 LAPACK 3.11.0 LAPACK: Linear Algebra PACKage
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## ◆ ssyrfsx()

 subroutine ssyrfsx ( character UPLO, character EQUED, integer N, integer NRHS, real, dimension( lda, * ) A, integer LDA, real, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, real, dimension( * ) S, 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 )

SSYRFSX

Purpose:
```    SSYRFSX improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric indefinite, 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 and S
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] UPLO ``` UPLO is CHARACTER*1 = 'U': Upper triangle of A is stored; = 'L': Lower triangle of A is stored.``` [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 = 'N': No equilibration = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S). 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 symmetric matrix A. 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.``` [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 factored form of the matrix A. AF contains the block diagonal matrix D and the multipliers used to obtain the factor U or L from the factorization A = U*D*U**T or A = L*D*L**T as computed by SSYTRF.``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [in] IPIV ``` IPIV is INTEGER array, dimension (N) Details of the interchanges and the block structure of D as determined by SSYTRF.``` [in,out] S ``` S is REAL array, dimension (N) The 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.``` [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 398 of file ssyrfsx.f.

402*
403* -- LAPACK computational routine --
404* -- LAPACK is a software package provided by Univ. of Tennessee, --
405* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
406*
407* .. Scalar Arguments ..
408 CHARACTER UPLO, EQUED
409 INTEGER INFO, LDA, LDAF, LDB, LDX, N, NRHS, NPARAMS,
410 \$ N_ERR_BNDS
411 REAL RCOND
412* ..
413* .. Array Arguments ..
414 INTEGER IPIV( * ), IWORK( * )
415 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
416 \$ X( LDX, * ), WORK( * )
417 REAL S( * ), PARAMS( * ), BERR( * ),
418 \$ ERR_BNDS_NORM( NRHS, * ),
419 \$ ERR_BNDS_COMP( NRHS, * )
420* ..
421*
422* ==================================================================
423*
424* .. Parameters ..
425 REAL ZERO, ONE
426 parameter( zero = 0.0e+0, one = 1.0e+0 )
427 REAL ITREF_DEFAULT, ITHRESH_DEFAULT,
428 \$ COMPONENTWISE_DEFAULT
429 REAL RTHRESH_DEFAULT, DZTHRESH_DEFAULT
430 parameter( itref_default = 1.0 )
431 parameter( ithresh_default = 10.0 )
432 parameter( componentwise_default = 1.0 )
433 parameter( rthresh_default = 0.5 )
434 parameter( dzthresh_default = 0.25 )
435 INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
436 \$ LA_LINRX_CWISE_I
437 parameter( la_linrx_itref_i = 1,
438 \$ la_linrx_ithresh_i = 2 )
439 parameter( la_linrx_cwise_i = 3 )
440 INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
441 \$ LA_LINRX_RCOND_I
442 parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
443 parameter( la_linrx_rcond_i = 3 )
444* ..
445* .. Local Scalars ..
446 CHARACTER(1) NORM
447 LOGICAL RCEQU
448 INTEGER J, PREC_TYPE, REF_TYPE, N_NORMS
449 REAL ANORM, RCOND_TMP
450 REAL ILLRCOND_THRESH, ERR_LBND, CWISE_WRONG
451 LOGICAL IGNORE_CWISE
452 INTEGER ITHRESH
453 REAL RTHRESH, UNSTABLE_THRESH
454* ..
455* .. External Subroutines ..
457* ..
458* .. Intrinsic Functions ..
459 INTRINSIC max, sqrt
460* ..
461* .. External Functions ..
462 EXTERNAL lsame, ilaprec
463 EXTERNAL slamch, slansy, sla_syrcond
464 REAL SLAMCH, SLANSY, SLA_SYRCOND
465 LOGICAL LSAME
466 INTEGER ILAPREC
467* ..
468* .. Executable Statements ..
469*
470* Check the input parameters.
471*
472 info = 0
473 ref_type = int( itref_default )
474 IF ( nparams .GE. la_linrx_itref_i ) THEN
475 IF ( params( la_linrx_itref_i ) .LT. 0.0 ) THEN
476 params( la_linrx_itref_i ) = itref_default
477 ELSE
478 ref_type = params( la_linrx_itref_i )
479 END IF
480 END IF
481*
482* Set default parameters.
