*> \brief ** ZSYSVX computes the solution to system of linear equations A * X = B for SY matrices**
*
* =========== DOCUMENTATION ===========
*
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*
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*
* Definition:
* ===========
*
* SUBROUTINE ZSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
* LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
* RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER FACT, UPLO
* INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
* DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
* COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ WORK( * ), X( LDX, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> ZSYSVX uses the diagonal pivoting factorization to compute 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.
*>
*> Error bounds on the solution and a condition estimate are also
*> provided.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'N', the diagonal pivoting method is used to factor A.
*> The form of the factorization is
*> A = U * D * U**T, if UPLO = 'U', or
*> A = L * D * L**T, if UPLO = 'L',
*> where U (or L) is a product of permutation and unit upper (lower)
*> triangular matrices, and D is symmetric and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*>
*> 2. If some D(i,i)=0, so that D is exactly singular, then the routine
*> returns with INFO = i. Otherwise, the factored form of A is used
*> to estimate the condition number of the matrix A. If the
*> reciprocal of the condition number is less than machine precision,
*> INFO = N+1 is returned as a warning, but the routine still goes on
*> to solve for X and compute error bounds as described below.
*>
*> 3. The system of equations is solved for X using the factored form
*> of A.
*>
*> 4. Iterative refinement is applied to improve the computed solution
*> matrix and calculate error bounds and backward error estimates
*> for it.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of A has been
*> supplied on entry.
*> = 'F': On entry, AF and IPIV contain the factored form
*> of A. A, AF and IPIV will not be modified.
*> = 'N': The matrix A will be copied to AF and factored.
*> \endverbatim
*>
*> \param[in] UPLO
*> \verbatim
*> UPLO is CHARACTER*1
*> = '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 matrices B and X. NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*> A is COMPLEX*16 array, dimension (LDA,N)
*> 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.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*> LDA is INTEGER
*> The leading dimension of the array A. LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] AF
*> \verbatim
*> AF is COMPLEX*16 array, dimension (LDAF,N)
*> If FACT = 'F', then AF is an input argument and on entry
*> contains the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L from the factorization
*> A = U*D*U**T or A = L*D*L**T as computed by ZSYTRF.
*>
*> If FACT = 'N', then AF is an output argument and on exit
*> returns the block diagonal matrix D and the multipliers used
*> to obtain the factor U or L from the factorization
*> A = U*D*U**T or A = L*D*L**T.
*> \endverbatim
*>
*> \param[in] LDAF
*> \verbatim
*> LDAF is INTEGER
*> The leading dimension of the array AF. LDAF >= max(1,N).
*> \endverbatim
*>
*> \param[in,out] IPIV
*> \verbatim
*> IPIV is INTEGER array, dimension (N)
*> If FACT = 'F', then IPIV is an input argument and on entry
*> contains details of the interchanges and the block structure
*> of D, as determined by ZSYTRF.
*> If IPIV(k) > 0, then rows and columns k and IPIV(k) were
*> interchanged and D(k,k) is a 1-by-1 diagonal block.
*> If UPLO = 'U' and IPIV(k) = IPIV(k-1) < 0, then rows and
*> columns k-1 and -IPIV(k) were interchanged and D(k-1:k,k-1:k)
*> is a 2-by-2 diagonal block. If UPLO = 'L' and IPIV(k) =
*> IPIV(k+1) < 0, then rows and columns k+1 and -IPIV(k) were
*> interchanged and D(k:k+1,k:k+1) is a 2-by-2 diagonal block.
*>
*> If FACT = 'N', then IPIV is an output argument and on exit
*> contains details of the interchanges and the block structure
*> of D, as determined by ZSYTRF.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*> B is COMPLEX*16 array, dimension (LDB,NRHS)
*> The N-by-NRHS right hand side matrix B.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*> LDB is INTEGER
*> The leading dimension of the array B. LDB >= max(1,N).
*> \endverbatim
*>
*> \param[out] X
*> \verbatim
*> X is COMPLEX*16 array, dimension (LDX,NRHS)
*> If INFO = 0 or INFO = N+1, the N-by-NRHS solution matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*> LDX is INTEGER
*> The leading dimension of the array X. LDX >= max(1,N).
*> \endverbatim
*>
*> \param[out] RCOND
*> \verbatim
*> RCOND is DOUBLE PRECISION
*> The estimate of the reciprocal condition number of the matrix
*> A. If RCOND is less than the machine precision (in
*> particular, if RCOND = 0), the matrix is singular to working
*> precision. This condition is indicated by a return code of
*> INFO > 0.
*> \endverbatim
*>
*> \param[out] FERR
*> \verbatim
*> FERR is DOUBLE PRECISION array, dimension (NRHS)
*> The estimated forward error bound for each solution vector
*> X(j) (the j-th column of the solution matrix X).
