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371 lines
11 KiB
371 lines
11 KiB
*> \brief \b CHETRS_3
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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 CHETRS_3 + dependencies
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*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chetrs_3.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/chetrs_3.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/chetrs_3.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 CHETRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
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* INFO )
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*
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* .. Scalar Arguments ..
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* CHARACTER UPLO
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* INTEGER INFO, LDA, LDB, N, NRHS
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* ..
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* .. Array Arguments ..
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* INTEGER IPIV( * )
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* COMPLEX A( LDA, * ), B( LDB, * ), E( * )
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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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*> CHETRS_3 solves a system of linear equations A * X = B with a complex
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*> Hermitian matrix A using the factorization computed
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*> by CHETRF_RK or CHETRF_BK:
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*>
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*> A = P*U*D*(U**H)*(P**T) or A = P*L*D*(L**H)*(P**T),
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*>
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*> where U (or L) is unit upper (or lower) triangular matrix,
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*> U**H (or L**H) is the conjugate of U (or L), P is a permutation
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*> matrix, P**T is the transpose of P, and D is Hermitian and block
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*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
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*>
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*> This algorithm is using 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] UPLO
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*> \verbatim
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*> UPLO is CHARACTER*1
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*> Specifies whether the details of the factorization are
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*> stored as an upper or lower triangular matrix:
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*> = 'U': Upper triangular, form is A = P*U*D*(U**H)*(P**T);
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*> = 'L': Lower triangular, form is A = P*L*D*(L**H)*(P**T).
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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] NRHS
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*> \verbatim
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*> NRHS is INTEGER
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*> The number of right hand sides, i.e., the number of columns
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*> of the matrix B. NRHS >= 0.
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*> \endverbatim
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*>
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*> \param[in] A
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*> \verbatim
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*> A is COMPLEX array, dimension (LDA,N)
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*> Diagonal of the block diagonal matrix D and factors U or L
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*> as computed by CHETRF_RK and CHETRF_BK:
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*> a) ONLY diagonal elements of the Hermitian block diagonal
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*> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
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*> (superdiagonal (or subdiagonal) elements of D
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*> should be provided on entry in array E), and
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*> b) If UPLO = 'U': factor U in the superdiagonal part of A.
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*> If UPLO = 'L': factor L in the subdiagonal part of A.
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*> \endverbatim
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*>
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*> \param[in] LDA
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*> \verbatim
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*> LDA is INTEGER
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*> The leading dimension of the array A. LDA >= max(1,N).
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*> \endverbatim
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*>
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*> \param[in] E
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*> \verbatim
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*> E is COMPLEX array, dimension (N)
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*> On entry, contains the superdiagonal (or subdiagonal)
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*> elements of the Hermitian block diagonal matrix D
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*> with 1-by-1 or 2-by-2 diagonal blocks, where
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*> If UPLO = 'U': E(i) = D(i-1,i),i=2:N, E(1) not referenced;
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*> If UPLO = 'L': E(i) = D(i+1,i),i=1:N-1, E(N) not referenced.
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*>
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*> NOTE: For 1-by-1 diagonal block D(k), where
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*> 1 <= k <= N, the element E(k) is not referenced in both
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*> UPLO = 'U' or UPLO = 'L' cases.
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*> \endverbatim
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*>
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*> \param[in] IPIV
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*> \verbatim
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*> IPIV is INTEGER array, dimension (N)
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*> Details of the interchanges and the block structure of D
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*> as determined by CHETRF_RK or CHETRF_BK.
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*> \endverbatim
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*>
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*> \param[in,out] B
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*> \verbatim
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*> B is COMPLEX array, dimension (LDB,NRHS)
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*> On entry, the right hand side matrix B.
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*> On exit, the solution matrix X.
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*> \endverbatim
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*>
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*> \param[in] LDB
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*> \verbatim
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*> LDB is INTEGER
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*> The leading dimension of the array B. LDB >= max(1,N).
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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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*> \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 complexHEcomputational
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*
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*> \par Contributors:
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* ==================
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*>
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*> \verbatim
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*>
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*> June 2017, Igor Kozachenko,
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*> Computer Science Division,
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*> University of California, Berkeley
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*>
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*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
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*> School of Mathematics,
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*> University of Manchester
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*>
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*> \endverbatim
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*
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* =====================================================================
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SUBROUTINE CHETRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
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$ 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 UPLO
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INTEGER INFO, LDA, LDB, N, NRHS
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* ..
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* .. Array Arguments ..
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INTEGER IPIV( * )
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COMPLEX A( LDA, * ), B( LDB, * ), E( * )
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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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COMPLEX ONE
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PARAMETER ( ONE = ( 1.0E+0,0.0E+0 ) )
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* ..
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* .. Local Scalars ..
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LOGICAL UPPER
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INTEGER I, J, K, KP
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REAL S
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COMPLEX AK, AKM1, AKM1K, BK, BKM1, DENOM
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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 CSSCAL, CSWAP, CTRSM, XERBLA
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* ..
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* .. Intrinsic Functions ..
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INTRINSIC ABS, CONJG, MAX, REAL
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* ..
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* .. Executable Statements ..
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*
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INFO = 0
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UPPER = LSAME( UPLO, 'U' )
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IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
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INFO = -1
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ELSE IF( N.LT.0 ) THEN
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INFO = -2
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ELSE IF( NRHS.LT.0 ) THEN
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INFO = -3
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ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
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INFO = -5
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ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
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INFO = -9
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END IF
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IF( INFO.NE.0 ) THEN
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CALL XERBLA( 'CHETRS_3', -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 .OR. NRHS.EQ.0 )
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$ RETURN
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*
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IF( UPPER ) THEN
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*
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* Begin Upper
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*
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* Solve A*X = B, where A = U*D*U**H.
