You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
314 lines
10 KiB
314 lines
10 KiB
2 years ago
|
*> \brief <b> CSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices</b>
|
||
|
|
*
|
||
|
|
* =========== DOCUMENTATION ===========
|
||
|
|
*
|
||
|
|
* Online html documentation available at
|
||
|
|
* http://www.netlib.org/lapack/explore-html/
|
||
|
|
*
|
||
|
|
*> \htmlonly
|
||
|
|
*> Download CSYSV_RK + dependencies
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/csysv_rk.f">
|
||
|
|
*> [TGZ]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/csysv_rk.f">
|
||
|
|
*> [ZIP]</a>
|
||
|
|
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/csysv_rk.f">
|
||
|
|
*> [TXT]</a>
|
||
|
|
*> \endhtmlonly
|
||
|
|
*
|
||
|
|
* Definition:
|
||
|
|
* ===========
|
||
|
|
*
|
||
|
|
* SUBROUTINE CSYSV_RK( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB,
|
||
|
|
* WORK, LWORK, INFO )
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
* CHARACTER UPLO
|
||
|
|
* INTEGER INFO, LDA, LDB, LWORK, N, NRHS
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
* INTEGER IPIV( * )
|
||
|
|
* COMPLEX A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
*
|
||
|
|
*> \par Purpose:
|
||
|
|
* =============
|
||
|
|
*>
|
||
|
|
*> \verbatim
|
||
|
|
*> CSYSV_RK computes the solution to a complex system of linear
|
||
|
|
*> equations A * X = B, where A is an N-by-N symmetric matrix
|
||
|
|
*> and X and B are N-by-NRHS matrices.
|
||
|
|
*>
|
||
|
|
*> The bounded Bunch-Kaufman (rook) diagonal pivoting method is used
|
||
|
|
*> to factor A as
|
||
|
|
*> A = P*U*D*(U**T)*(P**T), if UPLO = 'U', or
|
||
|
|
*> A = P*L*D*(L**T)*(P**T), if UPLO = 'L',
|
||
|
|
*> where U (or L) is unit upper (or lower) triangular matrix,
|
||
|
|
*> U**T (or L**T) is the transpose of U (or L), P is a permutation
|
||
|
|
*> matrix, P**T is the transpose of P, and D is symmetric and block
|
||
|
|
*> diagonal with 1-by-1 and 2-by-2 diagonal blocks.
|
||
|
|
*>
|
||
|
|
*> CSYTRF_RK is called to compute the factorization of a complex
|
||
|
|
*> symmetric matrix. The factored form of A is then used to solve
|
||
|
|
*> the system of equations A * X = B by calling BLAS3 routine CSYTRS_3.
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* Arguments:
|
||
|
|
* ==========
|
||
|
|
*
|
||
|
|
*> \param[in] UPLO
|
||
|
|
*> \verbatim
|
||
|
|
*> UPLO is CHARACTER*1
|
||
|
|
*> Specifies whether the upper or lower triangular part of the
|
||
|
|
*> symmetric matrix A is stored:
|
||
|
|
*> = 'U': Upper triangle of A is stored;
|
||
|
|
*> = 'L': Lower triangle of A is stored.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] N
|
||
|
|
*> \verbatim
|
||
|
|
*> N is INTEGER
|
||
|
|
*> The number of linear equations, i.e., the order of the
|
||
|
|
*> matrix A. N >= 0.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] NRHS
|
||
|
|
*> \verbatim
|
||
|
|
*> NRHS is INTEGER
|
||
|
|
*> The number of right hand sides, i.e., the number of columns
|
||
|
|
*> of the matrix B. NRHS >= 0.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in,out] A
|
||
|
|
*> \verbatim
|
||
|
|
*> A is COMPLEX array, dimension (LDA,N)
|
||
|
|
*> On entry, the symmetric matrix A.
|
||
|
|
*> If UPLO = 'U': the leading N-by-N upper triangular part
|
||
|
|
*> of A contains the upper triangular part of the matrix A,
|
||
|
|
*> and the strictly lower triangular part of A is not
|
||
|
|
*> referenced.
|
||
|
|
*>
|
||
|
|
*> If UPLO = 'L': the leading N-by-N lower triangular part
|
||
|
|
*> of A contains the lower triangular part of the matrix A,
|
||
|
|
*> and the strictly upper triangular part of A is not
|
||
|
|
*> referenced.
|
||
|
|
*>
|
||
|
|
*> On exit, if INFO = 0, diagonal of the block diagonal
|
||
|
|
*> matrix D and factors U or L as computed by CSYTRF_RK:
|
||
|
|
*> a) ONLY diagonal elements of the symmetric block diagonal
|
||
|
|
*> matrix D on the diagonal of A, i.e. D(k,k) = A(k,k);
|
||
|
|
*> (superdiagonal (or subdiagonal) elements of D
|
||
|
|
*> are stored on exit in array E), and
|
||
|
|
*> b) If UPLO = 'U': factor U in the superdiagonal part of A.
|
||
|
|
*> If UPLO = 'L': factor L in the subdiagonal part of A.
|
||
|
|
*>
|
||
|
|
*> For more info see the description of CSYTRF_RK routine.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] LDA
|
||
|
|
*> \verbatim
|
||
|
|
*> LDA is INTEGER
|
||
|
|
*> The leading dimension of the array A. LDA >= max(1,N).
