version 1.6, 2010/08/13 21:04:10
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version 1.15, 2014/01/27 09:28:39
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*> \brief \b ZLARFT forms the triangular factor T of a block reflector H = I - vtvH |
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* |
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* =========== DOCUMENTATION =========== |
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* |
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* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
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* |
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*> \htmlonly |
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*> Download ZLARFT + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zlarft.f"> |
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*> [TGZ]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zlarft.f"> |
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*> [ZIP]</a> |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zlarft.f"> |
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*> [TXT]</a> |
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*> \endhtmlonly |
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* |
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* Definition: |
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* =========== |
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* |
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* SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER DIRECT, STOREV |
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* INTEGER K, LDT, LDV, N |
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* .. |
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* .. Array Arguments .. |
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* COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * ) |
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* .. |
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* |
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* |
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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> ZLARFT forms the triangular factor T of a complex block reflector H |
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*> of order n, which is defined as a product of k elementary reflectors. |
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*> |
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*> If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; |
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*> |
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*> If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. |
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*> |
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*> If STOREV = 'C', the vector which defines the elementary reflector |
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*> H(i) is stored in the i-th column of the array V, and |
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*> |
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*> H = I - V * T * V**H |
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*> |
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*> If STOREV = 'R', the vector which defines the elementary reflector |
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*> H(i) is stored in the i-th row of the array V, and |
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*> |
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*> H = I - V**H * T * V |
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*> \endverbatim |
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* |
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* Arguments: |
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* ========== |
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* |
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*> \param[in] DIRECT |
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*> \verbatim |
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*> DIRECT is CHARACTER*1 |
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*> Specifies the order in which the elementary reflectors are |
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*> multiplied to form the block reflector: |
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*> = 'F': H = H(1) H(2) . . . H(k) (Forward) |
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*> = 'B': H = H(k) . . . H(2) H(1) (Backward) |
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*> \endverbatim |
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*> |
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*> \param[in] STOREV |
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*> \verbatim |
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*> STOREV is CHARACTER*1 |
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*> Specifies how the vectors which define the elementary |
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*> reflectors are stored (see also Further Details): |
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*> = 'C': columnwise |
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*> = 'R': rowwise |
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*> \endverbatim |
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*> |
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*> \param[in] N |
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*> \verbatim |
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*> N is INTEGER |
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*> The order of the block reflector H. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in] K |
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*> \verbatim |
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*> K is INTEGER |
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*> The order of the triangular factor T (= the number of |
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*> elementary reflectors). K >= 1. |
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*> \endverbatim |
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*> |
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*> \param[in] V |
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*> \verbatim |
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*> V is COMPLEX*16 array, dimension |
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*> (LDV,K) if STOREV = 'C' |
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*> (LDV,N) if STOREV = 'R' |
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*> The matrix V. See further details. |
