LAPACK  3.8.0 LAPACK: Linear Algebra PACKage

## ◆ sporfsx()

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

SPORFSX

Purpose:
```    SPORFSX improves the computed solution to a system of linear
equations when the coefficient matrix is symmetric positive
definite, 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 = 'N': No equilibration = 'Y': Both row and column equilibration, i.e., A has been replaced by diag(S) * A * diag(S).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 triangular factor U or L from the Cholesky factorization A = U**T*U or A = L*L**T, as computed by SPOTRF.``` [in] LDAF ``` LDAF is INTEGER The leading dimension of the array AF. LDAF >= max(1,N).``` [in,out] S ``` S is REAL array, dimension (N) The row 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 .LT. 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 .LE. 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 .LT. 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.```
Date
April 2012

Definition at line 396 of file sporfsx.f.

396 *
397 * -- LAPACK computational routine (version 3.7.0) --
398 * -- LAPACK is a software package provided by Univ. of Tennessee, --
399 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
400 * April 2012
401 *
402 * .. Scalar Arguments ..
403  CHARACTER uplo, equed
404  INTEGER info, lda, ldaf, ldb, ldx, n, nrhs, nparams,
405  \$ n_err_bnds
406  REAL rcond
407 * ..
408 * .. Array Arguments ..
409  INTEGER iwork( * )
410  REAL a( lda, * ), af( ldaf, * ), b( ldb, * ),
411  \$ x( ldx, * ), work( * )
412  REAL s( * ), params( * ), berr( * ),
413  \$ err_bnds_norm( nrhs, * ),
414  \$ err_bnds_comp( nrhs, * )
415 * ..
416 *
417 * ==================================================================
418 *
419 * .. Parameters ..
420  REAL zero, one
421  parameter( zero = 0.0e+0, one = 1.0e+0 )
422  REAL itref_default, ithresh_default,
423  \$ componentwise_default
424  REAL rthresh_default, dzthresh_default
425  parameter( itref_default = 1.0 )
426  parameter( ithresh_default = 10.0 )
427  parameter( componentwise_default = 1.0 )
428  parameter( rthresh_default = 0.5 )
429  parameter( dzthresh_default = 0.25 )
430  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
431  \$ la_linrx_cwise_i
432  parameter( la_linrx_itref_i = 1,
433  \$ la_linrx_ithresh_i = 2 )
434  parameter( la_linrx_cwise_i = 3 )
435  INTEGER la_linrx_trust_i, la_linrx_err_i,
436  \$ la_linrx_rcond_i
437  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
438  parameter( la_linrx_rcond_i = 3 )
439 * ..
440 * .. Local Scalars ..
441  CHARACTER(1) norm
442  LOGICAL rcequ
443  INTEGER j, prec_type, ref_type
444  INTEGER n_norms
445  REAL anorm, rcond_tmp
446  REAL illrcond_thresh, err_lbnd, cwise_wrong
447  LOGICAL ignore_cwise
448  INTEGER ithresh
449  REAL rthresh, unstable_thresh
450 * ..
451 * .. External Subroutines ..
453 * ..
454 * .. Intrinsic Functions ..
455  INTRINSIC max, sqrt
456 * ..
457 * .. External Functions ..
458  EXTERNAL lsame, ilaprec
459  EXTERNAL slamch, slansy, sla_porcond
460  REAL slamch, slansy, sla_porcond
461  LOGICAL lsame
462  INTEGER ilaprec
463 * ..
464 * .. Executable Statements ..
465 *
466 * Check the input parameters.
467 *
468  info = 0
469  ref_type = int( itref_default )
470  IF ( nparams .GE. la_linrx_itref_i ) THEN
471  IF ( params( la_linrx_itref_i ) .LT. 0.0 ) THEN
472  params( la_linrx_itref_i ) = itref_default
473  ELSE
474  ref_type = params( la_linrx_itref_i )
475  END IF
476  END IF
477 *
478 * Set default parameters.