*> If XTRUE is the true solution corresponding to X(j), FERR(j)
*> is an estimated upper bound for the magnitude of the largest
*> element in (X(j) - XTRUE) divided by the magnitude of the
*> largest element in X(j). The estimate is as reliable as
*> the estimate for RCOND, and is almost always a slight
*> overestimate of the true error.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is DOUBLE PRECISION array, dimension (NRHS)
*> The componentwise relative backward error of each solution
*> vector X(j) (i.e., the smallest relative change in
*> any element of A or B that makes X(j) an exact solution).
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The length of WORK. LWORK >= max(1,2*N), and for best
*> performance, when FACT = 'N', LWORK >= max(1,2*N,N*NB), where
*> NB is the optimal blocksize for ZSYTRF.
*>
*> If LWORK = -1, then a workspace query is assumed; the routine
*> only calculates the optimal size of the WORK array, 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] RWORK
*> \verbatim
*> RWORK is DOUBLE PRECISION array, dimension (N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit
*> < 0: if INFO = -i, the i-th argument had an illegal value
*> > 0: if INFO = i, and i is
*> <= N: D(i,i) is exactly zero. The factorization
*> has been completed but the factor D is exactly
*> singular, so the solution and error bounds could
*> not be computed. RCOND = 0 is returned.
*> = N+1: D is nonsingular, but RCOND is less than machine
*> precision, meaning that the matrix is singular
*> to working precision. Nevertheless, the
*> solution and error bounds are computed because
*> there are a number of situations where the
*> computed solution can be more accurate than the
*> value of RCOND would suggest.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16SYsolve
*
* =====================================================================
SUBROUTINE ZSYSVX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B,
$ LDB, X, LDX, RCOND, FERR, BERR, WORK, LWORK,
$ RWORK, 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 FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, LWORK, N, NRHS
DOUBLE PRECISION RCOND
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
DOUBLE PRECISION BERR( * ), FERR( * ), RWORK( * )
COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ WORK( * ), X( LDX, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D+0 )
* ..
* .. Local Scalars ..
LOGICAL LQUERY, NOFACT
INTEGER LWKOPT, NB
DOUBLE PRECISION ANORM
* ..
* .. External Functions ..
LOGICAL LSAME
INTEGER ILAENV
DOUBLE PRECISION DLAMCH, ZLANSY
EXTERNAL LSAME, ILAENV, DLAMCH, ZLANSY
* ..
* .. External Subroutines ..
EXTERNAL XERBLA, ZLACPY, ZSYCON, ZSYRFS, ZSYTRF, ZSYTRS
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
LQUERY = ( LWORK.EQ.-1 )
IF( .NOT.NOFACT .AND. .NOT.LSAME( FACT, 'F' ) ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( UPLO, 'U' ) .AND. .NOT.LSAME( UPLO, 'L' ) )
$ THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( NRHS.LT.0 ) THEN
INFO = -4
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -6
ELSE IF( LDAF.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -11
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( LWORK.LT.MAX( 1, 2*N ) .AND. .NOT.LQUERY ) THEN
INFO = -18
END IF
*
IF( INFO.EQ.0 ) THEN
LWKOPT = MAX( 1, 2*N )
IF( NOFACT ) THEN
NB = ILAENV( 1, 'ZSYTRF', UPLO, N, -1, -1, -1 )
LWKOPT = MAX( LWKOPT, N*NB )
END IF
WORK( 1 ) = LWKOPT
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'ZSYSVX', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
IF( NOFACT ) THEN
*
* Compute the factorization A = U*D*U**T or A = L*D*L**T.
*
CALL ZLACPY( UPLO, N, N, A, LDA, AF, LDAF )
CALL ZSYTRF( UPLO, N, AF, LDAF, IPIV, WORK, LWORK, INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 )THEN
RCOND = ZERO
RETURN
END IF
END IF
*
* Compute the norm of the matrix A.
*
ANORM = ZLANSY( 'I', UPLO, N, A, LDA, RWORK )
*
* Compute the reciprocal of the condition number of A.
*
CALL ZSYCON( UPLO, N, AF, LDAF, IPIV, ANORM, RCOND, WORK, INFO )
*
* Compute the solution vectors X.
*
CALL ZLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL ZSYTRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solutions and
* compute error bounds and backward error estimates for them.
*
CALL ZSYRFS( UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, B, LDB, X,
$ LDX, FERR, BERR, WORK, RWORK, INFO )
*
* Set INFO = N+1 if the matrix is singular to working precision.
*
IF( RCOND.LT.DLAMCH( 'Epsilon' ) )
$ INFO = N + 1
*
WORK( 1 ) = LWKOPT
*
RETURN
*
* End of ZSYSVX
*
END