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*> \endverbatim |
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*> |
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*> \param[in] LDV |
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*> \verbatim |
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*> LDV is INTEGER |
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*> The leading dimension of the array V. |
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*> If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K. |
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*> \endverbatim |
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*> |
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*> \param[in] TAU |
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*> \verbatim |
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*> TAU is COMPLEX*16 array, dimension (K) |
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*> TAU(i) must contain the scalar factor of the elementary |
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*> reflector H(i). |
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*> \endverbatim |
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*> |
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*> \param[out] T |
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*> \verbatim |
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*> T is COMPLEX*16 array, dimension (LDT,K) |
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*> The k by k triangular factor T of the block reflector. |
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*> If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is |
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*> lower triangular. The rest of the array is not used. |
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*> \endverbatim |
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*> |
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*> \param[in] LDT |
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*> \verbatim |
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*> LDT is INTEGER |
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*> The leading dimension of the array T. LDT >= K. |
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*> \endverbatim |
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* |
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* Authors: |
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* ======== |
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* |
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*> \author Univ. of Tennessee |
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*> \author Univ. of California Berkeley |
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*> \author Univ. of Colorado Denver |
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*> \author NAG Ltd. |
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* |
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*> \date September 2012 |
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* |
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*> \ingroup complex16OTHERauxiliary |
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* |
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*> \par Further Details: |
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* ===================== |
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*> |
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*> \verbatim |
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*> |
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*> The shape of the matrix V and the storage of the vectors which define |
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*> the H(i) is best illustrated by the following example with n = 5 and |
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*> k = 3. The elements equal to 1 are not stored. |
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*> |
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*> DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': |
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*> |
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*> V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) |
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*> ( v1 1 ) ( 1 v2 v2 v2 ) |
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*> ( v1 v2 1 ) ( 1 v3 v3 ) |
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*> ( v1 v2 v3 ) |
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*> ( v1 v2 v3 ) |
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*> |
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*> DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': |
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*> |
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*> V = ( v1 v2 v3 ) V = ( v1 v1 1 ) |
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*> ( v1 v2 v3 ) ( v2 v2 v2 1 ) |
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*> ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) |
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*> ( 1 v3 ) |
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*> ( 1 ) |
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*> \endverbatim |
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*> |
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* ===================================================================== |
SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT ) |
SUBROUTINE ZLARFT( DIRECT, STOREV, N, K, V, LDV, TAU, T, LDT ) |
* |
* |
* -- LAPACK auxiliary routine (version 3.2) -- |
* -- LAPACK auxiliary routine (version 3.4.2) -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- LAPACK is a software package provided by Univ. of Tennessee, -- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- |
* November 2006 |
* September 2012 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER DIRECT, STOREV |
CHARACTER DIRECT, STOREV |
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COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * ) |
COMPLEX*16 T( LDT, * ), TAU( * ), V( LDV, * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* ZLARFT forms the triangular factor T of a complex block reflector H |
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* of order n, which is defined as a product of k elementary reflectors. |
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* |
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* If DIRECT = 'F', H = H(1) H(2) . . . H(k) and T is upper triangular; |
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* |
