LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ sgerfsx()

 subroutine sgerfsx ( character TRANS, character EQUED, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) R, real, dimension( * ) C, real, dimension( ldb, * ) B, integer LDB, real, dimension( ldx , * ) X, integer LDX, real RCOND, real, dimension( * ) BERR, integer N_ERR_BNDS, real, dimension( nrhs, * ) ERR_BNDS_NORM, real, dimension( nrhs, * ) ERR_BNDS_COMP, integer NPARAMS, real, dimension( * ) PARAMS, real, dimension( * ) WORK, integer, dimension( * ) IWORK, integer INFO )

SGERFSX

Purpose:
```    SGERFSX 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.```
```     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.```
Parameters
 [in] TRANS ``` 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 = Transpose)``` [in] EQUED ``` 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 = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).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.``` [in] N ``` N is INTEGER The order of the matrix A. N >= 0.``` [in] NRHS ``` NRHS is INTEGER The number of right hand sides, i.e., the number of columns of the matrices B and X. NRHS >= 0.``` [in] A ``` A is REAL array, dimension (LDA,N) The original N-by-N matrix A.``` [in] LDA ``` LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).``` [in] AF ``` AF is REAL array, dimension (LDAF,N) The factors L and U from the factorization A = P*L*U as computed by SGETRF.``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) The pivot indices from SGETRF; for 1<=i<=N, row i of the matrix was interchanged with row IPIV(i).``` [in] R ``` R is REAL 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. If R is accessed, 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.``` [in] C ``` C is REAL 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. If C is accessed, 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.``` [in] B ``` B is REAL array, dimension (LDB,NRHS) The right hand side matrix B.``` [in] LDB ``` LDB is INTEGER The leading dimension of the array B. LDB >= max(1,N).``` [in,out] X ``` X is REAL array, dimension (LDX,NRHS) On entry, the solution matrix X, as computed by SGETRS. On exit, the improved solution matrix X.``` [in] LDX ``` LDX is INTEGER The leading dimension of the array X. LDX >= max(1,N).``` [out] RCOND ``` 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.``` [out] BERR ``` 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).``` [in] N_ERR_BNDS ``` 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.``` [out] ERR_BNDS_NORM ``` 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.``` [out] ERR_BNDS_COMP ``` 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.``` [in] NPARAMS ``` NPARAMS is INTEGER Specifies the number of parameters set in PARAMS. If <= 0, the PARAMS array is never referenced and default values are used.``` [in,out] PARAMS ``` 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)``` [out] WORK ` WORK is REAL array, dimension (4*N)` [out] IWORK ` IWORK is INTEGER array, dimension (N)` [out] INFO ``` 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.```

Definition at line 410 of file sgerfsx.f.

414 *
415 * -- LAPACK computational routine --
416 * -- LAPACK is a software package provided by Univ. of Tennessee, --
417 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
418 *
419 * .. Scalar Arguments ..
420  CHARACTER TRANS, EQUED
421  INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
422  \$ N_ERR_BNDS
423  REAL RCOND
424 * ..
425 * .. Array Arguments ..
426  INTEGER IPIV( * ), IWORK( * )
427  REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
428  \$ X( LDX , * ), WORK( * )
429  REAL R( * ), C( * ), PARAMS( * ), BERR( * ),
430  \$ ERR_BNDS_NORM( NRHS, * ),
431  \$ ERR_BNDS_COMP( NRHS, * )
432 * ..
433 *
434 * ==================================================================
435 *
436 * .. Parameters ..
437  REAL ZERO, ONE
438  parameter( zero = 0.0e+0, one = 1.0e+0 )
439  REAL ITREF_DEFAULT, ITHRESH_DEFAULT,
440  \$ COMPONENTWISE_DEFAULT
441  REAL RTHRESH_DEFAULT, DZTHRESH_DEFAULT
442  parameter( itref_default = 1.0 )
443  parameter( ithresh_default = 10.0 )
444  parameter( componentwise_default = 1.0 )
445  parameter( rthresh_default = 0.5 )
446  parameter( dzthresh_default = 0.25 )
447  INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
448  \$ LA_LINRX_CWISE_I
449  parameter( la_linrx_itref_i = 1,
450  \$ la_linrx_ithresh_i = 2 )
451  parameter( la_linrx_cwise_i = 3 )
452  INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
453  \$ LA_LINRX_RCOND_I
454  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
455  parameter( la_linrx_rcond_i = 3 )
456 * ..
