*> \brief <b> CHESVXX computes the solution to system of linear equations A * X = B for HE matrices</b>
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
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*
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*
* Definition:
* ===========
*
* SUBROUTINE CHESVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
* EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
* N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
* NPARAMS, PARAMS, WORK, RWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER EQUED, FACT, UPLO
* INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
* $ N_ERR_BNDS
* REAL RCOND, RPVGRW
* ..
* .. Array Arguments ..
* INTEGER IPIV( * )
* COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
* $ WORK( * ), X( LDX, * )
* REAL S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
* $ ERR_BNDS_NORM( NRHS, * ),
* $ ERR_BNDS_COMP( NRHS, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> CHESVXX 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 Hermitian matrix and X and B are N-by-NRHS
*> matrices.
*>
*> If requested, both normwise and maximum componentwise error bounds
*> are returned. CHESVXX will return a solution with a tiny
*> guaranteed error (O(eps) where eps is the working machine
*> precision) unless the matrix is very ill-conditioned, in which
*> case a warning is returned. Relevant condition numbers also are
*> calculated and returned.
*>
*> CHESVXX accepts user-provided factorizations and equilibration
*> factors; see the definitions of the FACT and EQUED options.
*> Solving with refinement and using a factorization from a previous
*> CHESVXX call will also produce a solution with either O(eps)
*> errors or warnings, but we cannot make that claim for general
*> user-provided factorizations and equilibration factors if they
*> differ from what CHESVXX would itself produce.
*> \endverbatim
*
*> \par Description:
* =================
*>
*> \verbatim
*>
*> The following steps are performed:
*>
*> 1. If FACT = 'E', real scaling factors are computed to equilibrate
*> the system:
*>
*> diag(S)*A*diag(S) *inv(diag(S))*X = diag(S)*B
*>
*> Whether or not the system will be equilibrated depends on the
*> scaling of the matrix A, but if equilibration is used, A is
*> overwritten by diag(S)*A*diag(S) and B by diag(S)*B.
*>
*> 2. If FACT = 'N' or 'E', the LU decomposition is used to factor
*> the matrix A (after equilibration if FACT = 'E') as
*>
*> 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 Hermitian and block diagonal with
*> 1-by-1 and 2-by-2 diagonal blocks.
*>
*> 3. 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 (see
*> argument RCOND). If the reciprocal of the condition number is
*> less than machine precision, the routine still goes on to solve
*> for X and compute error bounds as described below.
*>
*> 4. The system of equations is solved for X using the factored form
*> of A.
*>
*> 5. By default (unless PARAMS(LA_LINRX_ITREF_I) is set to zero),
*> the routine will use iterative refinement to try to get a small
*> error and error bounds. Refinement calculates the residual to at
*> least twice the working precision.
*>
*> 6. If equilibration was used, the matrix X is premultiplied by
*> diag(R) so that it solves the original system before
*> equilibration.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \verbatim
*> Some optional parameters are bundled in the PARAMS array. These
*> settings determine how refinement is performed, but often the
*> defaults are acceptable. If the defaults are acceptable, users
*> can pass NPARAMS = 0 which prevents the source code from accessing
*> the PARAMS argument.
*> \endverbatim
*>
*> \param[in] FACT
*> \verbatim
*> FACT is CHARACTER*1
*> Specifies whether or not the factored form of the matrix A is
*> supplied on entry, and if not, whether the matrix A should be
*> equilibrated before it is factored.
*> = 'F': On entry, AF and IPIV contain the factored form of A.
*> If EQUED is not 'N', the matrix A has been
*> equilibrated with scaling factors given by S.
*> A, AF, and IPIV are not modified.
*> = 'N': The matrix A will be copied to AF and factored.
*> = 'E': The matrix A will be equilibrated if necessary, then
*> 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,out] A
*> \verbatim
*> A is COMPLEX array, dimension (LDA,N)
*> The Hermitian 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 FACT = 'E' and EQUED = 'Y', A is overwritten by
*> diag(S)*A*diag(S).