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* If DIRECT = 'B', H = H(k) . . . H(2) H(1) and T is lower triangular. |
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* |
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* If STOREV = 'C', the vector which defines the elementary reflector |
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* H(i) is stored in the i-th column of the array V, and |
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* |
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* H = I - V * T * V' |
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* |
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* If STOREV = 'R', the vector which defines the elementary reflector |
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* H(i) is stored in the i-th row of the array V, and |
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* |
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* H = I - V' * T * V |
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* |
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* Arguments |
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* ========= |
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* |
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* DIRECT (input) CHARACTER*1 |
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* Specifies the order in which the elementary reflectors are |
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* multiplied to form the block reflector: |
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* = 'F': H = H(1) H(2) . . . H(k) (Forward) |
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* = 'B': H = H(k) . . . H(2) H(1) (Backward) |
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* |
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* STOREV (input) CHARACTER*1 |
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* Specifies how the vectors which define the elementary |
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* reflectors are stored (see also Further Details): |
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* = 'C': columnwise |
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* = 'R': rowwise |
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* |
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* N (input) INTEGER |
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* The order of the block reflector H. N >= 0. |
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* |
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* K (input) INTEGER |
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* The order of the triangular factor T (= the number of |
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* elementary reflectors). K >= 1. |
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* |
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* V (input/output) COMPLEX*16 array, dimension |
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* (LDV,K) if STOREV = 'C' |
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* (LDV,N) if STOREV = 'R' |
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* The matrix V. See further details. |
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* |
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* LDV (input) INTEGER |
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* The leading dimension of the array V. |
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* If STOREV = 'C', LDV >= max(1,N); if STOREV = 'R', LDV >= K. |
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* |
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* TAU (input) COMPLEX*16 array, dimension (K) |
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* TAU(i) must contain the scalar factor of the elementary |
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* reflector H(i). |
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* |
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* T (output) COMPLEX*16 array, dimension (LDT,K) |
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* The k by k triangular factor T of the block reflector. |
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* If DIRECT = 'F', T is upper triangular; if DIRECT = 'B', T is |
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* lower triangular. The rest of the array is not used. |
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* |
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* LDT (input) INTEGER |
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* The leading dimension of the array T. LDT >= K. |
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* |
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* Further Details |
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* =============== |
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* |
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* The shape of the matrix V and the storage of the vectors which define |
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* the H(i) is best illustrated by the following example with n = 5 and |
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* k = 3. The elements equal to 1 are not stored; the corresponding |
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* array elements are modified but restored on exit. The rest of the |
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* array is not used. |
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* |
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* DIRECT = 'F' and STOREV = 'C': DIRECT = 'F' and STOREV = 'R': |
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* |
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* V = ( 1 ) V = ( 1 v1 v1 v1 v1 ) |
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* ( v1 1 ) ( 1 v2 v2 v2 ) |
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* ( v1 v2 1 ) ( 1 v3 v3 ) |
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* ( v1 v2 v3 ) |
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* ( v1 v2 v3 ) |
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* |
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* DIRECT = 'B' and STOREV = 'C': DIRECT = 'B' and STOREV = 'R': |
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* |
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* V = ( v1 v2 v3 ) V = ( v1 v1 1 ) |
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* ( v1 v2 v3 ) ( v2 v2 v2 1 ) |
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* ( 1 v2 v3 ) ( v3 v3 v3 v3 1 ) |
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* ( 1 v3 ) |
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* ( 1 ) |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* .. |