479 *
480  illrcond_thresh = REAL( N ) * slamch( 'Epsilon' )
481  ithresh = int( ithresh_default )
482  rthresh = rthresh_default
483  unstable_thresh = dzthresh_default
484  ignore_cwise = componentwise_default .EQ. 0.0
485 *
486  IF ( nparams.GE.la_linrx_ithresh_i ) THEN
487  IF ( params( la_linrx_ithresh_i ).LT.0.0 ) THEN
488  params( la_linrx_ithresh_i ) = ithresh
489  ELSE
490  ithresh = int( params( la_linrx_ithresh_i ) )
491  END IF
492  END IF
493  IF ( nparams.GE.la_linrx_cwise_i ) THEN
494  IF ( params( la_linrx_cwise_i ).LT.0.0 ) THEN
495  IF ( ignore_cwise ) THEN
496  params( la_linrx_cwise_i ) = 0.0
497  ELSE
498  params( la_linrx_cwise_i ) = 1.0
499  END IF
500  ELSE
501  ignore_cwise = params( la_linrx_cwise_i ) .EQ. 0.0
502  END IF
503  END IF
504  IF ( ref_type .EQ. 0 .OR. n_err_bnds .EQ. 0 ) THEN
505  n_norms = 0
506  ELSE IF ( ignore_cwise ) THEN
507  n_norms = 1
508  ELSE
509  n_norms = 2
510  END IF
511 *
512  rcequ = lsame( equed, 'Y' )
513 *
514 * Test input parameters.
515 *
516  IF (.NOT.lsame(uplo, 'U') .AND. .NOT.lsame(uplo, 'L')) THEN
517  info = -1
518  ELSE IF( .NOT.rcequ .AND. .NOT.lsame( equed, 'N' ) ) THEN
519  info = -2
520  ELSE IF( n.LT.0 ) THEN
521  info = -3
522  ELSE IF( nrhs.LT.0 ) THEN
523  info = -4
524  ELSE IF( lda.LT.max( 1, n ) ) THEN
525  info = -6
526  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
527  info = -8
528  ELSE IF( ldb.LT.max( 1, n ) ) THEN
529  info = -11
530  ELSE IF( ldx.LT.max( 1, n ) ) THEN
531  info = -13
532  END IF
533  IF( info.NE.0 ) THEN
534  CALL xerbla( 'SPORFSX', -info )
535  RETURN
536  END IF
537 *
538 * Quick return if possible.
539 *
540  IF( n.EQ.0 .OR. nrhs.EQ.0 ) THEN
541  rcond = 1.0
542  DO j = 1, nrhs
543  berr( j ) = 0.0
544  IF ( n_err_bnds .GE. 1 ) THEN
545  err_bnds_norm( j, la_linrx_trust_i ) = 1.0
546  err_bnds_comp( j, la_linrx_trust_i ) = 1.0
547  END IF
548  IF ( n_err_bnds .GE. 2 ) THEN
549  err_bnds_norm( j, la_linrx_err_i ) = 0.0
550  err_bnds_comp( j, la_linrx_err_i ) = 0.0
551  END IF
552  IF ( n_err_bnds .GE. 3 ) THEN
553  err_bnds_norm( j, la_linrx_rcond_i ) = 1.0
554  err_bnds_comp( j, la_linrx_rcond_i ) = 1.0
555  END IF
556  END DO
557  RETURN
558  END IF
559 *
560 * Default to failure.
561 *
562  rcond = 0.0
563  DO j = 1, nrhs
564  berr( j ) = 1.0
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 ) = 1.0
571  err_bnds_comp( j, la_linrx_err_i ) = 1.0
572  END IF
573  IF ( n_err_bnds .GE. 3 ) THEN
574  err_bnds_norm( j, la_linrx_rcond_i ) = 0.0
575  err_bnds_comp( j, la_linrx_rcond_i ) = 0.0
576  END IF
577  END DO
578 *
579 * Compute the norm of A and the reciprocal of the condition
580 * number of A.
581 *
582  norm = 'I'
583  anorm = slansy( norm, uplo, n, a, lda, work )
584  CALL spocon( uplo, n, af, ldaf, anorm, rcond, work,
585  \$ iwork, info )
586 *
587 * Perform refinement on each right-hand side
588 *
589  IF ( ref_type .NE. 0 ) THEN
590
591  prec_type = ilaprec( 'D' )
592
593  CALL sla_porfsx_extended( prec_type, uplo, n,
594  \$ nrhs, a, lda, af, ldaf, rcequ, s, b,
595  \$ ldb, x, ldx, berr, n_norms, err_bnds_norm, err_bnds_comp,
596  \$ work( n+1 ), work( 1 ), work( 2*n+1 ), work( 1 ), rcond,
597  \$ ithresh, rthresh, unstable_thresh, ignore_cwise,
598  \$ info )
599  END IF
600
601  err_lbnd = max( 10.0, sqrt( REAL( N ) ) ) * slamch( 'Epsilon' )
602  IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 1 ) THEN
603 *
604 * Compute scaled normwise condition number cond(A*C).
