*> \brief \b ZLARZT forms the triangular factor T of a block reflector H = I - vtvH. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZLARZT + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZLARZT( 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 *> *> ZLARZT 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 *> *> Currently, only STOREV = 'R' and DIRECT = 'B' are supported. *> \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, not supported yet) *> = '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 (not supported yet) *> = '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,out] 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 complex16OTHERcomputational * *> \par Contributors: * ================== *> *> A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA * *> \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; 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_____ *> ( v1 v2 v3 ) / \ *> ( v1 v2 v3 ) ( v1 v1 v1 v1 v1 . . . . 1 ) *> V = ( v1 v2 v3 ) ( v2 v2 v2 v2 v2 . . . 1 ) *> ( v1 v2 v3 ) ( v3 v3 v3 v3 v3 . . 1 ) *> ( v1 v2 v3 ) *> . . . *> . . . *> 1 . . *> 1 . *> 1 *> *> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': *> *> ______V_____ *> 1 / \ *> . 1 ( 1 . . . . v1 v1 v1 v1 v1 ) *> . . 1 ( . 1 . . . v2 v2 v2 v2 v2 ) *> . . . ( . . 1 . . v3 v3 v3 v3 v3 ) *> . . . *> ( v1 v2 v3 ) *> ( v1 v2 v3 ) *> V = ( v1 v2 v3 ) *> ( v1 v2 v3 ) *> ( v1 v2 v3 ) *> \endverbatim *> * ===================================================================== SUBROUTINE ZLARZT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT ) * * -- LAPACK computational 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 ZERO PARAMETER ( ZERO = ( 0.0D+0, 0.0D+0 ) ) * .. * .. Local Scalars .. INTEGER I, INFO, J * .. * .. External Subroutines .. EXTERNAL XERBLA, ZGEMV, ZLACGV, ZTRMV * .. * .. External Functions .. LOGICAL LSAME EXTERNAL LSAME * .. * .. Executable Statements .. * * Check for currently supported options * INFO = 0 IF( .NOT.LSAME( DIRECT, 'B' ) ) THEN INFO = -1 ELSE IF( .NOT.LSAME( STOREV, 'R' ) ) THEN INFO = -2 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZLARZT', -INFO ) RETURN END IF * DO 20 I = K, 1, -1 IF( TAU( I ).EQ.ZERO ) THEN * * H(i) = I * DO 10 J = I, K T( J, I ) = ZERO 10 CONTINUE ELSE * * general case * IF( I.LT.K ) THEN * * T(i+1:k,i) = - tau(i) * V(i+1:k,1:n) * V(i,1:n)**H * CALL ZLACGV( N, V( I, 1 ), LDV ) CALL ZGEMV( 'No transpose', K-I, N, -TAU( I ), $ V( I+1, 1 ), LDV, V( I, 1 ), LDV, ZERO, $ T( I+1, I ), 1 ) CALL ZLACGV( N, V( I, 1 ), LDV ) * * 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 ) END IF T( I, I ) = TAU( I ) END IF 20 CONTINUE RETURN * * End of ZLARZT * END