457 * .. Local Scalars ..
458  CHARACTER(1) NORM
459  LOGICAL ROWEQU, COLEQU, NOTRAN
460  INTEGER J, TRANS_TYPE, PREC_TYPE, REF_TYPE
461  INTEGER N_NORMS
462  REAL ANORM, RCOND_TMP
463  REAL ILLRCOND_THRESH, ERR_LBND, CWISE_WRONG
464  LOGICAL IGNORE_CWISE
465  INTEGER ITHRESH
466  REAL RTHRESH, UNSTABLE_THRESH
467 * ..
468 * .. External Subroutines ..
470 * ..
471 * .. Intrinsic Functions ..
472  INTRINSIC max, sqrt
473 * ..
474 * .. External Functions ..
475  EXTERNAL lsame, ilatrans, ilaprec
476  EXTERNAL slamch, slange, sla_gercond
477  REAL SLAMCH, SLANGE, SLA_GERCOND
478  LOGICAL LSAME
479  INTEGER ILATRANS, ILAPREC
480 * ..
481 * .. Executable Statements ..
482 *
483 * Check the input parameters.
484 *
485  info = 0
486  trans_type = ilatrans( trans )
487  ref_type = int( itref_default )
488  IF ( nparams .GE. la_linrx_itref_i ) THEN
489  IF ( params( la_linrx_itref_i ) .LT. 0.0 ) THEN
490  params( la_linrx_itref_i ) = itref_default
491  ELSE
492  ref_type = params( la_linrx_itref_i )
493  END IF
494  END IF
495 *
496 * Set default parameters.
497 *
498  illrcond_thresh = real( n ) * slamch( 'Epsilon' )
499  ithresh = int( ithresh_default )
500  rthresh = rthresh_default
501  unstable_thresh = dzthresh_default
502  ignore_cwise = componentwise_default .EQ. 0.0
503 *
504  IF ( nparams.GE.la_linrx_ithresh_i ) THEN
505  IF ( params( la_linrx_ithresh_i ).LT.0.0 ) THEN
506  params( la_linrx_ithresh_i ) = ithresh
507  ELSE
508  ithresh = int( params( la_linrx_ithresh_i ) )
509  END IF
510  END IF
511  IF ( nparams.GE.la_linrx_cwise_i ) THEN
512  IF ( params( la_linrx_cwise_i ).LT.0.0 ) THEN
513  IF ( ignore_cwise ) THEN
514  params( la_linrx_cwise_i ) = 0.0
515  ELSE
516  params( la_linrx_cwise_i ) = 1.0
517  END IF
518  ELSE
519  ignore_cwise = params( la_linrx_cwise_i ) .EQ. 0.0
520  END IF
521  END IF
522  IF ( ref_type .EQ. 0 .OR. n_err_bnds .EQ. 0 ) THEN
523  n_norms = 0
524  ELSE IF ( ignore_cwise ) THEN
525  n_norms = 1
526  ELSE
527  n_norms = 2
528  END IF
529 *
530  notran = lsame( trans, 'N' )
531  rowequ = lsame( equed, 'R' ) .OR. lsame( equed, 'B' )
532  colequ = lsame( equed, 'C' ) .OR. lsame( equed, 'B' )
533 *
534 * Test input parameters.
535 *
536  IF( trans_type.EQ.-1 ) THEN
537  info = -1
538  ELSE IF( .NOT.rowequ .AND. .NOT.colequ .AND.
539  \$ .NOT.lsame( equed, 'N' ) ) THEN
540  info = -2
541  ELSE IF( n.LT.0 ) THEN
542  info = -3
543  ELSE IF( nrhs.LT.0 ) THEN
544  info = -4
545  ELSE IF( lda.LT.max( 1, n ) ) THEN
546  info = -6
547  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
548  info = -8
549  ELSE IF( ldb.LT.max( 1, n ) ) THEN
550  info = -13
551  ELSE IF( ldx.LT.max( 1, n ) ) THEN
552  info = -15
553  END IF
554  IF( info.NE.0 ) THEN
555  CALL xerbla( 'SGERFSX', -info )
556  RETURN
557  END IF
558 *
559 * Quick return if possible.