*> \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 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**H or A = L*D*L**H as computed by CHETRF.
*>
*> 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**H or A = L*D*L**H.
*> \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 CHETRF. 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 CHETRF.
*> \endverbatim
*>
*> \param[in,out] EQUED
*> \verbatim
*> EQUED is CHARACTER*1
*> Specifies the form of equilibration that was done.
*> = 'N': No equilibration (always true if FACT = 'N').
*> = 'Y': Both row and column equilibration, i.e., A has been
*> replaced by diag(S) * A * diag(S).
*> EQUED is an input argument if FACT = 'F'; otherwise, it is an
*> output argument.
*> \endverbatim
*>
*> \param[in,out] S
*> \verbatim
*> S is REAL array, dimension (N)
*> The scale factors for A. If EQUED = 'Y', A is multiplied on
*> the left and right by diag(S). S is an input argument if FACT =
*> 'F'; otherwise, S is an output argument. If FACT = 'F' and EQUED
*> = 'Y', each element of S must be positive. If S is output, each
*> element of S is a power of the radix. If S is input, each element
*> of S should be a power of the radix to ensure a reliable solution
*> and error estimates. Scaling by powers of the radix does not cause
*> rounding errors unless the result underflows or overflows.
*> Rounding errors during scaling lead to refining with a matrix that
*> is not equivalent to the input matrix, producing error estimates
*> that may not be reliable.
*> \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 EQUED = 'N', B is not modified;
*> if EQUED = 'Y', B is overwritten by diag(S)*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 array, dimension (LDX,NRHS)
*> If INFO = 0, the N-by-NRHS solution matrix X to the original
*> system of equations. Note that A and B are modified on exit if
*> EQUED .ne. 'N', and the solution to the equilibrated system is
*> inv(diag(S))*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 REAL
*> Reciprocal scaled condition number. This is an estimate of the
*> reciprocal Skeel condition number of the matrix A after
*> equilibration (if done). If this is less than the machine
*> precision (in particular, if it is zero), the matrix is singular
*> to working precision. Note that the error may still be small even
*> if this number is very small and the matrix appears ill-
*> conditioned.
*> \endverbatim
*>
*> \param[out] RPVGRW
*> \verbatim
*> RPVGRW is REAL
*> Reciprocal pivot growth. On exit, this contains the reciprocal
*> pivot growth factor norm(A)/norm(U). The "max absolute element"
*> norm is used. If this is much less than 1, then the stability of
*> the LU factorization of the (equilibrated) matrix A could be poor.
*> This also means that the solution X, estimated condition numbers,
*> and error bounds could be unreliable. If factorization fails with
*> 0<INFO<=N, then this contains the reciprocal pivot growth factor
*> for the leading INFO columns of A.
*> \endverbatim
*>
*> \param[out] BERR
*> \verbatim
*> BERR is REAL array, dimension (NRHS)
*> Componentwise relative backward error. This is 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[in] N_ERR_BNDS
*> \verbatim
*> N_ERR_BNDS is INTEGER
*> Number of error bounds to return for each right hand side
*> and each type (normwise or componentwise). See ERR_BNDS_NORM and
*> ERR_BNDS_COMP below.
*> \endverbatim
*>
*> \param[out] ERR_BNDS_NORM
*> \verbatim
*> ERR_BNDS_NORM is REAL array, dimension (NRHS, N_ERR_BNDS)
*> For each right-hand side, this array contains information about
*> various error bounds and condition numbers corresponding to the
*> normwise relative error, which is defined as follows:
*>
*> Normwise relative error in the ith solution vector:
*> max_j (abs(XTRUE(j,i) - X(j,i)))
*> ------------------------------
*> max_j abs(X(j,i))
*>
*> The array is indexed by the type of error information as described
*> below. There currently are up to three pieces of information
*> returned.
*>
*> The first index in ERR_BNDS_NORM(i,:) corresponds to the ith
*> right-hand side.
*>
*> The second index in ERR_BNDS_NORM(:,err) contains the following
*> three fields:
*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
*> reciprocal condition number is less than the threshold
*> sqrt(n) * slamch('Epsilon').