483*
484 illrcond_thresh = real( n )*slamch( 'Epsilon' )
485 ithresh = int( ithresh_default )
486 rthresh = rthresh_default
487 unstable_thresh = dzthresh_default
488 ignore_cwise = componentwise_default .EQ. 0.0
489*
490 IF ( nparams.GE.la_linrx_ithresh_i ) THEN
491 IF ( params( la_linrx_ithresh_i ).LT.0.0 ) THEN
492 params( la_linrx_ithresh_i ) = ithresh
493 ELSE
494 ithresh = int( params( la_linrx_ithresh_i ) )
495 END IF
496 END IF
497 IF ( nparams.GE.la_linrx_cwise_i ) THEN
498 IF ( params( la_linrx_cwise_i ).LT.0.0 ) THEN
499 IF ( ignore_cwise ) THEN
500 params( la_linrx_cwise_i ) = 0.0
501 ELSE
502 params( la_linrx_cwise_i ) = 1.0
503 END IF
504 ELSE
505 ignore_cwise = params( la_linrx_cwise_i ) .EQ. 0.0
506 END IF
507 END IF
508 IF ( ref_type .EQ. 0 .OR. n_err_bnds .EQ. 0 ) THEN
509 n_norms = 0
510 ELSE IF ( ignore_cwise ) THEN
511 n_norms = 1
512 ELSE
513 n_norms = 2
514 END IF
515*
516 rcequ = lsame( equed, 'Y' )
517*
518* Test input parameters.
519*
520 IF ( .NOT.lsame( uplo, 'U' ) .AND. .NOT.lsame( uplo, 'L' ) ) THEN
521 info = -1
522 ELSE IF( .NOT.rcequ .AND. .NOT.lsame( equed, 'N' ) ) THEN
523 info = -2
524 ELSE IF( n.LT.0 ) THEN
525 info = -3
526 ELSE IF( nrhs.LT.0 ) THEN
527 info = -4
528 ELSE IF( lda.LT.max( 1, n ) ) THEN
529 info = -6
530 ELSE IF( ldaf.LT.max( 1, n ) ) THEN
531 info = -8
532 ELSE IF( ldb.LT.max( 1, n ) ) THEN
533 info = -12
534 ELSE IF( ldx.LT.max( 1, n ) ) THEN
535 info = -14
536 END IF
537 IF( info.NE.0 ) THEN
538 CALL xerbla( 'SSYRFSX', -info )
539 RETURN
540 END IF
541*
542* Quick return if possible.
543*
544 IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
545 rcond = 1.0
546 DO j = 1, nrhs
547 berr( j ) = 0.0
548 IF ( n_err_bnds .GE. 1 ) THEN
549 err_bnds_norm( j, la_linrx_trust_i ) = 1.0
550 err_bnds_comp( j, la_linrx_trust_i ) = 1.0
551 END IF
552 IF ( n_err_bnds .GE. 2 ) THEN
553 err_bnds_norm( j, la_linrx_err_i ) = 0.0
554 err_bnds_comp( j, la_linrx_err_i ) = 0.0
555 END IF
556 IF ( n_err_bnds .GE. 3 ) THEN
557 err_bnds_norm( j, la_linrx_rcond_i ) = 1.0
558 err_bnds_comp( j, la_linrx_rcond_i ) = 1.0
559 END IF
560 END DO
561 RETURN
562 END IF
563*
564* Default to failure.
565*
566 rcond = 0.0
567 DO j = 1, nrhs
568 berr( j ) = 1.0
569 IF ( n_err_bnds .GE. 1 ) THEN
570 err_bnds_norm( j, la_linrx_trust_i ) = 1.0
571 err_bnds_comp( j, la_linrx_trust_i ) = 1.0
572 END IF
573 IF ( n_err_bnds .GE. 2 ) THEN
574 err_bnds_norm( j, la_linrx_err_i ) = 1.0
575 err_bnds_comp( j, la_linrx_err_i ) = 1.0
576 END IF
577 IF ( n_err_bnds .GE. 3 ) THEN
578 err_bnds_norm( j, la_linrx_rcond_i ) = 0.0
579 err_bnds_comp( j, la_linrx_rcond_i ) = 0.0
580 END IF
581 END DO
582*
583* Compute the norm of A and the reciprocal of the condition
584* number of A.
585*
586 norm = 'I'
587 anorm = slansy( norm, uplo, n, a, lda, work )
588 CALL ssycon( uplo, n, af, ldaf, ipiv, anorm, rcond, work,
589 \$ iwork, info )
590*
591* Perform refinement on each right-hand side
592*
593 IF ( ref_type .NE. 0 ) THEN
594
595 prec_type = ilaprec( 'D' )
596
597 CALL sla_syrfsx_extended( prec_type, uplo, n,
598 \$ nrhs, a, lda, af, ldaf, ipiv, rcequ, s, b,
599 \$ ldb, x, ldx, berr, n_norms, err_bnds_norm, err_bnds_comp,
600 \$ work( n+1 ), work( 1 ), work( 2*n+1 ), work( 1 ), rcond,
601 \$ ithresh, rthresh, unstable_thresh, ignore_cwise,
602 \$ info )
603 END IF
604
605 err_lbnd = max( 10.0, sqrt( real( n ) ) )*slamch( 'Epsilon' )
606 IF (n_err_bnds .GE. 1 .AND. n_norms .GE. 1) THEN
607*
608* Compute scaled normwise condition number cond(A*C).