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] E
|
||
|
|
*> \verbatim
|
||
|
|
*> E is COMPLEX array, dimension (N)
|
||
|
|
*> On exit, contains the output computed by the factorization
|
||
|
|
*> routine CSYTRF_RK, i.e. the superdiagonal (or subdiagonal)
|
||
|
|
*> elements of the symmetric block diagonal matrix D
|
||
|
|
*> with 1-by-1 or 2-by-2 diagonal blocks, where
|
||
|
|
*> If UPLO = 'U': E(i) = D(i-1,i), i=2:N, E(1) is set to 0;
|
||
|
|
*> If UPLO = 'L': E(i) = D(i+1,i), i=1:N-1, E(N) is set to 0.
|
||
|
|
*>
|
||
|
|
*> NOTE: For 1-by-1 diagonal block D(k), where
|
||
|
|
*> 1 <= k <= N, the element E(k) is set to 0 in both
|
||
|
|
*> UPLO = 'U' or UPLO = 'L' cases.
|
||
|
|
*>
|
||
|
|
*> For more info see the description of CSYTRF_RK routine.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] IPIV
|
||
|
|
*> \verbatim
|
||
|
|
*> IPIV is INTEGER array, dimension (N)
|
||
|
|
*> Details of the interchanges and the block structure of D,
|
||
|
|
*> as determined by CSYTRF_RK.
|
||
|
|
*>
|
||
|
|
*> For more info see the description of CSYTRF_RK routine.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in,out] B
|
||
|
|
*> \verbatim
|
||
|
|
*> B is COMPLEX array, dimension (LDB,NRHS)
|
||
|
|
*> On entry, the N-by-NRHS right hand side matrix B.
|
||
|
|
*> On exit, if INFO = 0, the N-by-NRHS solution matrix X.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] LDB
|
||
|
|
*> \verbatim
|
||
|
|
*> LDB is INTEGER
|
||
|
|
*> The leading dimension of the array B. LDB >= max(1,N).
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] WORK
|
||
|
|
*> \verbatim
|
||
|
|
*> WORK is COMPLEX array, dimension ( MAX(1,LWORK) ).
|
||
|
|
*> Work array used in the factorization stage.
|
||
|
|
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[in] LWORK
|
||
|
|
*> \verbatim
|
||
|
|
*> LWORK is INTEGER
|
||
|
|
*> The length of WORK. LWORK >= 1. For best performance
|
||
|
|
*> of factorization stage LWORK >= max(1,N*NB), where NB is
|
||
|
|
*> the optimal blocksize for CSYTRF_RK.
|
||
|
|
*>
|
||
|
|
*> If LWORK = -1, then a workspace query is assumed;
|
||
|
|
*> the routine only calculates the optimal size of the WORK
|
||
|
|
*> array for factorization stage, returns this value as
|
||
|
|
*> the first entry of the WORK array, and no error message
|
||
|
|
*> related to LWORK is issued by XERBLA.
|
||
|
|
*> \endverbatim
|
||
|
|
*>
|
||
|
|
*> \param[out] INFO
|
||
|
|
*> \verbatim
|
||
|
|
*> INFO is INTEGER
|
||
|
|
*> = 0: successful exit
|
||
|
|
*>
|
||
|
|
*> < 0: If INFO = -k, the k-th argument had an illegal value
|
||
|
|
*>
|
||
|
|
*> > 0: If INFO = k, the matrix A is singular, because:
|
||
|
|
*> If UPLO = 'U': column k in the upper
|
||
|
|
*> triangular part of A contains all zeros.
|
||
|
|
*> If UPLO = 'L': column k in the lower
|
||
|
|
*> triangular part of A contains all zeros.
|
||
|
|
*>
|
||
|
|
*> Therefore D(k,k) is exactly zero, and superdiagonal
|
||
|
|
*> elements of column k of U (or subdiagonal elements of
|
||
|
|
*> column k of L ) are all zeros. The factorization has
|
||
|
|
*> been completed, but the block diagonal matrix D is
|
||
|
|
*> exactly singular, and division by zero will occur if
|
||
|
|
*> it is used to solve a system of equations.
|
||
|
|
*>
|
||
|
|
*> NOTE: INFO only stores the first occurrence of
|
||
|
|
*> a singularity, any subsequent occurrence of singularity
|
||
|
|
*> is not stored in INFO even though the factorization
|
||
|
|
*> always completes.
