numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
| Name | Size | Mode | |
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| lapack/SRC/zgbrfsx.f | 29587B | -rw-r--r-- |
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*> \brief \b ZGBRFSX * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download ZGBRFSX + dependencies *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zgbrfsx.f"> *> [TGZ]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zgbrfsx.f"> *> [ZIP]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zgbrfsx.f"> *> [TXT]</a> *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, AFB, * LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, * BERR, N_ERR_BNDS, ERR_BNDS_NORM, * ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, * INFO ) * * .. Scalar Arguments .. * CHARACTER TRANS, EQUED * INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, * $ NPARAMS, N_ERR_BNDS * DOUBLE PRECISION RCOND * .. * .. Array Arguments .. * INTEGER IPIV( * ) * COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), * $ X( LDX , * ),WORK( * ) * DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ), * $ ERR_BNDS_NORM( NRHS, * ), * $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> ZGBRFSX improves the computed solution to a system of linear *> equations and provides error bounds and backward error estimates *> for the solution. In addition to normwise error bound, the code *> provides maximum componentwise error bound if possible. See *> comments for ERR_BNDS_NORM and ERR_BNDS_COMP for details of the *> error bounds. *> *> The original system of linear equations may have been equilibrated *> before calling this routine, as described by arguments EQUED, R *> and C below. In this case, the solution and error bounds returned *> are for the original unequilibrated system. *> \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] TRANS *> \verbatim *> TRANS is CHARACTER*1 *> Specifies the form of the system of equations: *> = 'N': A * X = B (No transpose) *> = 'T': A**T * X = B (Transpose) *> = 'C': A**H * X = B (Conjugate transpose) *> \endverbatim *> *> \param[in] EQUED *> \verbatim *> EQUED is CHARACTER*1 *> Specifies the form of equilibration that was done to A *> before calling this routine. This is needed to compute *> the solution and error bounds correctly. *> = 'N': No equilibration *> = 'R': Row equilibration, i.e., A has been premultiplied by *> diag(R). *> = 'C': Column equilibration, i.e., A has been postmultiplied *> by diag(C). *> = 'B': Both row and column equilibration, i.e., A has been *> replaced by diag(R) * A * diag(C). *> The right hand side B has been changed accordingly. *> \endverbatim *> *> \param[in] N *> \verbatim *> N is INTEGER *> The order of the matrix A. N >= 0. *> \endverbatim *> *> \param[in] KL *> \verbatim *> KL is INTEGER *> The number of subdiagonals within the band of A. KL >= 0. *> \endverbatim *> *> \param[in] KU *> \verbatim *> KU is INTEGER *> The number of superdiagonals within the band of A. KU >= 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] AB *> \verbatim *> AB is COMPLEX*16 array, dimension (LDAB,N) *> The original band matrix A, stored in rows 1 to KL+KU+1. *> The j-th column of A is stored in the j-th column of the *> array AB as follows: *> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). *> \endverbatim *> *> \param[in] LDAB *> \verbatim *> LDAB is INTEGER *> The leading dimension of the array AB. LDAB >= KL+KU+1. *> \endverbatim *> *> \param[in] AFB *> \verbatim *> AFB is COMPLEX*16 array, dimension (LDAFB,N) *> Details of the LU factorization of the band matrix A, as *> computed by ZGBTRF. U is stored as an upper triangular band *> matrix with KL+KU superdiagonals in rows 1 to KL+KU+1, and *> the multipliers used during the factorization