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443 lines
14 KiB
443 lines
14 KiB
2 years ago
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*> \brief \b CPFTRI
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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 CPFTRI + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cpftri.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/cpftri.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/cpftri.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 CPFTRI( 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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* COMPLEX 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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*> CPFTRI computes the inverse of a complex Hermitian positive definite
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*> matrix A using the Cholesky factorization A = U**H*U or A = L*L**H
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*> computed by CPFTRF.
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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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*> = 'C': The Conjugate-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 COMPLEX array, dimension ( N*(N+1)/2 );
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*> On entry, the Hermitian 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 = 'C' then RFP is
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*> the Conjugate-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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*> 'C'. 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 Hermitian 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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*> \ingroup complexOTHERcomputational
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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 Standard Packed Format when N is even.
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*> 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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*> conjugate-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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*> conjugate-transpose of the last three columns of AP lower.
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*> To denote conjugate we place -- above the element. This covers the
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*> 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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*> -- -- --
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*> 03 04 05 33 43 53
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*> -- --
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*> 13 14 15 00 44 54
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*> --
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*> 23 24 25 10 11 55
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*>
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*> 33 34 35 20 21 22
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*> --
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*> 00 44 45 30 31 32
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*> -- --
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*> 01 11 55 40 41 42
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*> -- -- --
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*> 02 12 22 50 51 52
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*>
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*> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
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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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*> -- -- -- -- -- -- -- -- -- --
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*> 03 13 23 33 00 01 02 33 00 10 20 30 40 50
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*> -- -- -- -- -- -- -- -- -- --
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*> 04 14 24 34 44 11 12 43 44 11 21 31 41 51
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*> -- -- -- -- -- -- -- -- -- --
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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 next consider Standard Packed Format when N is odd.
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*> 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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*> conjugate-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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*> conjugate-transpose of the last two columns of AP lower.
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*> To denote conjugate we place -- above the element. This covers the
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*> 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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*> -- --
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*> 02 03 04 00 33 43
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*> --
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*> 12 13 14 10 11 44
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*>
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*> 22 23 24 20 21 22
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*> --
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*> 00 33 34 30 31 32
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*> -- --
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*> 01 11 44 40 41 42
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*>
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*> Now let TRANSR = 'C'. RFP A in both UPLO cases is just the conjugate-
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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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*> -- -- -- -- -- -- -- -- --
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*> 02 12 22 00 01 00 10 20 30 40 50
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*> -- -- -- -- -- -- -- -- --
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*> 03 13 23 33 11 33 11 21 31 41 51
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*> -- -- -- -- -- -- -- -- --
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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 CPFTRI( TRANSR, UPLO, N, A, INFO )
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*
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* -- LAPACK computational routine --
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* -- LAPACK is a software package provided by Univ. of Tennessee, --
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* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
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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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COMPLEX A( 0: * )
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* ..
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*
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* =====================================================================
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*
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* .. Parameters ..
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REAL ONE
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COMPLEX CONE
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PARAMETER ( ONE = 1.0E+0, CONE = ( 1.0E+0, 0.0E+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL LOWER, NISODD, NORMALTRANSR
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INTEGER N1, N2, K
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* ..
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* .. External Functions ..
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LOGICAL LSAME
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EXTERNAL LSAME
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* ..
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* .. External Subroutines ..
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EXTERNAL XERBLA, CTFTRI, CLAUUM, CTRMM, CHERK
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC MOD
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* ..
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* .. Executable Statements ..
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*
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* Test the input parameters.
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*
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INFO = 0
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NORMALTRANSR = LSAME( TRANSR, 'N' )
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LOWER = LSAME( UPLO, 'L' )
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IF( .NOT.NORMALTRANSR .AND. .NOT.LSAME( TRANSR, 'C' ) ) THEN
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INFO = -1
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ELSE IF( .NOT.LOWER .AND. .NOT.LSAME( UPLO, 'U' ) ) THEN
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INFO = -2
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ELSE IF( N.LT.0 ) THEN
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INFO = -3
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CPFTRI', -INFO )
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RETURN
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END IF
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*
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* Quick return if possible
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*
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IF( N.EQ.0 )
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$ RETURN
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*
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* Invert the triangular Cholesky factor U or L.