*>
*> err = 2 "Guaranteed" error bound: The estimated forward error,
*> almost certainly within a factor of 10 of the true error
*> so long as the next entry is greater than the threshold
*> sqrt(n) * slamch('Epsilon'). This error bound should only
*> be trusted if the previous boolean is true.
*>
*> err = 3 Reciprocal condition number: Estimated normwise
*> reciprocal condition number. Compared with the threshold
*> sqrt(n) * slamch('Epsilon') to determine if the error
*> estimate is "guaranteed". These reciprocal condition
*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*> appropriately scaled matrix Z.
*> Let Z = S*A, where S scales each row by a power of the
*> radix so all absolute row sums of Z are approximately 1.
*>
*> See Lapack Working Note 165 for further details and extra
*> cautions.
*> \endverbatim
*>
*> \param[out] ERR_BNDS_COMP
*> \verbatim
*> ERR_BNDS_COMP is REAL array, dimension (NRHS, N_ERR_BNDS)
*> For each right-hand side, this array contains information about
*> various error bounds and condition numbers corresponding to the
*> componentwise relative error, which is defined as follows:
*>
*> Componentwise relative error in the ith solution vector:
*> abs(XTRUE(j,i) - X(j,i))
*> max_j ----------------------
*> abs(X(j,i))
*>
*> The array is indexed by the right-hand side i (on which the
*> componentwise relative error depends), and the type of error
*> information as described below. There currently are up to three
*> pieces of information returned for each right-hand side. If
*> componentwise accuracy is not requested (PARAMS(3) = 0.0), then
*> ERR_BNDS_COMP is not accessed. If N_ERR_BNDS < 3, then at most
*> the first (:,N_ERR_BNDS) entries are returned.
*>
*> The first index in ERR_BNDS_COMP(i,:) corresponds to the ith
*> right-hand side.
*>
*> The second index in ERR_BNDS_COMP(:,err) contains the following
*> three fields:
*> err = 1 "Trust/don't trust" boolean. Trust the answer if the
*> reciprocal condition number is less than the threshold
*> sqrt(n) * slamch('Epsilon').
*>
*> err = 2 "Guaranteed" error bound: The estimated forward error,
*> almost certainly within a factor of 10 of the true error
*> so long as the next entry is greater than the threshold
*> sqrt(n) * slamch('Epsilon'). This error bound should only
*> be trusted if the previous boolean is true.
*>
*> err = 3 Reciprocal condition number: Estimated componentwise
*> reciprocal condition number. Compared with the threshold
*> sqrt(n) * slamch('Epsilon') to determine if the error
*> estimate is "guaranteed". These reciprocal condition
*> numbers are 1 / (norm(Z^{-1},inf) * norm(Z,inf)) for some
*> appropriately scaled matrix Z.
*> Let Z = S*(A*diag(x)), where x is the solution for the
*> current right-hand side and S scales each row of
*> A*diag(x) by a power of the radix so all absolute row
*> sums of Z are approximately 1.
*>
*> See Lapack Working Note 165 for further details and extra
*> cautions.
*> \endverbatim
*>
*> \param[in] NPARAMS
*> \verbatim
*> NPARAMS is INTEGER
*> Specifies the number of parameters set in PARAMS. If <= 0, the
*> PARAMS array is never referenced and default values are used.
*> \endverbatim
*>
*> \param[in,out] PARAMS
*> \verbatim
*> PARAMS is REAL array, dimension NPARAMS
*> Specifies algorithm parameters. If an entry is < 0.0, then
*> that entry will be filled with default value used for that
*> parameter. Only positions up to NPARAMS are accessed; defaults
*> are used for higher-numbered parameters.
*>
*> PARAMS(LA_LINRX_ITREF_I = 1) : Whether to perform iterative
*> refinement or not.
*> Default: 1.0
*> = 0.0: No refinement is performed, and no error bounds are
*> computed.
*> = 1.0: Use the double-precision refinement algorithm,
*> possibly with doubled-single computations if the
*> compilation environment does not support DOUBLE
*> PRECISION.