are stored in *> rows KL+KU+2 to 2*KL+KU+1. *> \endverbatim *> *> \param[in] LDAFB *> \verbatim *> LDAFB is INTEGER *> The leading dimension of the array AFB. LDAFB >= 2*KL*KU+1. *> \endverbatim *> *> \param[in] IPIV *> \verbatim *> IPIV is INTEGER array, dimension (N) *> The pivot indices from ZGETRF; for 1<=i<=N, row i of the *> matrix was interchanged with row IPIV(i). *> \endverbatim *> *> \param[in,out] R *> \verbatim *> R is DOUBLE PRECISION array, dimension (N) *> The row scale factors for A. If EQUED = 'R' or 'B', A is *> multiplied on the left by diag(R); if EQUED = 'N' or 'C', R *> is not accessed. R is an input argument if FACT = 'F'; *> otherwise, R is an output argument. If FACT = 'F' and *> EQUED = 'R' or 'B', each element of R must be positive. *> If R is output, each element of R is a power of the radix. *> If R is input, each element of R 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] C *> \verbatim *> C is DOUBLE PRECISION array, dimension (N) *> The column scale factors for A. If EQUED = 'C' or 'B', A is *> multiplied on the right by diag(C); if EQUED = 'N' or 'R', C *> is not accessed. C is an input argument if FACT = 'F'; *> otherwise, C is an output argument. If FACT = 'F' and *> EQUED = 'C' or 'B', each element of C must be positive. *> If C is output, each element of C is a power of the radix. *> 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 COMPLEX*16 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] X *> \verbatim *> X is COMPLEX*16 array, dimension (LDX,NRHS) *> On entry, the solution matrix X, as computed by ZGETRS. *> On exit, the improved 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 *> 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] BERR *> \verbatim *> BERR is DOUBLE PRECISION 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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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) * dlamch('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) * dlamch('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) * dlamch('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 DOUBLE PRECISION 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: 1.0D+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*16 array, dimension (2*N) *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is DOUBLE PRECISION 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 gbrfsx * * ===================================================================== SUBROUTINE ZGBRFSX( TRANS, EQUED, N, KL, KU, NRHS, AB, LDAB, $ AFB, $ LDAFB, IPIV, R, C, B, LDB, X, LDX, RCOND, $ BERR, N_ERR_BNDS, ERR_BNDS_NORM, $ ERR_BNDS_COMP, NPARAMS, PARAMS, WORK, RWORK, $ INFO ) * * -- LAPACK computational 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 TRANS, EQUED INTEGER INFO, LDAB, LDAFB, LDB, LDX, N, KL, KU, NRHS, $ NPARAMS, N_ERR_BNDS DOUBLE PRECISION RCOND * .. * .. Array Arguments .. INTEGER IPIV( * ) COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ), $ X( LDX , * ),WORK( * ) DOUBLE PRECISION R( * ), C( * ), PARAMS( * ), BERR( * ), $ ERR_BNDS_NORM( NRHS, * ), $ ERR_BNDS_COMP( NRHS, * ), RWORK( * ) * .. * * ================================================================== * * .. Parameters .. DOUBLE PRECISION ZERO, ONE PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) DOUBLE PRECISION ITREF_DEFAULT, ITHRESH_DEFAULT DOUBLE PRECISION COMPONENTWISE_DEFAULT, RTHRESH_DEFAULT DOUBLE PRECISION DZTHRESH_DEFAULT PARAMETER ( ITREF_DEFAULT = 1.0D+0 ) PARAMETER ( ITHRESH_DEFAULT = 10.0D+0 ) PARAMETER ( COMPONENTWISE_DEFAULT = 1.0D+0 ) PARAMETER ( RTHRESH_DEFAULT = 0.5D+0 ) PARAMETER ( DZTHRESH_DEFAULT = 0.25D+0 ) 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 ) * .. * .. Local Scalars .. CHARACTER(1) NORM LOGICAL ROWEQU, COLEQU, NOTRAN, IGNORE_CWISE