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*
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CALL CTFTRI( TRANSR, UPLO, 'N', N, A, INFO )
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IF( INFO.GT.0 )
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$ RETURN
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*
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* If N is odd, set NISODD = .TRUE.
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* If N is even, set K = N/2 and NISODD = .FALSE.
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*
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IF( MOD( N, 2 ).EQ.0 ) THEN
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K = N / 2
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NISODD = .FALSE.
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ELSE
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NISODD = .TRUE.
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END IF
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*
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* Set N1 and N2 depending on LOWER
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*
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IF( LOWER ) THEN
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N2 = N / 2
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N1 = N - N2
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ELSE
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N1 = N / 2
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N2 = N - N1
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END IF
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*
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* Start execution of triangular matrix multiply: inv(U)*inv(U)^C or
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* inv(L)^C*inv(L). There are eight cases.
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*
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IF( NISODD ) THEN
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*
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* N is odd
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*
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IF( NORMALTRANSR ) THEN
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*
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* N is odd and TRANSR = 'N'
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*
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IF( LOWER ) THEN
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*
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* SRPA for LOWER, NORMAL and N is odd ( a(0:n-1,0:N1-1) )
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* T1 -> a(0,0), T2 -> a(0,1), S -> a(N1,0)
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* T1 -> a(0), T2 -> a(n), S -> a(N1)
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*
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CALL CLAUUM( 'L', N1, A( 0 ), N, INFO )
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CALL CHERK( 'L', 'C', N1, N2, ONE, A( N1 ), N, ONE,
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$ A( 0 ), N )
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CALL CTRMM( 'L', 'U', 'N', 'N', N2, N1, CONE, A( N ), N,
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$ A( N1 ), N )
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CALL CLAUUM( 'U', N2, A( N ), N, INFO )
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*
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ELSE
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*
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* SRPA for UPPER, NORMAL and N is odd ( a(0:n-1,0:N2-1)
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* T1 -> a(N1+1,0), T2 -> a(N1,0), S -> a(0,0)
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* T1 -> a(N2), T2 -> a(N1), S -> a(0)
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*
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CALL CLAUUM( 'L', N1, A( N2 ), N, INFO )
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CALL CHERK( 'L', 'N', N1, N2, ONE, A( 0 ), N, ONE,
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$ A( N2 ), N )
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CALL CTRMM( 'R', 'U', 'C', 'N', N1, N2, CONE, A( N1 ), N,
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$ A( 0 ), N )
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CALL CLAUUM( 'U', N2, A( N1 ), N, INFO )