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* Authors:
|
||
|
|
* ========
|
||
|
|
*
|
||
|
|
*> \author Univ. of Tennessee
|
||
|
|
*> \author Univ. of California Berkeley
|
||
|
|
*> \author Univ. of Colorado Denver
|
||
|
|
*> \author NAG Ltd.
|
||
|
|
*
|
||
|
|
*> \ingroup complexSYsolve
|
||
|
|
*
|
||
|
|
*> \par Contributors:
|
||
|
|
* ==================
|
||
|
|
*>
|
||
|
|
*> \verbatim
|
||
|
|
*>
|
||
|
|
*> December 2016, Igor Kozachenko,
|
||
|
|
*> Computer Science Division,
|
||
|
|
*> University of California, Berkeley
|
||
|
|
*>
|
||
|
|
*> September 2007, Sven Hammarling, Nicholas J. Higham, Craig Lucas,
|
||
|
|
*> School of Mathematics,
|
||
|
|
*> University of Manchester
|
||
|
|
*>
|
||
|
|
*> \endverbatim
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
SUBROUTINE CSYSV_RK( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, WORK,
|
||
|
|
$ LWORK, INFO )
|
||
|
|
*
|
||
|
|
* -- LAPACK driver routine --
|
||
|
|
* -- LAPACK is a software package provided by Univ. of Tennessee, --
|
||
|
|
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
|
||
|
|
*
|
||
|
|
* .. Scalar Arguments ..
|
||
|
|
CHARACTER UPLO
|
||
|
|
INTEGER INFO, LDA, LDB, LWORK, N, NRHS
|
||
|
|
* ..
|
||
|
|
* .. Array Arguments ..
|
||
|
|
INTEGER IPIV( * )
|
||
|
|
COMPLEX A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
|
||
|
|
* ..
|
||
|
|
*
|
||
|
|
* =====================================================================
|
||
|
|
*
|
||
|
|
* .. Local Scalars ..
|
||
|
|
LOGICAL LQUERY
|
||
|
|
INTEGER LWKOPT
|
||
|
|
* ..
|
||
|
|
* .. External Functions ..
|
||
|
|
LOGICAL LSAME
|
||
|
|
EXTERNAL LSAME
|
||
|
|
* ..
|
||
|
|
* .. External Subroutines ..
|
||
|
|
EXTERNAL XERBLA, CSYTRF_RK, CSYTRS_3
|
||
|
|
* ..
|
||
|
|
* .. Intrinsic Functions ..
|
||
|
|
INTRINSIC MAX
|
||
|
|
* ..
|
||
|
|
* .. Executable Statements ..
|
||
|
|
*
|
||
|
|
* Test the input parameters.
|
||
|
|
*
|
||
|
|
INFO = 0
|
||
|
|
LQUERY = ( LWORK.EQ.-1 )
|
||
|
|
IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
|
||
|
|
INFO = -1
|
||
|
|
ELSE IF( N.LT.0 ) THEN
|
||
|
|
INFO = -2
|
||
|
|
ELSE IF( NRHS.LT.0 ) THEN
|
||
|
|
INFO = -3
|
||
|
|
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
|
||
|
|
INFO = -5
|
||
|
|
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
|
||
|
|
INFO = -9
|
||
|
|
ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
|
||
|
|
INFO = -11
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( INFO.EQ.0 ) THEN
|
||
|
|
IF( N.EQ.0 ) THEN
|
||
|
|
LWKOPT = 1
|
||
|
|
ELSE
|
||
|
|
CALL CSYTRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, -1, INFO )
|
||
|
|
LWKOPT = INT( WORK( 1 ) )
|
||
|
|
END IF
|
||
|
|
WORK( 1 ) = LWKOPT
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
IF( INFO.NE.0 ) THEN
|
||
|
|
CALL XERBLA( 'CSYSV_RK ', -INFO )
|
||
|
|
RETURN
|
||
|
|
ELSE IF( LQUERY ) THEN
|
||
|
|
RETURN
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
* Compute the factorization A = U*D*U**T or A = L*D*L**T.
|
||
|
|
*
|
||
|
|
CALL CSYTRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, LWORK, INFO )
|
||
|
|
*
|
||
|
|
IF( INFO.EQ.0 ) THEN
|
||
|
|
*
|
||
|
|
* Solve the system A*X = B with BLAS3 solver, overwriting B with X.
|
||
|
|
*
|
||
|
|
CALL CSYTRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO )
|
||
|
|
*
|
||
|
|
END IF
|
||
|
|
*
|
||
|
|
WORK( 1 ) = LWKOPT
|
||
|
|
*
|
||
|
|
RETURN
|
||
|
|
*
|
||
|
|
* End of CSYSV_RK
|
||
|
|
*
|
||
|
|
END
|