INTEGER J, TRANS_TYPE, PREC_TYPE, REF_TYPE, N_NORMS, $ ITHRESH DOUBLE PRECISION ANORM, RCOND_TMP, ILLRCOND_THRESH, ERR_LBND, $ CWISE_WRONG, RTHRESH, UNSTABLE_THRESH * .. * .. External Subroutines .. EXTERNAL XERBLA, ZGBCON, ZLA_GBRFSX_EXTENDED * .. * .. Intrinsic Functions .. INTRINSIC MAX, SQRT, TRANSFER * .. * .. External Functions .. EXTERNAL LSAME, ILAPREC EXTERNAL DLAMCH, ZLANGB, ZLA_GBRCOND_X, $ ZLA_GBRCOND_C DOUBLE PRECISION DLAMCH, ZLANGB, ZLA_GBRCOND_X, ZLA_GBRCOND_C LOGICAL LSAME INTEGER ILATRANS, ILAPREC * .. * .. Executable Statements .. * * Check the input parameters. * INFO = 0 TRANS_TYPE = ILATRANS( TRANS ) REF_TYPE = INT( ITREF_DEFAULT ) IF ( NPARAMS .GE. LA_LINRX_ITREF_I ) THEN IF ( PARAMS( LA_LINRX_ITREF_I ) .LT. 0.0D+0 ) THEN PARAMS( LA_LINRX_ITREF_I ) = ITREF_DEFAULT ELSE REF_TYPE = PARAMS( LA_LINRX_ITREF_I ) END IF END IF * * Set default parameters. * ILLRCOND_THRESH = DBLE( N ) * DLAMCH( 'Epsilon' ) ITHRESH = INT( ITHRESH_DEFAULT ) RTHRESH = RTHRESH_DEFAULT UNSTABLE_THRESH = DZTHRESH_DEFAULT IGNORE_CWISE = COMPONENTWISE_DEFAULT .EQ. 0.0D+0 * IF ( NPARAMS.GE.LA_LINRX_ITHRESH_I ) THEN IF ( PARAMS( LA_LINRX_ITHRESH_I ).LT.0.0D+0 ) THEN PARAMS( LA_LINRX_ITHRESH_I ) = ITHRESH ELSE ITHRESH = INT( PARAMS( LA_LINRX_ITHRESH_I ) ) END IF END IF IF ( NPARAMS.GE.LA_LINRX_CWISE_I ) THEN IF ( PARAMS( LA_LINRX_CWISE_I ).LT.0.0D+0 ) THEN IF ( IGNORE_CWISE ) THEN PARAMS( LA_LINRX_CWISE_I ) = 0.0D+0 ELSE PARAMS( LA_LINRX_CWISE_I ) = 1.0D+0 END IF ELSE IGNORE_CWISE = PARAMS( LA_LINRX_CWISE_I ) .EQ. 0.0D+0 END IF END IF IF ( REF_TYPE .EQ. 0 .OR. N_ERR_BNDS .EQ. 0 ) THEN N_NORMS = 0 ELSE IF ( IGNORE_CWISE ) THEN N_NORMS = 1 ELSE N_NORMS = 2 END IF * NOTRAN = LSAME( TRANS, 'N' ) ROWEQU = LSAME( EQUED, 'R' ) .OR. LSAME( EQUED, 'B' ) COLEQU = LSAME( EQUED, 'C' ) .OR. LSAME( EQUED, 'B' ) * * Test input parameters. * IF( TRANS_TYPE.EQ.-1 ) THEN INFO = -1 ELSE IF( .NOT.ROWEQU .AND. .NOT.COLEQU .AND. $ .NOT.LSAME( EQUED, 'N' ) ) THEN INFO = -2 ELSE IF( N.LT.0 ) THEN INFO = -3 ELSE IF( KL.LT.0 ) THEN INFO = -4 ELSE IF( KU.LT.0 ) THEN INFO = -5 ELSE IF( NRHS.LT.0 ) THEN INFO = -6 ELSE IF( LDAB.LT.KL+KU+1 ) THEN INFO = -8 ELSE IF( LDAFB.LT.2*KL+KU+1 ) THEN INFO = -10 ELSE IF( LDB.LT.MAX( 1, N ) ) THEN INFO = -13 ELSE IF( LDX.LT.MAX( 1, N ) ) THEN INFO = -15 END IF IF( INFO.NE.0 ) THEN CALL XERBLA( 'ZGBRFSX', -INFO ) RETURN END IF * * Quick return if possible. * IF( N.EQ.0 .OR. NRHS.EQ.0 ) THEN RCOND = 1.0D+0 DO J = 1, NRHS BERR( J ) = 0.0D+0 IF ( N_ERR_BNDS .GE. 1 ) THEN ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0 ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0 END IF IF ( N_ERR_BNDS .GE. 2 ) THEN ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 0.0D+0 ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 0.0D+0 END IF IF ( N_ERR_BNDS .GE. 3 ) THEN ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 1.0D+0 ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 1.0D+0 END IF END DO RETURN END IF * * Default to failure. * RCOND = 0.0D+0 DO J = 1, NRHS BERR( J ) = 1.0D+0 IF ( N_ERR_BNDS .GE. 1 ) THEN ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0 ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0 END IF IF ( N_ERR_BNDS .GE. 2 ) THEN ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0 ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0 END IF IF ( N_ERR_BNDS .GE. 3 ) THEN ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = 0.0D+0 ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = 0.0D+0 END IF END DO * * Compute the norm of A and the reciprocal of the condition * number of A. * IF( NOTRAN ) THEN NORM = 'I' ELSE NORM = '1' END IF ANORM = ZLANGB( NORM, N, KL, KU, AB, LDAB, RWORK ) CALL ZGBCON( NORM, N, KL, KU, AFB, LDAFB, IPIV, ANORM, RCOND, $ WORK, RWORK, INFO ) * * Perform