560 *
561  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
562  rcond = 1.0
563  DO j = 1, nrhs
564  berr( j ) = 0.0
565  IF ( n_err_bnds .GE. 1 ) THEN
566  err_bnds_norm( j, la_linrx_trust_i) = 1.0
567  err_bnds_comp( j, la_linrx_trust_i ) = 1.0
568  END IF
569  IF ( n_err_bnds .GE. 2 ) THEN
570  err_bnds_norm( j, la_linrx_err_i) = 0.0
571  err_bnds_comp( j, la_linrx_err_i ) = 0.0
572  END IF
573  IF ( n_err_bnds .GE. 3 ) THEN
574  err_bnds_norm( j, la_linrx_rcond_i) = 1.0
575  err_bnds_comp( j, la_linrx_rcond_i ) = 1.0
576  END IF
577  END DO
578  RETURN
579  END IF
580 *
581 * Default to failure.
582 *
583  rcond = 0.0
584  DO j = 1, nrhs
585  berr( j ) = 1.0
586  IF ( n_err_bnds .GE. 1 ) THEN
587  err_bnds_norm( j, la_linrx_trust_i ) = 1.0
588  err_bnds_comp( j, la_linrx_trust_i ) = 1.0
589  END IF
590  IF ( n_err_bnds .GE. 2 ) THEN
591  err_bnds_norm( j, la_linrx_err_i ) = 1.0
592  err_bnds_comp( j, la_linrx_err_i ) = 1.0
593  END IF
594  IF ( n_err_bnds .GE. 3 ) THEN
595  err_bnds_norm( j, la_linrx_rcond_i ) = 0.0
596  err_bnds_comp( j, la_linrx_rcond_i ) = 0.0
597  END IF
598  END DO
599 *
600 * Compute the norm of A and the reciprocal of the condition
601 * number of A.
602 *
603  IF( notran ) THEN
604  norm = 'I'
605  ELSE
606  norm = '1'
607  END IF
608  anorm = slange( norm, n, n, a, lda, work )
609  CALL sgecon( norm, n, af, ldaf, anorm, rcond, work, iwork, info )
610 *
611 * Perform refinement on each right-hand side
612 *
613  IF ( ref_type .NE. 0 ) THEN
614
615  prec_type = ilaprec( 'D' )
616
617  IF ( notran ) THEN
618  CALL sla_gerfsx_extended( prec_type, trans_type, n,
619  \$ nrhs, a, lda, af, ldaf, ipiv, colequ, c, b,
620  \$ ldb, x, ldx, berr, n_norms, err_bnds_norm,
621  \$ err_bnds_comp, work(n+1), work(1), work(2*n+1),
622  \$ work(1), rcond, ithresh, rthresh, unstable_thresh,
623  \$ ignore_cwise, info )
624  ELSE
625  CALL sla_gerfsx_extended( prec_type, trans_type, n,
626  \$ nrhs, a, lda, af, ldaf, ipiv, rowequ, r, b,
627  \$ ldb, x, ldx, berr, n_norms, err_bnds_norm,
628  \$ err_bnds_comp, work(n+1), work(1), work(2*n+1),
629  \$ work(1), rcond, ithresh, rthresh, unstable_thresh,
630  \$ ignore_cwise, info )
631  END IF
632  END IF
633
634  err_lbnd = max( 10.0, sqrt( real( n ) ) ) * slamch( 'Epsilon' )
635  IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 1 ) THEN
636 *
637 * Compute scaled normwise condition number cond(A*C).