* .. |
* .. Local Scalars .. |
* .. Local Scalars .. |
INTEGER I, J, PREVLASTV, LASTV |
INTEGER I, J, PREVLASTV, LASTV |
COMPLEX*16 VII |
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* .. |
* .. |
* .. External Subroutines .. |
* .. External Subroutines .. |
EXTERNAL ZGEMV, ZLACGV, ZTRMV |
EXTERNAL ZGEMV, ZLACGV, ZTRMV |
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* |
* |
IF( LSAME( DIRECT, 'F' ) ) THEN |
IF( LSAME( DIRECT, 'F' ) ) THEN |
PREVLASTV = N |
PREVLASTV = N |
DO 20 I = 1, K |
DO I = 1, K |
PREVLASTV = MAX( PREVLASTV, I ) |
PREVLASTV = MAX( PREVLASTV, I ) |
IF( TAU( I ).EQ.ZERO ) THEN |
IF( TAU( I ).EQ.ZERO ) THEN |
* |
* |
* H(i) = I |
* H(i) = I |
* |
* |
DO 10 J = 1, I |
DO J = 1, I |
T( J, I ) = ZERO |
T( J, I ) = ZERO |
10 CONTINUE |
END DO |
ELSE |
ELSE |
* |
* |
* general case |
* general case |
* |
* |
VII = V( I, I ) |
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V( I, I ) = ONE |
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IF( LSAME( STOREV, 'C' ) ) THEN |
IF( LSAME( STOREV, 'C' ) ) THEN |
! Skip any trailing zeros. |
* Skip any trailing zeros. |
DO LASTV = N, I+1, -1 |
DO LASTV = N, I+1, -1 |
IF( V( LASTV, I ).NE.ZERO ) EXIT |
IF( V( LASTV, I ).NE.ZERO ) EXIT |
END DO |
END DO |
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DO J = 1, I-1 |
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T( J, I ) = -TAU( I ) * CONJG( V( I , J ) ) |
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END DO |
J = MIN( LASTV, PREVLASTV ) |
J = MIN( LASTV, PREVLASTV ) |
* |
* |
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)' * V(i:j,i) |
* T(1:i-1,i) := - tau(i) * V(i:j,1:i-1)**H * V(i:j,i) |
* |
* |
CALL ZGEMV( 'Conjugate transpose', J-I+1, I-1, |
CALL ZGEMV( 'Conjugate transpose', J-I, I-1, |
$ -TAU( I ), V( I, 1 ), LDV, V( I, I ), 1, |
$ -TAU( I ), V( I+1, 1 ), LDV, |
$ ZERO, T( 1, I ), 1 ) |
$ V( I+1, I ), 1, ONE, T( 1, I ), 1 ) |
ELSE |
ELSE |
! Skip any trailing zeros. |
* Skip any trailing zeros. |
DO LASTV = N, I+1, -1 |
DO LASTV = N, I+1, -1 |
IF( V( I, LASTV ).NE.ZERO ) EXIT |
IF( V( I, LASTV ).NE.ZERO ) EXIT |
END DO |
END DO |
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DO J = 1, I-1 |
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T( J, I ) = -TAU( I ) * V( J , I ) |
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END DO |
J = MIN( LASTV, PREVLASTV ) |
J = MIN( LASTV, PREVLASTV ) |
* |
* |
* T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)' |
* T(1:i-1,i) := - tau(i) * V(1:i-1,i:j) * V(i,i:j)**H |
* |
* |
IF( I.LT.J ) |
CALL ZGEMM( 'N', 'C', I-1, 1, J-I, -TAU( I ), |
$ CALL ZLACGV( J-I, V( I, I+1 ), LDV ) |
$ V( 1, I+1 ), LDV, V( I, I+1 ), LDV, |
CALL ZGEMV( 'No transpose', I-1, J-I+1, -TAU( I ), |
$ ONE, T( 1, I ), LDT ) |
$ V( 1, I ), LDV, V( I, I ), LDV, ZERO, |
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$ T( 1, I ), 1 ) |
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IF( I.LT.J ) |
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$ CALL ZLACGV( J-I, V( I, I+1 ), LDV ) |
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END IF |
END IF |
V( I, I ) = VII |
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* |
* |
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i) |
* T(1:i-1,i) := T(1:i-1,1:i-1) * T(1:i-1,i) |
* |
* |
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PREVLASTV = LASTV |
PREVLASTV = LASTV |
END IF |
END IF |
END IF |
END IF |
20 CONTINUE |
END DO |
ELSE |
ELSE |
PREVLASTV = 1 |
PREVLASTV = 1 |
DO 40 I = K, 1, -1 |
DO I = K, 1, -1 |
IF( TAU( I ).EQ.ZERO ) THEN |
IF( TAU( I ).EQ.ZERO ) THEN |
* |
* |
* H(i) = I |
* H(i) = I |
* |
* |
DO 30 J = I, K |
DO J = I, K |
T( J, I ) = ZERO |
T( J, I ) = ZERO |
30 CONTINUE |
END DO |
ELSE |
ELSE |
* |
* |
* general case |
* general case |
* |
* |
IF( I.LT.K ) THEN |
IF( I.LT.K ) THEN |
IF( LSAME( STOREV, 'C' ) ) THEN |
IF( LSAME( STOREV, 'C' ) ) THEN |
VII = V( N-K+I, I ) |
* Skip any leading zeros. |
V( N-K+I, I ) = ONE |
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! Skip any leading zeros. |
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DO LASTV = 1, I-1 |
DO LASTV = 1, I-1 |
IF( V( LASTV, I ).NE.ZERO ) EXIT |
IF( V( LASTV, I ).NE.ZERO ) EXIT |
END DO |
END DO |
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DO J = I+1, K |
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T( J, I ) = -TAU( I ) * CONJG( V( N-K+I , J ) ) |
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END DO |
J = MAX( LASTV, PREVLASTV ) |
J = MAX( LASTV, PREVLASTV ) |
* |
* |
* T(i+1:k,i) := |
* T(i+1:k,i) = -tau(i) * V(j:n-k+i,i+1:k)**H * V(j:n-k+i,i) |
* - tau(i) * V(j:n-k+i,i+1:k)' * V(j:n-k+i,i) |
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* |
* |
CALL ZGEMV( 'Conjugate transpose', N-K+I-J+1, K-I, |
CALL ZGEMV( 'Conjugate transpose', N-K+I-J, K-I, |
$ -TAU( I ), V( J, I+1 ), LDV, V( J, I ), |
$ -TAU( I ), V( J, I+1 ), LDV, V( J, I ), |
$ 1, ZERO, T( I+1, I ), 1 ) |
$ 1, ONE, T( I+1, I ), 1 ) |
V( N-K+I, I ) = VII |
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ELSE |
ELSE |
VII = V( I, N-K+I ) |
* Skip any leading zeros. |
V( I, N-K+I ) = ONE |
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! Skip any leading zeros. |
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DO LASTV = 1, I-1 |
DO LASTV = 1, I-1 |
IF( V( I, LASTV ).NE.ZERO ) EXIT |
IF( V( I, LASTV ).NE.ZERO ) EXIT |
END DO |
END DO |
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DO J = I+1, K |
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T( J, I ) = -TAU( I ) * V( J, N-K+I ) |
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END DO |
J = MAX( LASTV, PREVLASTV ) |
J = MAX( LASTV, PREVLASTV ) |
* |
* |
* T(i+1:k,i) := |
* T(i+1:k,i) = -tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)**H |
* - tau(i) * V(i+1:k,j:n-k+i) * V(i,j:n-k+i)' |
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* |
* |
CALL ZLACGV( N-K+I-1-J+1, V( I, J ), LDV ) |
CALL ZGEMM( 'N', 'C', K-I, 1, N-K+I-J, -TAU( I ), |
CALL ZGEMV( 'No transpose', K-I, N-K+I-J+1, |
$ V( I+1, J ), LDV, V( I, J ), LDV, |
$ -TAU( I ), V( I+1, J ), LDV, V( I, J ), LDV, |
$ ONE, T( I+1, I ), LDT ) |
$ ZERO, T( I+1, I ), 1 ) |
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CALL ZLACGV( N-K+I-1-J+1, V( I, J ), LDV ) |
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V( I, N-K+I ) = VII |
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END IF |
END IF |
* |
* |
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i) |
* T(i+1:k,i) := T(i+1:k,i+1:k) * T(i+1:k,i) |
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END IF |
END IF |
T( I, I ) = TAU( I ) |
T( I, I ) = TAU( I ) |
END IF |
END IF |
40 CONTINUE |
END DO |
END IF |
END IF |
RETURN |
RETURN |
* |
* |