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*
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END IF
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*
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ELSE
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*
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* N is odd and TRANSR = 'C'
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*
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IF( LOWER ) THEN
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*
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* SRPA for LOWER, TRANSPOSE, and N is odd
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* T1 -> a(0), T2 -> a(1), S -> a(0+N1*N1)
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*
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CALL CLAUUM( 'U', N1, A( 0 ), N1, INFO )
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CALL CHERK( 'U', 'N', N1, N2, ONE, A( N1*N1 ), N1, ONE,
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$ A( 0 ), N1 )
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CALL CTRMM( 'R', 'L', 'N', 'N', N1, N2, CONE, A( 1 ), N1,
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$ A( N1*N1 ), N1 )
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CALL CLAUUM( 'L', N2, A( 1 ), N1, INFO )
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*
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ELSE
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*
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* SRPA for UPPER, TRANSPOSE, and N is odd
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* T1 -> a(0+N2*N2), T2 -> a(0+N1*N2), S -> a(0)
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*
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CALL CLAUUM( 'U', N1, A( N2*N2 ), N2, INFO )
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||
|
|
CALL CHERK( 'U', 'C', N1, N2, ONE, A( 0 ), N2, ONE,
|
||
|
|
$ A( N2*N2 ), N2 )
|
||
|
|
CALL CTRMM( 'L', 'L', 'C', 'N', N2, N1, CONE, A( N1*N2 ),
|
||
|
|
$ N2, A( 0 ), N2 )
|
||
|
|
CALL CLAUUM( 'L', N2, A( N1*N2 ), N2, INFO )
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* N is even
|
||
|
|
*
|
||
|
|
IF( NORMALTRANSR ) THEN
|
||
|
|
*
|
||
|
|
* N is even and TRANSR = 'N'
|
||
|
|
*
|
||
|
|
IF( LOWER ) THEN
|
||
|
|
*
|
||
|
|
* SRPA for LOWER, NORMAL, and N is even ( a(0:n,0:k-1) )
|
||
|
|
* T1 -> a(1,0), T2 -> a(0,0), S -> a(k+1,0)
|
||
|
|
* T1 -> a(1), T2 -> a(0), S -> a(k+1)
|
||
|
|
*
|
||
|
|
CALL CLAUUM( 'L', K, A( 1 ), N+1, INFO )
|
||
|
|
CALL CHERK( 'L', 'C', K, K, ONE, A( K+1 ), N+1, ONE,
|
||
|
|
$ A( 1 ), N+1 )
|
||
|
|
CALL CTRMM( 'L', 'U', 'N', 'N', K, K, CONE, A( 0 ), N+1,
|
||
|
|
$ A( K+1 ), N+1 )
|
||
|
|
CALL CLAUUM( 'U', K, A( 0 ), N+1, INFO )
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* SRPA for UPPER, NORMAL, and N is even ( a(0:n,0:k-1) )
|
||
|
|
* T1 -> a(k+1,0) , T2 -> a(k,0), S -> a(0,0)
|
||
|
|
* T1 -> a(k+1), T2 -> a(k), S -> a(0)
|
||
|
|
*
|
||
|
|
CALL CLAUUM( 'L', K, A( K+1 ), N+1, INFO )
|
||
|
|
CALL CHERK( 'L', 'N', K, K, ONE, A( 0 ), N+1, ONE,
|
||
|
|
$ A( K+1 ), N+1 )
|
||
|
|
CALL CTRMM( 'R', 'U', 'C', 'N', K, K, CONE, A( K ), N+1,
|
||
|
|
$ A( 0 ), N+1 )
|
||
|
|
CALL CLAUUM( 'U', K, A( K ), N+1, INFO )
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* N is even and TRANSR = 'C'
|
||
|
|
*
|
||
|
|
IF( LOWER ) THEN
|
||
|
|
*
|
||
|
|
* SRPA for LOWER, TRANSPOSE, and N is even (see paper)
|
||
|
|
* T1 -> B(0,1), T2 -> B(0,0), S -> B(0,k+1),
|
||
|
|
* T1 -> a(0+k), T2 -> a(0+0), S -> a(0+k*(k+1)); lda=k
|
||
|
|
*
|
||
|
|
CALL CLAUUM( 'U', K, A( K ), K, INFO )
|
||
|
|
CALL CHERK( 'U', 'N', K, K, ONE, A( K*( K+1 ) ), K, ONE,
|
||
|
|
$ A( K ), K )
|
||
|
|
CALL CTRMM( 'R', 'L', 'N', 'N', K, K, CONE, A( 0 ), K,
|
||
|
|
$ A( K*( K+1 ) ), K )
|
||
|
|
CALL CLAUUM( 'L', K, A( 0 ), K, INFO )
|
||
|
|
*
|
||
|
|
ELSE
|
||
|
|
*
|
||
|
|
* SRPA for UPPER, TRANSPOSE, and N is even (see paper)
|
||
|
|
* T1 -> B(0,k+1), T2 -> B(0,k), S -> B(0,0),
|
||
|
|
* T1 -> a(0+k*(k+1)), T2 -> a(0+k*k), S -> a(0+0)); lda=k
|
||
|
|
*
|
||
|
|
CALL CLAUUM( 'U', K, A( K*( K+1 ) ), K, INFO )
|
||
|
|
CALL CHERK( 'U', 'C', K, K, ONE, A( 0 ), K, ONE,
|
||
|
|
$ A( K*( K+1 ) ), K )
|
||
|
|
CALL CTRMM( 'L', 'L', 'C', 'N', K, K, CONE, A( K*K ), K,
|
||
|
|
$ A( 0 ), K )
|
||
|
|
CALL CLAUUM( 'L', K, A( K*K ), K, INFO )
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of CPFTRI
|
||
|
|
*
|
||
|
|
END
|