version 1.6, 2011/07/22 07:38:09
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version 1.15, 2018/05/29 07:18:04
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SUBROUTINE DPFTRF( TRANSR, UPLO, N, A, INFO ) |
*> \brief \b DPFTRF |
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* |
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* =========== DOCUMENTATION =========== |
* |
* |
* -- LAPACK routine (version 3.3.1) -- |
* 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 |
* -- April 2011 -- |
*> Download DPFTRF + dependencies |
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dpftrf.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/dpftrf.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/dpftrf.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 DPFTRF( 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 N, INFO |
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* .. |
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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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*> \par Purpose: |
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* ============= |
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*> |
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*> \verbatim |
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*> |
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*> DPFTRF computes the Cholesky factorization of a real symmetric |
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*> positive definite matrix A. |
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*> |
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*> The factorization has the form |
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*> A = U**T * U, if UPLO = 'U', or |
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*> A = L * L**T, if UPLO = 'L', |
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*> where U is an upper triangular matrix and L is lower triangular. |
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*> |
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*> This is the block version of the algorithm, calling Level 3 BLAS. |
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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 RFP A is stored; |
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*> = 'L': Lower triangle of RFP 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, if INFO = 0, the factor U or L from the Cholesky |
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*> factorization RFP A = U**T*U or RFP A = L*L**T. |
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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 leading minor of order i is not |
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*> positive definite, and the factorization could not be |
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*> completed. |
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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 DPFTRF( 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 |
* |
* |
* .. |
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* .. Scalar Arguments .. |
* .. Scalar Arguments .. |
CHARACTER TRANSR, UPLO |
CHARACTER TRANSR, UPLO |
INTEGER N, INFO |
INTEGER N, INFO |
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* .. Array Arguments .. |
* .. Array Arguments .. |
DOUBLE PRECISION A( 0: * ) |
DOUBLE PRECISION A( 0: * ) |
* |
* |
* Purpose |
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* ======= |
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* |
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* DPFTRF computes the Cholesky factorization of a real symmetric |
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* positive definite matrix A. |
|
* |
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* The factorization has the form |
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* A = U**T * U, if UPLO = 'U', or |
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* A = L * L**T, if UPLO = 'L', |
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* where U is an upper triangular matrix and L is lower triangular. |
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* |
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* This is the block version of the algorithm, calling Level 3 BLAS. |
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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 RFP A is stored; |
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* = 'L': Lower triangle of RFP 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, if INFO = 0, the factor U or L from the Cholesky |
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* factorization RFP A = U**T*U or RFP A = L*L**T. |
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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 leading minor of order i is not |
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* positive definite, and the factorization could not be |
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* completed. |
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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 .. |