i)

*

                  CALL DGEMV( 'Transpose', J-I+1, I-1, -TAU( I ),

     $                        V( I, 1 ), LDV, V( I, I ), 1, ZERO,

     $                        T( 1, I ), 1 )

               ELSE

!                 Skip any trailing zeros.

                  DO LASTV = N, I+1, -1

                     IF( V( I, LASTV ).NE.ZERO ) EXIT

                  END DO

                  J = MIN( LASTV, PREVLASTV )

*

*                 T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)'

*

                  CALL DGEMV( 'No transpose', I-1, J-I+1, -TAU( I ),

     $                        V( 1, I ), LDV, V( I, I ), LDV, ZERO,

     $                        T( 1, I ), 1 )

               END IF

               V( I, I ) = VII

*

*              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

   20    CONTINUE

      ELSE

         PREVLASTV = 1

         DO 40 I = K, 1, -1

            IF( TAU( I ).EQ.ZERO ) THEN

*

*              H(i)  =  I

*

               DO 30 J = I, K

                  T( J, I ) = ZERO

   30          CONTINUE

            ELSE

*

*              general case

*

               IF( I.LT.K ) THEN

                  IF( LSAME( STOREV, 'C' ) ) THEN

                     VII = V( N-K+I, I )

                     V( N-K+I, I ) = ONE

!                    Skip any leading zeros.

                     DO LASTV = 1, I-1

                        IF( V( LASTV, I ).NE.ZERO ) EXIT

                     END DO

                     J = MAX( LASTV, PREVLASTV )

*

*                    T(i+1:k,i) :=

*                            - tau(i) * V(j:n-k+i,i+1:k)' * V(j:n-k+i,i)

*

                     CALL DGEMV( 'Transpose', N-K+I-J+1, K-I, -TAU( I ),

     $                           V( J, I+1 ), LDV, V( J, I ), 1, ZERO,

     $                           T( I+1, I ), 1 )

                     V( N-K+I, I ) = VII

                  ELSE

                     VII = V( I, N-K+I )

                     V( I, N-K+I ) = ONE

!                    Skip any leading zeros.

                     DO LASTV = 1, I-1

                        IF( V( I, LASTV ).NE.ZERO ) EXIT

                     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)'

*

                     CALL DGEMV( 'No transpose', K-I, N-K+I-J+1,

     $                    -TAU( I ), V( I+1, J ), LDV, V( I, J ), LDV,

     $                    ZERO, T( I+1, I ), 1 )

                     V( I, N-K+I ) = VII

                  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

   40    CONTINUE

      END IF

      RETURN

*

*     End of DLARFT

*

      END