version 1.5, 2010/12/21 13:53:35
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version 1.15, 2018/05/29 07:18:04
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SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO ) |
*> \brief \b DPFTRI |
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* |
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* =========== DOCUMENTATION =========== |
* |
* |
* -- LAPACK routine (version 3.3.0) -- |
* Online html documentation available at |
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* http://www.netlib.org/lapack/explore-html/ |
* |
* |
* -- Contributed by Fred Gustavson of the IBM Watson Research Center -- |
*> \htmlonly |
* November 2010 |
*> Download DPFTRI + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpftri.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/dpftri.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/dpftri.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 DPFTRI( TRANSR, UPLO, N, A, INFO ) |
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* |
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* .. Scalar Arguments .. |
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* CHARACTER TRANSR, UPLO |
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* INTEGER INFO, N |
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* .. Array Arguments .. |
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* DOUBLE PRECISION A( 0: * ) |
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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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*> DPFTRI computes the inverse of a (real) symmetric positive definite |
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*> matrix A using the Cholesky factorization A = U**T*U or A = L*L**T |
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*> computed by DPFTRF. |
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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] TRANSR |
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*> \verbatim |
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*> TRANSR is CHARACTER*1 |
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*> = 'N': The Normal TRANSR of RFP A is stored; |
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*> = 'T': The Transpose TRANSR of RFP A is stored. |
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*> \endverbatim |
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*> |
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*> \param[in] UPLO |
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*> \verbatim |
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*> UPLO is CHARACTER*1 |
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*> = 'U': Upper triangle of A is stored; |
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*> = 'L': Lower triangle of A is stored. |
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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 matrix A. N >= 0. |
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*> \endverbatim |
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*> |
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*> \param[in,out] A |
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*> \verbatim |
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*> A is DOUBLE PRECISION array, dimension ( N*(N+1)/2 ) |
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*> On entry, the symmetric matrix A in RFP format. RFP format is |
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*> described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' |
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*> then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is |
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*> (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is |
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*> the transpose of RFP A as defined when |
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*> TRANSR = 'N'. The contents of RFP A are defined by UPLO as |
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*> follows: If UPLO = 'U' the RFP A contains the nt elements of |
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*> upper packed A. If UPLO = 'L' the RFP A contains the elements |
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*> of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = |
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*> 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N |
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*> is odd. See the Note below for more details. |
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*> |
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*> On exit, the symmetric inverse of the original matrix, in the |
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*> same storage format. |
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*> \endverbatim |
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*> |
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*> \param[out] INFO |
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*> \verbatim |
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*> INFO is INTEGER |
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*> = 0: successful exit |
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*> < 0: if INFO = -i, the i-th argument had an illegal value |
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*> > 0: if INFO = i, the (i,i) element of the factor U or L is |
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*> zero, and the inverse could not be computed. |
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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 December 2016 |
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* |
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*> \ingroup doubleOTHERcomputational |
