*> \brief SPOSVXX computes the solution to system of linear equations A * X = B for PO matrices * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SPOSVXX + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, * S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, * N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, * NPARAMS, PARAMS, WORK, IWORK, 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 IWORK( * ) * REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), * \$ X( LDX, * ), WORK( * ) * REAL S( * ), PARAMS( * ), BERR( * ), * \$ ERR_BNDS_NORM( NRHS, * ), * \$ ERR_BNDS_COMP( NRHS, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SPOSVXX uses the Cholesky factorization A = U**T*U or A = L*L**T *> to compute the solution to a real system of linear equations *> A * X = B, where A is an N-by-N symmetric positive definite matrix *> and X and B are N-by-NRHS matrices. *> *> If requested, both normwise and maximum componentwise error bounds *> are returned. SPOSVXX 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. *> *> SPOSVXX 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 *> SPOSVXX 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 SPOSVXX 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 Cholesky decomposition is used to *> factor the matrix A (after equilibration if FACT = 'E') as *> A = U**T* U, if UPLO = 'U', or *> A = L * L**T, if UPLO = 'L', *> where U is an upper triangular matrix and L is a lower triangular *> matrix. *> *> 3. If the leading i-by-i principal minor is not positive definite, *> 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(S) 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 contains the factored form of A. *> If EQUED is not 'N', the matrix A has been *> equilibrated with scaling factors given by S. *> A and AF 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 REAL array, dimension (LDA,N) *> On entry, the symmetric matrix A, except if FACT = 'F' and EQUED = *> 'Y', then A must contain the equilibrated matrix *> diag(S)*A*diag(S). 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. A is *> not modified if FACT = 'F' or 'N', or if FACT = 'E' and EQUED = *> 'N' on exit. *> *> 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 REAL array, dimension (LDAF,N) *> If FACT = 'F', then AF is an input argument and on entry *> contains the triangular factor U or L from the Cholesky *> factorization A = U**T*U or A = L*L**T, in the same storage *> format as A. If EQUED .ne. 'N', then AF is the factored *> form of the equilibrated matrix diag(S)*A*diag(S). *> *> If FACT = 'N', then AF is an output argument and on exit *> returns the triangular factor U or L from the Cholesky *> factorization A = U**T*U or A = L*L**T of the original *> matrix A. *> *> If FACT = 'E', then AF is an output argument and on exit *> returns the triangular factor U or L from the Cholesky *> factorization A = U**T*U or A = L*L**T of the equilibrated *> matrix A (see the description of A for the form of the *> equilibrated matrix). *> \endverbatim *> *> \param[in] LDAF *> \verbatim *> LDAF is INTEGER *> The leading dimension of the array AF. LDAF >= max(1,N). *> \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 row 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 REAL 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 REAL 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 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 .LT. 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 .LE. 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 .LT. 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 REAL array, dimension (4*N) *> \endverbatim *> *> \param[out] IWORK *> \verbatim *> IWORK is INTEGER array, dimension (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. * *> \date April 2012 * *> \ingroup realPOsolve * * ===================================================================== SUBROUTINE SPOSVXX( FACT, UPLO, N, NRHS, A, LDA, AF, LDAF, EQUED, \$ S, B, LDB, X, LDX, RCOND, RPVGRW, BERR, \$ N_ERR_BNDS, ERR_BNDS_NORM, ERR_BNDS_COMP, \$ NPARAMS, PARAMS, WORK, IWORK, 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..-- * April 2012 * * .. Scalar Arguments .. CHARACTER EQUED, FACT, UPLO INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS, \$ N_ERR_BNDS REAL RCOND, RPVGRW * .. * .. Array Arguments .. INTEGER IWORK( * ) REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), \$ X( LDX, * ), WORK( * ) REAL S( * ), PARAMS( * ), BERR( * ), \$ 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, SLA_PORPVGRW LOGICAL LSAME REAL SLAMCH, SLA_PORPVGRW * .. * .. External Subroutines .. EXTERNAL SPOEQUB, SPOTRF, SPOTRS, SLACPY, SLAQSY, \$ XERBLA, SLASCL2, SPORFSX * .. * .. 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 SPORFSX. * RPVGRW = ZERO * * Test the input parameters. PARAMS is not tested until SPORFSX. * 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( 'SPOSVXX', -INFO ) RETURN END IF * IF( EQUIL ) THEN * * Compute row and column scalings to equilibrate the matrix A. * CALL SPOEQUB( N, A, LDA, S, SCOND, AMAX, INFEQU ) IF( INFEQU.EQ.0 ) THEN * * Equilibrate the matrix. * CALL SLAQSY( UPLO, N, A, LDA, S, SCOND, AMAX, EQUED ) RCEQU = LSAME( EQUED, 'Y' ) END IF END IF * * Scale the right-hand side. * IF( RCEQU ) CALL SLASCL2( N, NRHS, S, B, LDB ) * IF( NOFACT .OR. EQUIL ) THEN * * Compute the Cholesky factorization of A. * CALL SLACPY( UPLO, N, N, A, LDA, AF, LDAF ) CALL SPOTRF( UPLO, N, AF, LDAF, INFO ) * * Return if INFO is non-zero. * IF( INFO.NE.0 ) THEN * * Pivot in column INFO is exactly 0 * Compute the reciprocal pivot growth factor of the * leading rank-deficient INFO columns of A. * RPVGRW = SLA_PORPVGRW( UPLO, INFO, A, LDA, AF, LDAF, WORK ) RETURN ENDIF END IF * * Compute the reciprocal growth factor RPVGRW. * RPVGRW = SLA_PORPVGRW( UPLO, N, A, LDA, AF, LDAF, WORK ) * * Compute the solution matrix X. * CALL SLACPY( 'Full', N, NRHS, B, LDB, X, LDX ) CALL SPOTRS( UPLO, N, NRHS, AF, LDAF, X, LDX, INFO ) * * Use iterative refinement to improve the computed solution and * compute error bounds and backward error estimates for it. * CALL SPORFSX( UPLO, EQUED, N, NRHS, A, LDA, AF, LDAF, \$ S, B, LDB, X, LDX, RCOND, BERR, N_ERR_BNDS, ERR_BNDS_NORM, \$ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, IWORK, INFO ) * * Scale solutions. * IF ( RCEQU ) THEN CALL SLASCL2 ( N, NRHS, S, X, LDX ) END IF * RETURN * * End of SPOSVXX * END