*> \brief \b SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLA_GERFSX_EXTENDED + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE SLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A, * LDA, AF, LDAF, IPIV, COLEQU, C, B, * LDB, Y, LDY, BERR_OUT, N_NORMS, * ERRS_N, ERRS_C, RES, * AYB, DY, Y_TAIL, RCOND, ITHRESH, * RTHRESH, DZ_UB, IGNORE_CWISE, * INFO ) * * .. Scalar Arguments .. * INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, * $ TRANS_TYPE, N_NORMS, ITHRESH * LOGICAL COLEQU, IGNORE_CWISE * REAL RTHRESH, DZ_UB * .. * .. Array Arguments .. * INTEGER IPIV( * ) * REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), * $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) * REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), * $ ERRS_N( NRHS, * ), * $ ERRS_C( NRHS, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> SLA_GERFSX_EXTENDED improves the computed solution to a system of *> linear equations by performing extra-precise iterative refinement *> and provides error bounds and backward error estimates for the solution. *> This subroutine is called by SGERFSX to perform iterative refinement. *> In addition to normwise error bound, the code provides maximum *> componentwise error bound if possible. See comments for ERRS_N *> and ERRS_C for details of the error bounds. Note that this *> subroutine is only resonsible for setting the second fields of *> ERRS_N and ERRS_C. *> \endverbatim * * Arguments: * ========== * *> \param[in] PREC_TYPE *> \verbatim *> PREC_TYPE is INTEGER *> Specifies the intermediate precision to be used in refinement. *> The value is defined by ILAPREC(P) where P is a CHARACTER and *> P = 'S': Single *> = 'D': Double *> = 'I': Indigenous *> = 'X', 'E': Extra *> \endverbatim *> *> \param[in] TRANS_TYPE *> \verbatim *> TRANS_TYPE is INTEGER *> Specifies the transposition operation on A. *> The value is defined by ILATRANS(T) where T is a CHARACTER and *> T = 'N': No transpose *> = 'T': Transpose *> = 'C': Conjugate transpose *> \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. *> \endverbatim *> *> \param[in] A *> \verbatim *> A is REAL array, dimension (LDA,N) *> On entry, the N-by-N matrix A. *> \endverbatim *> *> \param[in] LDA *> \verbatim *> LDA is INTEGER *> The leading dimension of the array A. LDA >= max(1,N). *> \endverbatim *> *> \param[in] AF *> \verbatim *> AF is REAL array, dimension (LDAF,N) *> The factors L and U from the factorization *> A = P*L*U as computed by SGETRF. *> \endverbatim *> *> \param[in] LDAF *> \verbatim *> LDAF is INTEGER *> The leading dimension of the array AF. LDAF >= max(1,N). *> \endverbatim *> *> \param[in] IPIV *> \verbatim *> IPIV is INTEGER array, dimension (N) *> The pivot indices from the factorization A = P*L*U *> as computed by SGETRF; row i of the matrix was interchanged *> with row IPIV(i). *> \endverbatim *> *> \param[in] COLEQU *> \verbatim *> COLEQU is LOGICAL *> If .TRUE. then column equilibration was done to A before calling *> this routine. This is needed to compute the solution and error *> bounds correctly. *> \endverbatim *> *> \param[in] C *> \verbatim *> C is REAL array, dimension (N) *> The column scale factors for A. If COLEQU = .FALSE., C *> is not accessed. If C is input, each element of C 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] B *> \verbatim *> B is REAL array, dimension (LDB,NRHS) *> The 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[in,out] Y *> \verbatim *> Y is REAL array, dimension (LDY,NRHS) *> On entry, the solution matrix X, as computed by SGETRS. *> On exit, the improved solution matrix Y. *> \endverbatim *> *> \param[in] LDY *> \verbatim *> LDY is INTEGER *> The leading dimension of the array Y. LDY >= max(1,N). *> \endverbatim *> *> \param[out] BERR_OUT *> \verbatim *> BERR_OUT is REAL array, dimension (NRHS) *> On exit, BERR_OUT(j) contains the componentwise relative backward *> error for right-hand-side j from the formula *> max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) *> where abs(Z) is the componentwise absolute value of the matrix *> or vector Z. This is computed by SLA_LIN_BERR. *> \endverbatim *> *> \param[in] N_NORMS *> \verbatim *> N_NORMS is INTEGER *> Determines which error bounds to return (see ERRS_N *> and ERRS_C). *> If N_NORMS >= 1 return normwise error bounds. *> If N_NORMS >= 2 return componentwise error bounds. *> \endverbatim *> *> \param[in,out] ERRS_N *> \verbatim *> ERRS_N 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 ERRS_N(i,:) corresponds to the ith *> right-hand side. *> *> The second index in ERRS_N(:,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. *> *> This subroutine is only responsible for setting the second field *> above. *> See Lapack Working Note 165 for further details and extra *> cautions. *> \endverbatim *> *> \param[in,out] ERRS_C *> \verbatim *> ERRS_C 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 *> ERRS_C is not accessed. If N_ERR_BNDS .LT. 3, then at most *> the first (:,N_ERR_BNDS) entries are returned. *> *> The first index in ERRS_C(i,:) corresponds to the ith *> right-hand side. *> *> The second index in ERRS_C(:,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. *> *> This subroutine is only responsible for setting the second field *> above. *> See Lapack Working Note 165 for further details and extra *> cautions. *> \endverbatim *> *> \param[in] RES *> \verbatim *> RES is REAL array, dimension (N) *> Workspace to hold the intermediate residual. *> \endverbatim *> *> \param[in] AYB *> \verbatim *> AYB is REAL array, dimension (N) *> Workspace. This can be the same workspace passed for Y_TAIL. *> \endverbatim *> *> \param[in] DY *> \verbatim *> DY is REAL array, dimension (N) *> Workspace to hold the intermediate solution. *> \endverbatim *> *> \param[in] Y_TAIL *> \verbatim *> Y_TAIL is REAL array, dimension (N) *> Workspace to hold the trailing bits of the intermediate solution. *> \endverbatim *> *> \param[in] 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[in] ITHRESH *> \verbatim *> ITHRESH is INTEGER *> The maximum number of residual computations allowed for *> refinement. The default is 10. For '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 *> ERRS_N and ERRS_C may no longer be trustworthy. *> \endverbatim *> *> \param[in] RTHRESH *> \verbatim *> RTHRESH is REAL *> Determines when to stop refinement if the error estimate stops *> decreasing. Refinement will stop when the next solution no longer *> satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is *> the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The *> default value is 0.5. For 'aggressive' set to 0.9 to permit *> convergence on extremely ill-conditioned matrices. See LAWN 165 *> for more details. *> \endverbatim *> *> \param[in] DZ_UB *> \verbatim *> DZ_UB is REAL *> Determines when to start considering componentwise convergence. *> Componentwise convergence is only considered after each component *> of the solution Y is stable, which we definte as the relative *> change in each component being less than DZ_UB. The default value *> is 0.25, requiring the first bit to be stable. See LAWN 165 for *> more details. *> \endverbatim *> *> \param[in] IGNORE_CWISE *> \verbatim *> IGNORE_CWISE is LOGICAL *> If .TRUE. then ignore componentwise convergence. Default value *> is .FALSE.. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: Successful exit. *> < 0: if INFO = -i, the ith argument to SGETRS had an illegal *> value *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup realGEcomputational * * ===================================================================== SUBROUTINE SLA_GERFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, NRHS, A, $ LDA, AF, LDAF, IPIV, COLEQU, C, B, $ LDB, Y, LDY, BERR_OUT, N_NORMS, $ ERRS_N, ERRS_C, RES, $ AYB, DY, Y_TAIL, RCOND, ITHRESH, $ RTHRESH, DZ_UB, IGNORE_CWISE, $ INFO ) * * -- LAPACK computational 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 .. INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE, $ TRANS_TYPE, N_NORMS, ITHRESH LOGICAL COLEQU, IGNORE_CWISE REAL RTHRESH, DZ_UB * .. * .. Array Arguments .. INTEGER IPIV( * ) REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ), $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * ) REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ), $ ERRS_N( NRHS, * ), $ ERRS_C( NRHS, * ) * .. * * ===================================================================== * * .. Local Scalars .. CHARACTER TRANS INTEGER CNT, I, J, X_STATE, Z_STATE, Y_PREC_STATE REAL YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT, $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX, $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z, $ EPS, HUGEVAL, INCR_THRESH LOGICAL INCR_PREC * .. * .. Parameters .. INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE, $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL, $ EXTRA_Y PARAMETER ( UNSTABLE_STATE = 0, WORKING_STATE = 1, $ CONV_STATE = 2, NOPROG_STATE = 3 ) PARAMETER ( BASE_RESIDUAL = 0, EXTRA_RESIDUAL = 1, $ EXTRA_Y = 2 ) 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 ) INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I, $ LA_LINRX_CWISE_I PARAMETER ( LA_LINRX_ITREF_I = 1, $ LA_LINRX_ITHRESH_I = 2 ) PARAMETER ( LA_LINRX_CWISE_I = 3 ) INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I, $ LA_LINRX_RCOND_I PARAMETER ( LA_LINRX_TRUST_I = 1, LA_LINRX_ERR_I = 2 ) PARAMETER ( LA_LINRX_RCOND_I = 3 ) * .. * .. External Subroutines .. EXTERNAL SAXPY, SCOPY, SGETRS, SGEMV, BLAS_SGEMV_X, $ BLAS_SGEMV2_X, SLA_GEAMV, SLA_WWADDW, SLAMCH, $ CHLA_TRANSTYPE, SLA_LIN_BERR REAL SLAMCH CHARACTER CHLA_TRANSTYPE * .. * .. Intrinsic Functions .. INTRINSIC ABS, MAX, MIN * .. * .. Executable Statements .. * IF ( INFO.NE.0 ) RETURN TRANS = CHLA_TRANSTYPE(TRANS_TYPE) EPS = SLAMCH( 'Epsilon' ) HUGEVAL = SLAMCH( 'Overflow' ) * Force HUGEVAL to Inf HUGEVAL = HUGEVAL * HUGEVAL * Using HUGEVAL may lead to spurious underflows. INCR_THRESH = REAL( N ) * EPS * DO J = 1, NRHS Y_PREC_STATE = EXTRA_RESIDUAL IF ( Y_PREC_STATE .EQ. EXTRA_Y ) THEN DO I = 1, N Y_TAIL( I ) = 0.0 END DO END IF DXRAT = 0.0 DXRATMAX = 0.0 DZRAT = 0.0 DZRATMAX = 0.0 FINAL_DX_X = HUGEVAL FINAL_DZ_Z = HUGEVAL PREVNORMDX = HUGEVAL PREV_DZ_Z = HUGEVAL DZ_Z = HUGEVAL DX_X = HUGEVAL X_STATE = WORKING_STATE Z_STATE = UNSTABLE_STATE INCR_PREC = .FALSE. DO CNT = 1, ITHRESH * * Compute residual RES = B_s - op(A_s) * Y, * op(A) = A, A**T, or A**H depending on TRANS (and type). * CALL SCOPY( N, B( 1, J ), 1, RES, 1 ) IF ( Y_PREC_STATE .EQ. BASE_RESIDUAL ) THEN CALL SGEMV( TRANS, N, N, -1.0, A, LDA, Y( 1, J ), 1, $ 1.0, RES, 1 ) ELSE IF ( Y_PREC_STATE .EQ. EXTRA_RESIDUAL ) THEN CALL BLAS_SGEMV_X( TRANS_TYPE, N, N, -1.0, A, LDA, $ Y( 1, J ), 1, 1.0, RES, 1, PREC_TYPE ) ELSE CALL BLAS_SGEMV2_X( TRANS_TYPE, N, N, -1.0, A, LDA, $ Y( 1, J ), Y_TAIL, 1, 1.0, RES, 1, PREC_TYPE ) END IF ! XXX: RES is no longer needed. CALL SCOPY( N, RES, 1, DY, 