609*
610 IF ( rcequ ) THEN
611 rcond_tmp = sla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
612 \$ -1, s, info, work, iwork )
613 ELSE
614 rcond_tmp = sla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
615 \$ 0, s, info, work, iwork )
616 END IF
617 DO j = 1, nrhs
618*
619* Cap the error at 1.0.
620*
621 IF (n_err_bnds .GE. la_linrx_err_i
622 \$ .AND. err_bnds_norm( j, la_linrx_err_i ) .GT. 1.0)
623 \$ err_bnds_norm( j, la_linrx_err_i ) = 1.0
624*
625* Threshold the error (see LAWN).
626*
627 IF ( rcond_tmp .LT. illrcond_thresh ) THEN
628 err_bnds_norm( j, la_linrx_err_i ) = 1.0
629 err_bnds_norm( j, la_linrx_trust_i ) = 0.0
630 IF ( info .LE. n ) info = n + j
631 ELSE IF (err_bnds_norm( j, la_linrx_err_i ) .LT. err_lbnd)
632 \$ THEN
633 err_bnds_norm( j, la_linrx_err_i ) = err_lbnd
634 err_bnds_norm( j, la_linrx_trust_i ) = 1.0
635 END IF
636*
637* Save the condition number.
638*
639 IF (n_err_bnds .GE. la_linrx_rcond_i) THEN
640 err_bnds_norm( j, la_linrx_rcond_i ) = rcond_tmp
641 END IF
642 END DO
643 END IF
644
645 IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 2 ) THEN
646*
647* Compute componentwise condition number cond(A*diag(Y(:,J))) for
648* each right-hand side using the current solution as an estimate of
649* the true solution. If the componentwise error estimate is too
650* large, then the solution is a lousy estimate of truth and the
651* estimated RCOND may be too optimistic. To avoid misleading users,
652* the inverse condition number is set to 0.0 when the estimated
653* cwise error is at least CWISE_WRONG.
654*
655 cwise_wrong = sqrt( slamch( 'Epsilon' ) )
656 DO j = 1, nrhs
657 IF ( err_bnds_comp( j, la_linrx_err_i ) .LT. cwise_wrong )
658 \$ THEN
659 rcond_tmp = sla_syrcond( uplo, n, a, lda, af, ldaf, ipiv,
660 \$ 1, x(1,j), info, work, iwork )
661 ELSE
662 rcond_tmp = 0.0
663 END IF
664*
665* Cap the error at 1.0.
666*
667 IF ( n_err_bnds .GE. la_linrx_err_i
668 \$ .AND. err_bnds_comp( j, la_linrx_err_i ) .GT. 1.0 )
669 \$ err_bnds_comp( j, la_linrx_err_i ) = 1.0
670*
671* Threshold the error (see LAWN).
672*
673 IF ( rcond_tmp .LT. illrcond_thresh ) THEN
674 err_bnds_comp( j, la_linrx_err_i ) = 1.0
675 err_bnds_comp( j, la_linrx_trust_i ) = 0.0
676 IF ( .NOT. ignore_cwise
677 \$ .AND. info.LT.n + j ) info = n + j
678 ELSE IF ( err_bnds_comp( j, la_linrx_err_i )
679 \$ .LT. err_lbnd ) THEN
680 err_bnds_comp( j, la_linrx_err_i ) = err_lbnd
681 err_bnds_comp( j, la_linrx_trust_i ) = 1.0
682 END IF
683*
684* Save the condition number.
685*
686 IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
687 err_bnds_comp( j, la_linrx_rcond_i ) = rcond_tmp
688 END IF
689
690 END DO
691 END IF
692*
693 RETURN
694*
695* End of SSYRFSX
696*
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
real function slansy(NORM, UPLO, N, A, LDA, WORK)
SLANSY returns the value of the 1-norm, or the Frobenius norm, or the infinity norm,...
Definition: slansy.f:122
real function sla_syrcond(UPLO, N, A, LDA, AF, LDAF, IPIV, CMODE, C, INFO, WORK, IWORK)
SLA_SYRCOND estimates the Skeel condition number for a symmetric indefinite matrix.
Definition: sla_syrcond.f:146
subroutine sla_syrfsx_extended(PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, IPIV, COLEQU, C, B, LDB, Y, LDY, BERR_OUT, N_NORMS, ERR_BNDS_NORM, ERR_BNDS_COMP, RES, AYB, DY, Y_TAIL, RCOND, ITHRESH, RTHRESH, DZ_UB, IGNORE_CWISE, INFO)
SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric inde...
subroutine ssycon(UPLO, N, A, LDA, IPIV, ANORM, RCOND, WORK, IWORK, INFO)
SSYCON
Definition: ssycon.f:130
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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