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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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*> We first consider Rectangular Full Packed (RFP) Format when N is |
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*> even. We give an example where N = 6. |
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*> |
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*> AP is Upper AP is Lower |
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*> |
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*> 00 01 02 03 04 05 00 |
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*> 11 12 13 14 15 10 11 |
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*> 22 23 24 25 20 21 22 |
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*> 33 34 35 30 31 32 33 |
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*> 44 45 40 41 42 43 44 |
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*> 55 50 51 52 53 54 55 |
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*> |
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*> |
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*> Let TRANSR = 'N'. RFP holds AP as follows: |
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*> For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last |
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*> three columns of AP upper. The lower triangle A(4:6,0:2) consists of |
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*> the transpose of the first three columns of AP upper. |
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*> For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first |
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*> three columns of AP lower. The upper triangle A(0:2,0:2) consists of |
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*> the transpose of the last three columns of AP lower. |
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*> This covers the case N even and TRANSR = 'N'. |
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*> |
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*> RFP A RFP A |
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*> |
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*> 03 04 05 33 43 53 |
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*> 13 14 15 00 44 54 |
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*> 23 24 25 10 11 55 |
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*> 33 34 35 20 21 22 |
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*> 00 44 45 30 31 32 |
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*> 01 11 55 40 41 42 |
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*> 02 12 22 50 51 52 |
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*> |
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*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the |
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*> transpose of RFP A above. One therefore gets: |
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*> |
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*> |
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*> RFP A RFP A |
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*> |
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*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50 |
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*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51 |
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*> 05 15 25 35 45 55 22 53 54 55 22 32 42 52 |
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*> |
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*> |
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*> We then consider Rectangular Full Packed (RFP) Format when N is |
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*> odd. We give an example where N = 5. |
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*> |
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*> AP is Upper AP is Lower |
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*> |
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*> 00 01 02 03 04 00 |
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*> 11 12 13 14 10 11 |
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*> 22 23 24 20 21 22 |
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*> 33 34 30 31 32 33 |
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*> 44 40 41 42 43 44 |
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*> |
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*> |
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*> Let TRANSR = 'N'. RFP holds AP as follows: |
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*> For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last |
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*> three columns of AP upper. The lower triangle A(3:4,0:1) consists of |
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*> the transpose of the first two columns of AP upper. |
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*> For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first |
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*> three columns of AP lower. The upper triangle A(0:1,1:2) consists of |
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*> the transpose of the last two columns of AP lower. |
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*> This covers the case N odd and TRANSR = 'N'. |
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*> |
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*> RFP A RFP A |
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*> |
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*> 02 03 04 00 33 43 |
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*> 12 13 14 10 11 44 |
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*> 22 23 24 20 21 22 |
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*> 00 33 34 30 31 32 |
