LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ sla_syrfsx_extended()

subroutine sla_syrfsx_extended ( integer  PREC_TYPE,
character  UPLO,
integer  N,
integer  NRHS,
real, dimension( lda, * )  A,
integer  LDA,
real, dimension( ldaf, * )  AF,
integer  LDAF,
integer, dimension( * )  IPIV,
logical  COLEQU,
real, dimension( * )  C,
real, dimension( ldb, * )  B,
integer  LDB,
real, dimension( ldy, * )  Y,
integer  LDY,
real, dimension( * )  BERR_OUT,
integer  N_NORMS,
real, dimension( nrhs, * )  ERR_BNDS_NORM,
real, dimension( nrhs, * )  ERR_BNDS_COMP,
real, dimension( * )  RES,
real, dimension( * )  AYB,
real, dimension( * )  DY,
real, dimension( * )  Y_TAIL,
real  RCOND,
integer  ITHRESH,
real  RTHRESH,
real  DZ_UB,
logical  IGNORE_CWISE,
integer  INFO 
)

SLA_SYRFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

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Purpose:
 SLA_SYRFSX_EXTENDED improves the computed solution to a system of
 linear equations by performing extra-precise iterative refinement
 and provides error bounds and backward error estimates for the solution.
 This subroutine is called by SSYRFSX to perform iterative refinement.
 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. Note that this
 subroutine is only resonsible for setting the second fields of
 ERR_BNDS_NORM and ERR_BNDS_COMP.
Parameters
[in]PREC_TYPE
          PREC_TYPE is INTEGER
     Specifies the intermediate precision to be used in refinement.
     The value is defined by ILAPREC(P) where P is a CHARACTER and P
          = 'S':  Single
          = 'D':  Double
          = 'I':  Indigenous
          = 'X' or 'E':  Extra
[in]UPLO
          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.
[in]N
          N is INTEGER
     The number of linear equations, i.e., 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
     matrix B.
[in]A
          A is REAL array, dimension (LDA,N)
     On entry, the 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 block diagonal matrix D and the multipliers used to
     obtain the factor U or L 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]COLEQU
          COLEQU is LOGICAL
     If .TRUE. then column equilibration was done to A before calling
     this routine. This is needed to compute the solution and error
     bounds correctly.
[in]C
          C is REAL array, dimension (N)
     The column scale factors for A. If COLEQU = .FALSE., C
     is not accessed. 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.
[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]Y
          Y is REAL array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by SSYTRS.
     On exit, the improved solution matrix Y.
[in]LDY
          LDY is INTEGER
     The leading dimension of the array Y.  LDY >= max(1,N).
[out]BERR_OUT
          BERR_OUT is REAL array, dimension (NRHS)
     On exit, BERR_OUT(j) contains the componentwise relative backward
     error for right-hand-side j from the formula
         max(i) ( abs(RES(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
     where abs(Z) is the componentwise absolute value of the matrix
     or vector Z. This is computed by SLA_LIN_BERR.
[in]N_NORMS
          N_NORMS is INTEGER
     Determines which error bounds to return (see ERR_BNDS_NORM
     and ERR_BNDS_COMP).
     If N_NORMS >= 1 return normwise error bounds.
     If N_NORMS >= 2 return componentwise error bounds.
[in,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.

     This subroutine is only responsible for setting the second field
     above.
     See Lapack Working Note 165 for further details and extra
     cautions.
[in,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.

     This subroutine is only responsible for setting the second field
     above.
     See Lapack Working Note 165 for further details and extra
     cautions.
[in]RES
          RES is REAL array, dimension (N)
     Workspace to hold the intermediate residual.
[in]AYB
          AYB is REAL array, dimension (N)
     Workspace. This can be the same workspace passed for Y_TAIL.
[in]DY
          DY is REAL array, dimension (N)
     Workspace to hold the intermediate solution.
[in]Y_TAIL
          Y_TAIL is REAL array, dimension (N)
     Workspace to hold the trailing bits of the intermediate solution.
[in]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.
[in]ITHRESH
          ITHRESH is INTEGER
     The maximum number of residual computations allowed for
     refinement. The default is 10. For '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.
[in]RTHRESH
          RTHRESH is REAL
     Determines when to stop refinement if the error estimate stops
     decreasing. Refinement will stop when the next solution no longer
     satisfies norm(dx_{i+1}) < RTHRESH * norm(dx_i) where norm(Z) is
     the infinity norm of Z. RTHRESH satisfies 0 < RTHRESH <= 1. The
     default value is 0.5. For 'aggressive' set to 0.9 to permit
     convergence on extremely ill-conditioned matrices. See LAWN 165
     for more details.
[in]DZ_UB
          DZ_UB is REAL
     Determines when to start considering componentwise convergence.
     Componentwise convergence is only considered after each component
     of the solution Y is stable, which we define as the relative
     change in each component being less than DZ_UB. The default value
     is 0.25, requiring the first bit to be stable. See LAWN 165 for
     more details.
[in]IGNORE_CWISE
          IGNORE_CWISE is LOGICAL
     If .TRUE. then ignore componentwise convergence. Default value
     is .FALSE..
[out]INFO
          INFO is INTEGER
       = 0:  Successful exit.
       < 0:  if INFO = -i, the ith argument to SLA_SYRFSX_EXTENDED had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 389 of file sla_syrfsx_extended.f.

