| *> \brief \b SLARFT | |
| * | |
| * =========== 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. | |
| * | |
| *> \date April 2012 | |
| * | |
| *> \ingroup realOTHERauxiliary | |
| * | |
| *> \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 (version 3.4.1) -- | |
| * -- LAPACK is a software package provided by Univ. of Tennessee, -- | |
| * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- | |
| * April 2012 | |
| * | |
| * .. 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 | |