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*> 01 11 44 40 41 42 |
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*> |
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*> Now let TRANSR = 'T'. RFP A in both UPLO cases is just the |
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*> transpose of RFP A above. One therefore gets: |
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*> |
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*> RFP A RFP A |
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*> |
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*> 02 12 22 00 01 00 10 20 30 40 50 |
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*> 03 13 23 33 11 33 11 21 31 41 51 |
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*> 04 14 24 34 44 43 44 22 32 42 52 |
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*> \endverbatim |
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*> |
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* ===================================================================== |
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SUBROUTINE DPFTRI( TRANSR, UPLO, N, A, INFO ) |
* |
* |
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* -- LAPACK computational routine (version 3.7.0) -- |
* -- 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..-- |
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* December 2016 |
* |
* |
* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER TRANSR, UPLO |
CHARACTER TRANSR, UPLO |
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DOUBLE PRECISION A( 0: * ) |
DOUBLE PRECISION A( 0: * ) |
* .. |
* .. |
* |
* |
* Purpose |
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* ======= |
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* |
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* DPFTRI computes the inverse of a (real) symmetric positive definite |
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* matrix A using the Cholesky factorization A = U**T*U or A = L*L**T |
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* computed by DPFTRF. |
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* |
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* Arguments |
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* ========= |
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* |
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* TRANSR (input) CHARACTER*1 |
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* = 'N': The Normal TRANSR of RFP A is stored; |
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* = 'T': The Transpose TRANSR of RFP A is stored. |
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* |
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* UPLO (input) CHARACTER*1 |
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* = 'U': Upper triangle of A is stored; |
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* = 'L': Lower triangle of A is stored. |
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* |
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* N (input) INTEGER |
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* The order of the matrix A. N >= 0. |
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* |
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* A (input/output) DOUBLE PRECISION array, dimension ( N*(N+1)/2 ) |
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* On entry, the symmetric matrix A in RFP format. RFP format is |
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* described by TRANSR, UPLO, and N as follows: If TRANSR = 'N' |
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* then RFP A is (0:N,0:k-1) when N is even; k=N/2. RFP A is |
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* (0:N-1,0:k) when N is odd; k=N/2. IF TRANSR = 'T' then RFP is |
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* the transpose of RFP A as defined when |
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* TRANSR = 'N'. The contents of RFP A are defined by UPLO as |
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* follows: If UPLO = 'U' the RFP A contains the nt elements of |
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* upper packed A. If UPLO = 'L' the RFP A contains the elements |
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* of lower packed A. The LDA of RFP A is (N+1)/2 when TRANSR = |
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* 'T'. When TRANSR is 'N' the LDA is N+1 when N is even and N |
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* is odd. See the Note below for more details. |
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* |
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* On exit, the symmetric inverse of the original matrix, in the |
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* same storage format. |
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* |
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* INFO (output) INTEGER |
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* = 0: successful exit |
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* < 0: if INFO = -i, the i-th argument had an illegal value |
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* > 0: if INFO = i, the (i,i) element of the factor U or L is |
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* zero, and the inverse could not be computed. |
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* |
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* Further Details |
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* =============== |
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* |
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* We first consider Rectangular Full Packed (RFP) Format when N is |
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* even. We give an example where N = 6. |
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* |