*> (other values are reserved for future use)
*>
*> PARAMS(LA_LINRX_ITHRESH_I = 2) : Maximum number of residual
*> computations allowed for refinement.
*> Default: 10
*> Aggressive: Set to 100 to permit convergence using approximate
*> factorizations or factorizations other than LU. If
*> the factorization uses a technique other than
*> Gaussian elimination, the guarantees in
*> err_bnds_norm and err_bnds_comp may no longer be
*> trustworthy.
*>
*> PARAMS(LA_LINRX_CWISE_I = 3) : Flag determining if the code
*> will attempt to find a solution with small componentwise
*> relative error in the double-precision algorithm. Positive
*> is true, 0.0 is false.
*> Default: 1.0 (attempt componentwise convergence)
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is COMPLEX array, dimension (5*N)
*> \endverbatim
*>
*> \param[out] RWORK
*> \verbatim
*> RWORK is REAL array, dimension (2*N)
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: Successful exit. The solution to every right-hand side is
*> guaranteed.
*> < 0: If INFO = -i, the i-th argument had an illegal value
*> > 0 and <= N: U(INFO,INFO) is exactly zero. The factorization
*> has been completed, but the factor U is exactly singular, so
*> the solution and error bounds could not be computed. RCOND = 0
*> is returned.
*> = N+J: The solution corresponding to the Jth right-hand side is
*> not guaranteed. The solutions corresponding to other right-
*> hand sides K with K > J may not be guaranteed as well, but
*> only the first such right-hand side is reported. If a small
*> componentwise error is not requested (PARAMS(3) = 0.0) then
*> the Jth right-hand side is the first with a normwise error
*> bound that is not guaranteed (the smallest J such
*> that ERR_BNDS_NORM(J,1) = 0.0). By default (PARAMS(3) = 1.0)
*> the Jth right-hand side is the first with either a normwise or
*> componentwise error bound that is not guaranteed (the smallest
*> J such that either ERR_BNDS_NORM(J,1) = 0.0 or
*> ERR_BNDS_COMP(J,1) = 0.0). See the definition of
*> ERR_BNDS_NORM(:,1) and ERR_BNDS_COMP(:,1). To get information
*> about all of the right-hand sides check ERR_BNDS_NORM or
*> ERR_BNDS_COMP.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complexHEsolve
*
* =====================================================================
SUBROUTINE CHESVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV,
$ EQUED, S, B, LDB, X, LDX, RCOND, RPVGRW, BERR,
$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP,
$ NPARAMS, PARAMS, WORK, 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 EQUED, FACT, UPLO
INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
$ N_ERR_BNDS
REAL RCOND, RPVGRW
* ..
* .. Array Arguments ..
INTEGER IPIV( * )
COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
$ WORK( * ), X( LDX, * )
REAL S( * ), PARAMS( * ), BERR( * ), RWORK( * ),
$ ERR_BNDS_NORM( NRHS, * ),
$ ERR_BNDS_COMP( NRHS, * )
* ..
*
* ==================================================================
*
* .. Parameters ..
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 )
INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
INTEGER CMP_ERR_I, PIV_GROWTH_I
PARAMETER ( FINAL_NRM_ERR_I = 1, FINAL_CMP_ERR_I = 2,
$ BERR_I = 3 )
PARAMETER ( RCOND_I = 4, NRM_RCOND_I = 5, NRM_ERR_I = 6 )
PARAMETER ( CMP_RCOND_I = 7, CMP_ERR_I = 8,
$ PIV_GROWTH_I = 9 )
* ..
* .. Local Scalars ..
LOGICAL EQUIL, NOFACT, RCEQU
INTEGER INFEQU, J
REAL AMAX, BIGNUM, SMIN, SMAX, SCOND, SMLNUM
* ..
* .. External Functions ..
EXTERNAL LSAME, SLAMCH, CLA_HERPVGRW
LOGICAL LSAME
REAL SLAMCH, CLA_HERPVGRW
* ..
* .. External Subroutines ..
