LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ 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 = 'B': Both row and column equilibration, i.e., A has been replaced by diag(R) * A * diag(C).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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