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* AP is Upper AP is Lower |
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* |
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* 00 01 02 03 04 05 00 |
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* 11 12 13 14 15 10 11 |
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* 22 23 24 25 20 21 22 |
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* 33 34 35 30 31 32 33 |
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* 44 45 40 41 42 43 44 |
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* 55 50 51 52 53 54 55 |
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* |
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* |
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* Let TRANSR = 'N'. RFP holds AP as follows: |
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* For UPLO = 'U' the upper trapezoid A(0:5,0:2) consists of the last |
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* three columns of AP upper. The lower triangle A(4:6,0:2) consists of |
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* the transpose of the first three columns of AP upper. |
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* For UPLO = 'L' the lower trapezoid A(1:6,0:2) consists of the first |
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* three columns of AP lower. The upper triangle A(0:2,0:2) consists of |
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* the transpose of the last three columns of AP lower. |
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* This covers the case N even and TRANSR = 'N'. |
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* |
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* RFP A RFP A |
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* |
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* 03 04 05 33 43 53 |
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* 13 14 15 00 44 54 |
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* 23 24 25 10 11 55 |
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* 33 34 35 20 21 22 |
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* 00 44 45 30 31 32 |
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* 01 11 55 40 41 42 |
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* 02 12 22 50 51 52 |
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* |
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* Now let TRANSR = 'T'. RFP A in both UPLO cases is just the |
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* transpose of RFP A above. One therefore gets: |
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* |
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* |
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* RFP A RFP A |
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* |
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* 03 13 23 33 00 01 02 33 00 10 20 30 40 50 |
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* 04 14 24 34 44 11 12 43 44 11 21 31 41 51 |
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* 05 15 25 35 45 55 22 53 54 55 22 32 42 52 |
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* |
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* |
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* We then consider Rectangular Full Packed (RFP) Format when N is |
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* odd. We give an example where N = 5. |
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* |
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* AP is Upper AP is Lower |
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* |
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* 00 01 02 03 04 00 |
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* 11 12 13 14 10 11 |
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* 22 23 24 20 21 22 |
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* 33 34 30 31 32 33 |
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* 44 40 41 42 43 44 |
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* |
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* |
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* Let TRANSR = 'N'. RFP holds AP as follows: |
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* For UPLO = 'U' the upper trapezoid A(0:4,0:2) consists of the last |
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* three columns of AP upper. The lower triangle A(3:4,0:1) consists of |
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* the transpose of the first two columns of AP upper. |
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* For UPLO = 'L' the lower trapezoid A(0:4,0:2) consists of the first |
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* three columns of AP lower. The upper triangle A(0:1,1:2) consists of |
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* the transpose of the last two columns of AP lower. |
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* This covers the case N odd and TRANSR = 'N'. |
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* |
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* RFP A RFP A |
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* |
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* 02 03 04 00 33 43 |
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* 12 13 14 10 11 44 |
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* 22 23 24 20 21 22 |
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* 00 33 34 30 31 32 |
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* 01 11 44 40 41 42 |
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* |
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* Now let TRANSR = 'T'. RFP A in both UPLO cases is just the |
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* transpose of RFP A above. One therefore gets: |
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* |
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* RFP A RFP A |