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*
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* P**T * B
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*
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* Interchange rows K and IPIV(K) of matrix B in the same order
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* that the formation order of IPIV(I) vector for Upper case.
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*
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* (We can do the simple loop over IPIV with decrement -1,
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* since the ABS value of IPIV(I) represents the row index
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* of the interchange with row i in both 1x1 and 2x2 pivot cases)
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*
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DO K = N, 1, -1
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KP = ABS( IPIV( K ) )
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IF( KP.NE.K ) THEN
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CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
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END IF
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END DO
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*
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* Compute (U \P**T * B) -> B [ (U \P**T * B) ]
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*
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CALL CTRSM( 'L', 'U', 'N', 'U', N, NRHS, ONE, A, LDA, B, LDB )
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*
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* Compute D \ B -> B [ D \ (U \P**T * B) ]
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*
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I = N
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DO WHILE ( I.GE.1 )
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IF( IPIV( I ).GT.0 ) THEN
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S = REAL( ONE ) / REAL( A( I, I ) )
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CALL CSSCAL( NRHS, S, B( I, 1 ), LDB )
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ELSE IF ( I.GT.1 ) THEN
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AKM1K = E( I )
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AKM1 = A( I-1, I-1 ) / AKM1K
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AK = A( I, I ) / CONJG( AKM1K )
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DENOM = AKM1*AK - ONE
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DO J = 1, NRHS
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BKM1 = B( I-1, J ) / AKM1K
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BK = B( I, J ) / CONJG( AKM1K )
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B( I-1, J ) = ( AK*BKM1-BK ) / DENOM
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B( I, J ) = ( AKM1*BK-BKM1 ) / DENOM
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END DO
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I = I - 1
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END IF
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I = I - 1
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END DO
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*
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* Compute (U**H \ B) -> B [ U**H \ (D \ (U \P**T * B) ) ]
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*
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CALL CTRSM( 'L', 'U', 'C', 'U', N, NRHS, ONE, A, LDA, B, LDB )
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*
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* P * B [ P * (U**H \ (D \ (U \P**T * B) )) ]
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*
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* Interchange rows K and IPIV(K) of matrix B in reverse order
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* from the formation order of IPIV(I) vector for Upper case.
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*
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* (We can do the simple loop over IPIV with increment 1,
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* since the ABS value of IPIV(I) represents the row index
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* of the interchange with row i in both 1x1 and 2x2 pivot cases)
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*
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DO K = 1, N, 1
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KP = ABS( IPIV( K ) )
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IF( KP.NE.K ) THEN
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CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
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END IF
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END DO
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*
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ELSE
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*
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* Begin Lower
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*
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* Solve A*X = B, where A = L*D*L**H.
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*
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* P**T * B
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* Interchange rows K and IPIV(K) of matrix B in the same order
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* that the formation order of IPIV(I) vector for Lower case.
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*
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* (We can do the simple loop over IPIV with increment 1,
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* since the ABS value of IPIV(I) represents the row index
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* of the interchange with row i in both 1x1 and 2x2 pivot cases)
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*
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DO K = 1, N, 1
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KP = ABS( IPIV( K ) )
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IF( KP.NE.K ) THEN
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CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
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END IF
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END DO
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*
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* Compute (L \P**T * B) -> B [ (L \P**T * B) ]
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*
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CALL CTRSM( 'L', 'L', 'N', 'U', N, NRHS, ONE, A, LDA, B, LDB )
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*
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* Compute D \ B -> B [ D \ (L \P**T * B) ]
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*
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I = 1
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DO WHILE ( I.LE.N )
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IF( IPIV( I ).GT.0 ) THEN
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S = REAL( ONE ) / REAL( A( I, I ) )
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CALL CSSCAL( NRHS, S, B( I, 1 ), LDB )
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ELSE IF( I.LT.N ) THEN
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AKM1K = E( I )
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AKM1 = A( I, I ) / CONJG( AKM1K )
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AK = A( I+1, I+1 ) / AKM1K
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DENOM = AKM1*AK - ONE
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DO J = 1, NRHS
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BKM1 = B( I, J ) / CONJG( AKM1K )
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BK = B( I+1, J ) / AKM1K
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B( I, J ) = ( AK*BKM1-BK ) / DENOM
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B( I+1, J ) = ( AKM1*BK-BKM1 ) / DENOM
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END DO
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I = I + 1
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END IF
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I = I + 1
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END DO
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*
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* Compute (L**H \ B) -> B [ L**H \ (D \ (L \P**T * B) ) ]
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*
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CALL CTRSM('L', 'L', 'C', 'U', N, NRHS, ONE, A, LDA, B, LDB )
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*
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* P * B [ P * (L**H \ (D \ (L \P**T * B) )) ]
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*
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* Interchange rows K and IPIV(K) of matrix B in reverse order
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* from the formation order of IPIV(I) vector for Lower case.
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*
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* (We can do the simple loop over IPIV with decrement -1,
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* since the ABS value of IPIV(I) represents the row index
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* of the interchange with row i in both 1x1 and 2x2 pivot cases)
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*
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DO K = N, 1, -1
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KP = ABS( IPIV( K ) )
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IF( KP.NE.K ) THEN
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CALL CSWAP( NRHS, B( K, 1 ), LDB, B( KP, 1 ), LDB )
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END IF
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END DO
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*
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* END Lower
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*
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END IF
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*
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RETURN
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*
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* End of CHETRS_3
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*
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END
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