638 *
639  IF ( colequ .AND. notran ) THEN
640  rcond_tmp = sla_gercond( trans, n, a, lda, af, ldaf, ipiv,
641  \$ -1, c, info, work, iwork )
642  ELSE IF ( rowequ .AND. .NOT. notran ) THEN
643  rcond_tmp = sla_gercond( trans, n, a, lda, af, ldaf, ipiv,
644  \$ -1, r, info, work, iwork )
645  ELSE
646  rcond_tmp = sla_gercond( trans, n, a, lda, af, ldaf, ipiv,
647  \$ 0, r, info, work, iwork )
648  END IF
649  DO j = 1, nrhs
650 *
651 * Cap the error at 1.0.
652 *
653  IF ( n_err_bnds .GE. la_linrx_err_i
654  \$ .AND. err_bnds_norm( j, la_linrx_err_i ) .GT. 1.0 )
655  \$ err_bnds_norm( j, la_linrx_err_i ) = 1.0
656 *
657 * Threshold the error (see LAWN).
658 *
659  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
660  err_bnds_norm( j, la_linrx_err_i ) = 1.0
661  err_bnds_norm( j, la_linrx_trust_i ) = 0.0
662  IF ( info .LE. n ) info = n + j
663  ELSE IF ( err_bnds_norm( j, la_linrx_err_i ) .LT. err_lbnd )
664  \$ THEN
665  err_bnds_norm( j, la_linrx_err_i ) = err_lbnd
666  err_bnds_norm( j, la_linrx_trust_i ) = 1.0
667  END IF
668 *
669 * Save the condition number.
670 *
671  IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
672  err_bnds_norm( j, la_linrx_rcond_i ) = rcond_tmp
673  END IF
674  END DO
675  END IF
676
677  IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 2 ) THEN
678 *
679 * Compute componentwise condition number cond(A*diag(Y(:,J))) for
680 * each right-hand side using the current solution as an estimate of
681 * the true solution. If the componentwise error estimate is too
682 * large, then the solution is a lousy estimate of truth and the
683 * estimated RCOND may be too optimistic. To avoid misleading users,
684 * the inverse condition number is set to 0.0 when the estimated
685 * cwise error is at least CWISE_WRONG.
686 *
687  cwise_wrong = sqrt( slamch( 'Epsilon' ) )
688  DO j = 1, nrhs
689  IF ( err_bnds_comp( j, la_linrx_err_i ) .LT. cwise_wrong )
690  \$ THEN
691  rcond_tmp = sla_gercond( trans, n, a, lda, af, ldaf,
692  \$ ipiv, 1, x(1,j), info, work, iwork )
693  ELSE
694  rcond_tmp = 0.0
695  END IF
696 *
697 * Cap the error at 1.0.
698 *
699  IF ( n_err_bnds .GE. la_linrx_err_i
700  \$ .AND. err_bnds_comp( j, la_linrx_err_i ) .GT. 1.0 )
701  \$ err_bnds_comp( j, la_linrx_err_i ) = 1.0
702 *
703 * Threshold the error (see LAWN).
704 *
705  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
706  err_bnds_comp( j, la_linrx_err_i ) = 1.0
707  err_bnds_comp( j, la_linrx_trust_i ) = 0.0
708  IF ( params( la_linrx_cwise_i ) .EQ. 1.0
709  \$ .AND. info.LT.n + j ) info = n + j
710  ELSE IF ( err_bnds_comp( j, la_linrx_err_i )
711  \$ .LT. err_lbnd ) THEN
712  err_bnds_comp( j, la_linrx_err_i ) = err_lbnd
713  err_bnds_comp( j, la_linrx_trust_i ) = 1.0
714  END IF
715 *
716 * Save the condition number.
717 *
718  IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
719  err_bnds_comp( j, la_linrx_rcond_i ) = rcond_tmp
720  END IF
721  END DO
722  END IF
723 *
724  RETURN
725 *
726 * End of SGERFSX
727 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
integer function ilaprec(PREC)
ILAPREC
Definition: ilaprec.f:58
integer function ilatrans(TRANS)
ILATRANS
Definition: ilatrans.f:58
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:114
real function sla_gercond(TRANS, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)
SLA_GERCOND estimates the Skeel condition number for a general matrix.
Definition: sla_gercond.f:150
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)
SLA_GERFSX_EXTENDED improves the computed solution to a system of linear equations for general matric...
subroutine sgecon(NORM, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SGECON
Definition: sgecon.f:124
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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