refinement on each right-hand side * IF ( REF_TYPE .NE. 0 .AND. INFO .EQ. 0 ) THEN PREC_TYPE = ILAPREC( 'E' ) IF ( NOTRAN ) THEN CALL ZLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, $ KU, $ NRHS, AB, LDAB, AFB, LDAFB, IPIV, COLEQU, C, B, $ LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM, $ ERR_BNDS_COMP, WORK, RWORK, WORK(N+1), $ TRANSFER (RWORK(1:2*N), (/ (ZERO, ZERO) /), N), $ RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH, IGNORE_CWISE, $ INFO ) ELSE CALL ZLA_GBRFSX_EXTENDED( PREC_TYPE, TRANS_TYPE, N, KL, $ KU, $ NRHS, AB, LDAB, AFB, LDAFB, IPIV, ROWEQU, R, B, $ LDB, X, LDX, BERR, N_NORMS, ERR_BNDS_NORM, $ ERR_BNDS_COMP, WORK, RWORK, WORK(N+1), $ TRANSFER (RWORK(1:2*N), (/ (ZERO, ZERO) /), N), $ RCOND, ITHRESH, RTHRESH, UNSTABLE_THRESH, IGNORE_CWISE, $ INFO ) END IF END IF ERR_LBND = MAX( 10.0D+0, $ SQRT( DBLE( N ) ) ) * DLAMCH( 'Epsilon' ) IF (N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 1) THEN * * Compute scaled normwise condition number cond(A*C). * IF ( COLEQU .AND. NOTRAN ) THEN RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, $ AFB, $ LDAFB, IPIV, C, .TRUE., INFO, WORK, RWORK ) ELSE IF ( ROWEQU .AND. .NOT. NOTRAN ) THEN RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, $ AFB, $ LDAFB, IPIV, R, .TRUE., INFO, WORK, RWORK ) ELSE RCOND_TMP = ZLA_GBRCOND_C( TRANS, N, KL, KU, AB, LDAB, $ AFB, $ LDAFB, IPIV, C, .FALSE., INFO, WORK, RWORK ) END IF DO J = 1, NRHS * * Cap the error at 1.0. * IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I $ .AND. ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .GT. 1.0D+0) $ ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0 * * Threshold the error (see LAWN). * IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = 1.0D+0 ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 0.0D+0 IF ( INFO .LE. N ) INFO = N + J ELSE IF ( ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) .LT. ERR_LBND ) $ THEN ERR_BNDS_NORM( J, LA_LINRX_ERR_I ) = ERR_LBND ERR_BNDS_NORM( J, LA_LINRX_TRUST_I ) = 1.0D+0 END IF * * Save the condition number. * IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN ERR_BNDS_NORM( J, LA_LINRX_RCOND_I ) = RCOND_TMP END IF END DO END IF IF (N_ERR_BNDS .GE. 1 .AND. N_NORMS .GE. 2) THEN * * Compute componentwise condition number cond(A*diag(Y(:,J))) for * each right-hand side using the current solution as an estimate of * the true solution. If the componentwise error estimate is too * large, then the solution is a lousy estimate of truth and the * estimated RCOND may be too optimistic. To avoid misleading users, * the inverse condition number is set to 0.0 when the estimated * cwise error is at least CWISE_WRONG. * CWISE_WRONG = SQRT( DLAMCH( 'Epsilon' ) ) DO J = 1, NRHS IF (ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .LT. CWISE_WRONG ) $ THEN RCOND_TMP = ZLA_GBRCOND_X( TRANS, N, KL, KU, AB, LDAB, $ AFB, LDAFB, IPIV, X( 1, J ), INFO, WORK, RWORK ) ELSE RCOND_TMP = 0.0D+0 END IF * * Cap the error at 1.0. * IF ( N_ERR_BNDS .GE. LA_LINRX_ERR_I $ .AND. ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) .GT. 1.0D+0 ) $ ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0 * * Threshold the error (see LAWN). * IF ( RCOND_TMP .LT. ILLRCOND_THRESH ) THEN ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = 1.0D+0 ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 0.0D+0 IF ( PARAMS( LA_LINRX_CWISE_I ) .EQ. 1.0D+0 $ .AND. INFO.LT.N + J ) INFO = N + J ELSE IF ( ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) $ .LT. ERR_LBND ) THEN ERR_BNDS_COMP( J, LA_LINRX_ERR_I ) = ERR_LBND ERR_BNDS_COMP( J, LA_LINRX_TRUST_I ) = 1.0D+0 END IF * * Save the condition number. * IF ( N_ERR_BNDS .GE. LA_LINRX_RCOND_I ) THEN ERR_BNDS_COMP( J, LA_LINRX_RCOND_I ) = RCOND_TMP END IF END DO END IF * RETURN * * End of ZGBRFSX * END