1 ) CALL SGETRS( TRANS, N, 1, AF, LDAF, IPIV, DY, N, INFO ) * * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT. * NORMX = 0.0 NORMY = 0.0 NORMDX = 0.0 DZ_Z = 0.0 YMIN = HUGEVAL * DO I = 1, N YK = ABS( Y( I, J ) ) DYK = ABS( DY( I ) ) IF ( YK .NE. 0.0 ) THEN DZ_Z = MAX( DZ_Z, DYK / YK ) ELSE IF ( DYK .NE. 0.0 ) THEN DZ_Z = HUGEVAL END IF YMIN = MIN( YMIN, YK ) NORMY = MAX( NORMY, YK ) IF ( COLEQU ) THEN NORMX = MAX( NORMX, YK * C( I ) ) NORMDX = MAX( NORMDX, DYK * C( I ) ) ELSE NORMX = NORMY NORMDX = MAX( NORMDX, DYK ) END IF END DO IF ( NORMX .NE. 0.0 ) THEN DX_X = NORMDX / NORMX ELSE IF ( NORMDX .EQ. 0.0 ) THEN DX_X = 0.0 ELSE DX_X = HUGEVAL END IF DXRAT = NORMDX / PREVNORMDX DZRAT = DZ_Z / PREV_DZ_Z * * Check termination criteria * IF (.NOT.IGNORE_CWISE $ .AND. YMIN*RCOND .LT. INCR_THRESH*NORMY $ .AND. Y_PREC_STATE .LT. EXTRA_Y) $ INCR_PREC = .TRUE. IF ( X_STATE .EQ. NOPROG_STATE .AND. DXRAT .LE. RTHRESH ) $ X_STATE = WORKING_STATE IF ( X_STATE .EQ. WORKING_STATE ) THEN IF ( DX_X .LE. EPS ) THEN X_STATE = CONV_STATE ELSE IF ( DXRAT .GT. RTHRESH ) THEN IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN INCR_PREC = .TRUE. ELSE X_STATE = NOPROG_STATE END IF ELSE IF ( DXRAT .GT. DXRATMAX ) DXRATMAX = DXRAT END IF IF ( X_STATE .GT. WORKING_STATE ) FINAL_DX_X = DX_X END IF IF ( Z_STATE .EQ. UNSTABLE_STATE .AND. DZ_Z .LE. DZ_UB ) $ Z_STATE = WORKING_STATE IF ( Z_STATE .EQ. NOPROG_STATE .AND. DZRAT .LE. RTHRESH ) $ Z_STATE = WORKING_STATE IF ( Z_STATE .EQ. WORKING_STATE ) THEN IF ( DZ_Z .LE. EPS ) THEN Z_STATE = CONV_STATE ELSE IF ( DZ_Z .GT. DZ_UB ) THEN Z_STATE = UNSTABLE_STATE DZRATMAX = 0.0 FINAL_DZ_Z = HUGEVAL ELSE IF ( DZRAT .GT. RTHRESH ) THEN IF ( Y_PREC_STATE .NE. EXTRA_Y ) THEN INCR_PREC = .TRUE. ELSE Z_STATE = NOPROG_STATE END IF ELSE IF ( DZRAT .GT. DZRATMAX ) DZRATMAX = DZRAT END IF IF ( Z_STATE .GT. WORKING_STATE ) FINAL_DZ_Z = DZ_Z END IF * * Exit if both normwise and componentwise stopped working, * but if componentwise is unstable, let it go at least two * iterations. * IF ( X_STATE.NE.WORKING_STATE ) THEN IF ( IGNORE_CWISE) GOTO 666 IF ( Z_STATE.EQ.NOPROG_STATE .OR. Z_STATE.EQ.CONV_STATE ) $ GOTO 666 IF ( Z_STATE.EQ.UNSTABLE_STATE .AND. CNT.GT.1 ) GOTO 666 END IF IF ( INCR_PREC ) THEN INCR_PREC = .FALSE. Y_PREC_STATE = Y_PREC_STATE + 1 DO I = 1, N Y_TAIL( I ) = 0.0 END DO END IF PREVNORMDX = NORMDX PREV_DZ_Z = DZ_Z * * Update soluton. * IF ( Y_PREC_STATE .LT. EXTRA_Y ) THEN CALL SAXPY( N, 1.0, DY, 1, Y( 1, J ), 1 ) ELSE CALL SLA_WWADDW( N, Y( 1, J ), Y_TAIL, DY ) END IF END DO * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT. 666 CONTINUE * * Set final_* when cnt hits ithresh. * IF ( X_STATE .EQ. WORKING_STATE ) FINAL_DX_X = DX_X IF ( Z_STATE .EQ. WORKING_STATE ) FINAL_DZ_Z = DZ_Z * * Compute error bounds * IF (N_NORMS .GE. 1) THEN ERRS_N( J, LA_LINRX_ERR_I ) = $ FINAL_DX_X / (1 - DXRATMAX) END IF IF ( N_NORMS .GE. 2 ) THEN ERRS_C( J, LA_LINRX_ERR_I ) = $ FINAL_DZ_Z / (1 - DZRATMAX) END IF * * Compute componentwise relative backward error from formula * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) ) * where abs(Z) is the componentwise absolute value of the matrix * or vector Z. * * Compute residual RES = B_s - op(A_s) * Y, * op(A) = A, A**T, or A**H depending on TRANS (and type). * CALL SCOPY( N, B( 1, J ), 1, RES, 1 ) CALL SGEMV( TRANS, N, N, -1.0, A, LDA, Y(1,J), 1, 1.0, RES, 1 ) DO I = 1, N AYB( I ) = ABS( B( I, J ) ) END DO * * Compute abs(op(A_s))*abs(Y) + abs(B_s). * CALL SLA_GEAMV ( TRANS_TYPE, N, N, 1.0, $ A, LDA, Y(1, J), 1, 1.0, AYB, 1 ) CALL SLA_LIN_BERR ( N, N, 1, RES, AYB, BERR_OUT( J ) ) * * End of loop for each RHS. * END DO * RETURN END