EXTERNAL CHEEQUB, CHETRF, CHETRS, CLACPY,
$ CLAQHE, XERBLA, CLASCL2, CHERFSX
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* ..
* .. Executable Statements ..
*
INFO = 0
NOFACT = LSAME( FACT, 'N' )
EQUIL = LSAME( FACT, 'E' )
SMLNUM = SLAMCH( 'Safe minimum' )
BIGNUM = ONE / SMLNUM
IF( NOFACT .OR. EQUIL ) THEN
EQUED = 'N'
RCEQU = .FALSE.
ELSE
RCEQU = LSAME( EQUED, 'Y' )
ENDIF
*
* Default is failure. If an input parameter is wrong or
* factorization fails, make everything look horrible. Only the
* pivot growth is set here, the rest is initialized in CHERFSX.
*
RPVGRW = ZERO
*
* Test the input parameters. PARAMS is not tested until CHERFSX.
*
IF( .NOT.NOFACT .AND. .NOT.EQUIL .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( LSAME( FACT, 'F' ) .AND. .NOT.
$ ( RCEQU .OR. LSAME( EQUED, 'N' ) ) ) THEN
INFO = -9
ELSE
IF ( RCEQU ) THEN
SMIN = BIGNUM
SMAX = ZERO
DO 10 J = 1, N
SMIN = MIN( SMIN, S( J ) )
SMAX = MAX( SMAX, S( J ) )
10 CONTINUE
IF( SMIN.LE.ZERO ) THEN
INFO = -10
ELSE IF( N.GT.0 ) THEN
SCOND = MAX( SMIN, SMLNUM ) / MIN( SMAX, BIGNUM )
ELSE
SCOND = ONE
END IF
END IF
IF( INFO.EQ.0 ) THEN
IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -12
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -14
END IF
END IF
END IF
*
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'CHESVXX', -INFO )
RETURN
END IF
*
IF( EQUIL ) THEN
*
* Compute row and column scalings to equilibrate the matrix A.
*
CALL CHEEQUB( UPLO, N, A, LDA, S, SCOND, AMAX, WORK, INFEQU )
IF( INFEQU.EQ.0 ) THEN
*
* Equilibrate the matrix.
*
CALL CLAQHE( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED )
RCEQU = LSAME( EQUED, 'Y' )
END IF
END IF
*
* Scale the right-hand side.
*
IF( RCEQU ) CALL CLASCL2( N, NRHS, S, B, LDB )
*
IF( NOFACT .OR. EQUIL ) THEN
*
* Compute the LDL^H or UDU^H factorization of A.
*
CALL CLACPY( UPLO, N, N, A, LDA, AF, LDAF )
CALL CHETRF( UPLO, N, AF, LDAF, IPIV, WORK, 5*MAX(1,N), INFO )
*
* Return if INFO is non-zero.
*
IF( INFO.GT.0 ) THEN
*
* Pivot in column INFO is exactly 0
* Compute the reciprocal pivot growth factor of the
* leading rank-deficient INFO columns of A.
*
IF( N.GT.0 )
$ RPVGRW = CLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF,
$ IPIV, RWORK )
RETURN
END IF
END IF
*
* Compute the reciprocal pivot growth factor RPVGRW.
*
IF( N.GT.0 )
$ RPVGRW = CLA_HERPVGRW( UPLO, N, INFO, A, LDA, AF, LDAF, IPIV,
$ RWORK )
*
* Compute the solution matrix X.
*
CALL CLACPY( 'Full', N, NRHS, B, LDB, X, LDX )
CALL CHETRS( UPLO, N, NRHS, AF, LDAF, IPIV, X, LDX, INFO )
*
* Use iterative refinement to improve the computed solution and
* compute error bounds and backward error estimates for it.
*
CALL CHERFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, IPIV,
$ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM,
$ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, INFO )
*
* Scale solutions.
*
IF ( RCEQU ) THEN
CALL CLASCL2 ( N, NRHS, S, X, LDX )
END IF
*
RETURN
*
* End of CHESVXX
*
END