*> \brief <b> ZSYSV_RK computes the solution to system of linear equations A * X = B for SY matrices</b>
*
* =========== DOCUMENTATION ===========
*
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*
* Definition:
* ===========
*
* SUBROUTINE ZSYSV_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*16 A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*> ZSYSV_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.
*>
*> ZSYTRF_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 ZSYTRS_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*16 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 ZSYTRF_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 ZSYTRF_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*16 array, dimension (N)
*> On exit, contains the output computed by the factorization
*> routine ZSYTRF_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 ZSYTRF_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 ZSYTRF_RK.
*>
*> For more info see the description of ZSYTRF_RK routine.
*> \endverbatim
*>
*> \param[in,out] B
*> \verbatim
*> B is COMPLEX*16 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*16 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 ZSYTRF_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.
*
*> \date December 2016
*
*> \ingroup complex16SYsolve
*
*> \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 ZSYSV_RK( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, WORK,
$ LWORK, INFO )
*
* -- LAPACK driver routine (version 3.7.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* December 2016
*
* .. Scalar Arguments ..
CHARACTER UPLO
INTEGER INFO, LDA, LDB, LWORK, N, NRHS
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX*16 A( LDA, * ), B( LDB, * ), E( * ), WORK( * )
* ..
*
* =====================================================================
*
* .. Local Scalars ..
LOGICAL LQUERY
INTEGER LWKOPT
* ..
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZSYTRF_RK, ZSYTRS_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 ZSYTRF_RK( UPLO, N, A, LDA, E, IPIV, WORK, -1, INFO )
LWKOPT = WORK(1)
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZSYSV_RK ', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Compute the factorization A = P*U*D*(U**T)*(P**T) or
* A = P*U*D*(U**T)*(P**T).
*
CALL ZSYTRF_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 ZSYTRS_3( UPLO, N, NRHS, A, LDA, E, IPIV, B, LDB, INFO )
*
END IF
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of ZSYSV_RK
*
END