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* |
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* 02 12 22 00 01 00 10 20 30 40 50 |
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* 03 13 23 33 11 33 11 21 31 41 51 |
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* 04 14 24 34 44 43 44 22 32 42 52 |
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* |
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* ===================================================================== |
* ===================================================================== |
* |
* |
* .. Parameters .. |
* .. Parameters .. |
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* Quick return if possible |
* Quick return if possible |
* |
* |
IF( N.EQ.0 ) |
IF( N.EQ.0 ) |
+ RETURN |
$ RETURN |
* |
* |
* Invert the triangular Cholesky factor U or L. |
* Invert the triangular Cholesky factor U or L. |
* |
* |
CALL DTFTRI( TRANSR, UPLO, 'N', N, A, INFO ) |
CALL DTFTRI( TRANSR, UPLO, 'N', N, A, INFO ) |
IF( INFO.GT.0 ) |
IF( INFO.GT.0 ) |
+ RETURN |
$ RETURN |
* |
* |
* If N is odd, set NISODD = .TRUE. |
* If N is odd, set NISODD = .TRUE. |
* If N is even, set K = N/2 and NISODD = .FALSE. |
* If N is even, set K = N/2 and NISODD = .FALSE. |
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* |
* |
CALL DLAUUM( 'L', N1, A( 0 ), N, INFO ) |
CALL DLAUUM( 'L', N1, A( 0 ), N, INFO ) |
CALL DSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE, |
CALL DSYRK( 'L', 'T', N1, N2, ONE, A( N1 ), N, ONE, |
+ A( 0 ), N ) |
$ A( 0 ), N ) |
CALL DTRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ), N, |
CALL DTRMM( 'L', 'U', 'N', 'N', N2, N1, ONE, A( N ), N, |
+ A( N1 ), N ) |
$ A( N1 ), N ) |
CALL DLAUUM( 'U', N2, A( N ), N, INFO ) |
CALL DLAUUM( 'U', N2, A( N ), N, INFO ) |
* |
* |
ELSE |
ELSE |
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* |
* |
CALL DLAUUM( 'L', N1, A( N2 ), N, INFO ) |
CALL DLAUUM( 'L', N1, A( N2 ), N, INFO ) |
CALL DSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE, |
CALL DSYRK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE, |
+ A( N2 ), N ) |
$ A( N2 ), N ) |
CALL DTRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ), N, |
CALL DTRMM( 'R', 'U', 'T', 'N', N1, N2, ONE, A( N1 ), N, |
+ A( 0 ), N ) |
$ A( 0 ), N ) |
CALL DLAUUM( 'U', N2, A( N1 ), N, INFO ) |
CALL DLAUUM( 'U', N2, A( N1 ), N, INFO ) |
* |
* |
END IF |
END IF |
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* |
* |
CALL DLAUUM( 'U', N1, A( 0 ), N1, INFO ) |
CALL DLAUUM( 'U', N1, A( 0 ), N1, INFO ) |
CALL DSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE, |
CALL DSYRK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE, |
+ A( 0 ), N1 ) |
$ A( 0 ), N1 ) |
CALL DTRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ), N1, |
CALL DTRMM( 'R', 'L', 'N', 'N', N1, N2, ONE, A( 1 ), N1, |
+ A( N1*N1 ), N1 ) |
$ A( N1*N1 ), N1 ) |
CALL DLAUUM( 'L', N2, A( 1 ), N1, INFO ) |
CALL DLAUUM( 'L', N2, A( 1 ), N1, INFO ) |
* |
* |
ELSE |
ELSE |
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* |
* |
CALL DLAUUM( 'U', N1, A( N2*N2 ), N2, INFO ) |
CALL DLAUUM( 'U', N1, A( N2*N2 ), N2, INFO ) |
CALL DSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE, |
CALL DSYRK( 'U', 'T', N1, N2, ONE, A( 0 ), N2, ONE, |
+ A( N2*N2 ), N2 ) |
$ A( N2*N2 ), N2 ) |
CALL DTRMM( 'L', 'L', 'T', 'N', N2, N1, ONE, A( N1*N2 ), |
CALL DTRMM( 'L', 'L', 'T', 'N', N2, N1, ONE, A( N1*N2 ), |
+ N2, A( 0 ), N2 ) |
$ N2, A( 0 ), N2 ) |
CALL DLAUUM( 'L', N2, A( N1*N2 ), N2, INFO ) |
CALL DLAUUM( 'L', N2, A( N1*N2 ), N2, INFO ) |
* |
* |
END IF |
END IF |
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* |
* |
CALL DLAUUM( 'L', K, A( 1 ), N+1, INFO ) |
CALL DLAUUM( 'L', K, A( 1 ), N+1, INFO ) |
CALL DSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE, |
CALL DSYRK( 'L', 'T', K, K, ONE, A( K+1 ), N+1, ONE, |
+ A( 1 ), N+1 ) |
$ A( 1 ), N+1 ) |
CALL DTRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ), N+1, |
CALL DTRMM( 'L', 'U', 'N', 'N', K, K, ONE, A( 0 ), N+1, |
+ A( K+1 ), N+1 ) |
$ A( K+1 ), N+1 ) |
CALL DLAUUM( 'U', K, A( 0 ), N+1, INFO ) |
CALL DLAUUM( 'U', K, A( 0 ), N+1, INFO ) |
* |
* |
ELSE |
ELSE |
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* |
* |
CALL DLAUUM( 'L', K, A( K+1 ), N+1, INFO ) |
CALL DLAUUM( 'L', K, A( K+1 ), N+1, INFO ) |
CALL DSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE, |
CALL DSYRK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE, |
+ A( K+1 ), N+1 ) |
$ A( K+1 ), N+1 ) |
CALL DTRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ), N+1, |
CALL DTRMM( 'R', 'U', 'T', 'N', K, K, ONE, A( K ), N+1, |
+ A( 0 ), N+1 ) |
$ A( 0 ), N+1 ) |
CALL DLAUUM( 'U', K, A( K ), N+1, INFO ) |
CALL DLAUUM( 'U', K, A( K ), N+1, INFO ) |
* |
* |
END IF |
END IF |
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* |
* |
CALL DLAUUM( 'U', K, A( K ), K, INFO ) |
CALL DLAUUM( 'U', K, A( K ), K, INFO ) |
CALL DSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE, |
CALL DSYRK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE, |
+ A( K ), K ) |
$ A( K ), K ) |
CALL DTRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K, |
CALL DTRMM( 'R', 'L', 'N', 'N', K, K, ONE, A( 0 ), K, |
+ A( K*( K+1 ) ), K ) |
$ A( K*( K+1 ) ), K ) |
CALL DLAUUM( 'L', K, A( 0 ), K, INFO ) |
CALL DLAUUM( 'L', K, A( 0 ), K, INFO ) |
* |
* |
ELSE |
ELSE |
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* |
* |
CALL DLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO ) |
CALL DLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO ) |
CALL DSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE, |
CALL DSYRK( 'U', 'T', K, K, ONE, A( 0 ), K, ONE, |
+ A( K*( K+1 ) ), K ) |
$ A( K*( K+1 ) ), K ) |
CALL DTRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ), K, |
CALL DTRMM( 'L', 'L', 'T', 'N', K, K, ONE, A( K*K ), K, |
+ A( 0 ), K ) |
$ A( 0 ), K ) |
CALL DLAUUM( 'L', K, A( K*K ), K, INFO ) |
CALL DLAUUM( 'L', K, A( K*K ), K, INFO ) |
* |
* |
END IF |
END IF |