605 *
606  IF ( rcequ ) THEN
607  rcond_tmp = sla_porcond( uplo, n, a, lda, af, ldaf,
608  \$ -1, s, info, work, iwork )
609  ELSE
610  rcond_tmp = sla_porcond( uplo, n, a, lda, af, ldaf,
611  \$ 0, s, info, work, iwork )
612  END IF
613  DO j = 1, nrhs
614 *
615 * Cap the error at 1.0.
616 *
617  IF ( n_err_bnds .GE. la_linrx_err_i
618  \$ .AND. err_bnds_norm( j, la_linrx_err_i ) .GT. 1.0 )
619  \$ err_bnds_norm( j, la_linrx_err_i ) = 1.0
620 *
621 * Threshold the error (see LAWN).
622 *
623  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
624  err_bnds_norm( j, la_linrx_err_i ) = 1.0
625  err_bnds_norm( j, la_linrx_trust_i ) = 0.0
626  IF ( info .LE. n ) info = n + j
627  ELSE IF ( err_bnds_norm( j, la_linrx_err_i ) .LT. err_lbnd )
628  \$ THEN
629  err_bnds_norm( j, la_linrx_err_i ) = err_lbnd
630  err_bnds_norm( j, la_linrx_trust_i ) = 1.0
631  END IF
632 *
633 * Save the condition number.
634 *
635  IF (n_err_bnds .GE. la_linrx_rcond_i) THEN
636  err_bnds_norm( j, la_linrx_rcond_i ) = rcond_tmp
637  END IF
638  END DO
639  END IF
640
641  IF ( n_err_bnds .GE. 1 .AND. n_norms .GE. 2 ) THEN
642 *
643 * Compute componentwise condition number cond(A*diag(Y(:,J))) for
644 * each right-hand side using the current solution as an estimate of
645 * the true solution. If the componentwise error estimate is too
646 * large, then the solution is a lousy estimate of truth and the
647 * estimated RCOND may be too optimistic. To avoid misleading users,
648 * the inverse condition number is set to 0.0 when the estimated
649 * cwise error is at least CWISE_WRONG.
650 *
651  cwise_wrong = sqrt( slamch( 'Epsilon' ) )
652  DO j = 1, nrhs
653  IF (err_bnds_comp( j, la_linrx_err_i ) .LT. cwise_wrong )
654  \$ THEN
655  rcond_tmp = sla_porcond( uplo, n, a, lda, af, ldaf, 1,
656  \$ x( 1, j ), info, work, iwork )
657  ELSE
658  rcond_tmp = 0.0
659  END IF
660 *
661 * Cap the error at 1.0.
662 *
663  IF ( n_err_bnds .GE. la_linrx_err_i
664  \$ .AND. err_bnds_comp( j, la_linrx_err_i ) .GT. 1.0 )
665  \$ err_bnds_comp( j, la_linrx_err_i ) = 1.0
666 *
667 * Threshold the error (see LAWN).
668 *
669  IF ( rcond_tmp .LT. illrcond_thresh ) THEN
670  err_bnds_comp( j, la_linrx_err_i ) = 1.0
671  err_bnds_comp( j, la_linrx_trust_i ) = 0.0
672  IF ( params( la_linrx_cwise_i ) .EQ. 1.0
673  \$ .AND. info.LT.n + j ) info = n + j
674  ELSE IF ( err_bnds_comp( j, la_linrx_err_i )
675  \$ .LT. err_lbnd ) THEN
676  err_bnds_comp( j, la_linrx_err_i ) = err_lbnd
677  err_bnds_comp( j, la_linrx_trust_i ) = 1.0
678  END IF
679 *
680 * Save the condition number.
681 *
682  IF ( n_err_bnds .GE. la_linrx_rcond_i ) THEN
683  err_bnds_comp( j, la_linrx_rcond_i ) = rcond_tmp
684  END IF
685
686  END DO
687  END IF
688 *
689  RETURN
690 *
691 * End of SPORFSX
692 *
real function sla_porcond(UPLO, N, A, LDA, AF, LDAF, CMODE, C, INFO, WORK, IWORK)
SLA_PORCOND estimates the Skeel condition number for a symmetric positive-definite matrix...
Definition: sla_porcond.f:142
subroutine sla_porfsx_extended(PREC_TYPE, UPLO, N, NRHS, A, LDA, AF, LDAF, 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_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or H...
integer function ilaprec(PREC)
ILAPREC
Definition: ilaprec.f:60
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine spocon(UPLO, N, A, LDA, ANORM, RCOND, WORK, IWORK, INFO)
SPOCON
Definition: spocon.f:123
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
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, or the element of largest absolute value of a real symmetric matrix.
Definition: slansy.f:124
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