396 *
397 * -- LAPACK computational routine --
398 * -- LAPACK is a software package provided by Univ. of Tennessee, --
399 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
400 *
401 * .. Scalar Arguments ..
402  INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
403  $ N_NORMS, ITHRESH
404  CHARACTER UPLO
405  LOGICAL COLEQU, IGNORE_CWISE
406  REAL RTHRESH, DZ_UB
407 * ..
408 * .. Array Arguments ..
409  INTEGER IPIV( * )
410  REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
411  $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
412  REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ),
413  $ ERR_BNDS_NORM( NRHS, * ),
414  $ ERR_BNDS_COMP( NRHS, * )
415 * ..
416 *
417 * =====================================================================
418 *
419 * .. Local Scalars ..
420  INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE
421  REAL YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
422  $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
423  $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
424  $ EPS, HUGEVAL, INCR_THRESH
425  LOGICAL INCR_PREC, UPPER
426 * ..
427 * .. Parameters ..
428  INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
429  $ NOPROG_STATE, Y_PREC_STATE, BASE_RESIDUAL,
430  $ EXTRA_RESIDUAL, EXTRA_Y
431  parameter( unstable_state = 0, working_state = 1,
432  $ conv_state = 2, noprog_state = 3 )
433  parameter( base_residual = 0, extra_residual = 1,
434  $ extra_y = 2 )
435  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
436  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
437  INTEGER CMP_ERR_I, PIV_GROWTH_I
438  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
439  $ berr_i = 3 )
440  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
441  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
442  $ piv_growth_i = 9 )
443  INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
444  $ LA_LINRX_CWISE_I
445  parameter( la_linrx_itref_i = 1,
446  $ la_linrx_ithresh_i = 2 )
447  parameter( la_linrx_cwise_i = 3 )
448  INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
449  $ LA_LINRX_RCOND_I
450  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
451  parameter( la_linrx_rcond_i = 3 )
452 * ..
453 * .. External Functions ..
454  LOGICAL LSAME
455  EXTERNAL ilauplo
456  INTEGER ILAUPLO
457 * ..
458 * .. External Subroutines ..
459  EXTERNAL saxpy, scopy, ssytrs, ssymv, blas_ssymv_x,
460  $ blas_ssymv2_x, sla_syamv, sla_wwaddw,
461  $ sla_lin_berr
462  REAL SLAMCH
463 * ..
464 * .. Intrinsic Functions ..
465  INTRINSIC abs, max, min
466 * ..
467 * .. Executable Statements ..
468 *
469  info = 0
470  upper = lsame( uplo, 'U' )
471  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
472  info = -2
473  ELSE IF( n.LT.0 ) THEN
474  info = -3
475  ELSE IF( nrhs.LT.0 ) THEN
476  info = -4
477  ELSE IF( lda.LT.max( 1, n ) ) THEN
478  info = -6
479  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
480  info = -8
481  ELSE IF( ldb.LT.max( 1, n ) ) THEN
482  info = -13
483  ELSE IF( ldy.LT.max( 1, n ) ) THEN
484  info = -15
485  END IF
486  IF( info.NE.0 ) THEN
487  CALL xerbla( 'SLA_SYRFSX_EXTENDED', -info )
488  RETURN
489  END IF
490  eps = slamch( 'Epsilon' )
491  hugeval = slamch( 'Overflow' )
492 * Force HUGEVAL to Inf
493  hugeval = hugeval * hugeval
494 * Using HUGEVAL may lead to spurious underflows.
495  incr_thresh = real( n )*eps
496 
497  IF ( lsame( uplo, 'L' ) ) THEN
498  uplo2 = ilauplo( 'L' )
499  ELSE
500  uplo2 = ilauplo( 'U' )
501  ENDIF
502 
503  DO j = 1, nrhs
504  y_prec_state = extra_residual
505  IF ( y_prec_state .EQ. extra_y ) THEN
506  DO i = 1, n
507  y_tail( i ) = 0.0
508  END DO
509  END IF
510 
511  dxrat = 0.0
512  dxratmax = 0.0
513  dzrat = 0.0
514  dzratmax = 0.0
515  final_dx_x = hugeval
516  final_dz_z = hugeval
517  prevnormdx = hugeval
518  prev_dz_z = hugeval
519  dz_z = hugeval
520  dx_x = hugeval
521 
522  x_state = working_state
523  z_state = unstable_state
524  incr_prec = .false.
525 
526  DO cnt = 1, ithresh
527 *
528 * Compute residual RES = B_s - op(A_s) * Y,
529 * op(A) = A, A**T, or A**H depending on TRANS (and type).
530 *
531  CALL scopy( n, b( 1, j ), 1, res, 1 )
532  IF (y_prec_state .EQ. base_residual) THEN
533  CALL ssymv( uplo, n, -1.0, a, lda, y(1,j), 1,
534  $ 1.0, res, 1 )
535  ELSE IF (y_prec_state .EQ. extra_residual) THEN
536  CALL blas_ssymv_x( uplo2, n, -1.0, a, lda,
537  $ y( 1, j ), 1, 1.0, res, 1, prec_type )
538  ELSE
539  CALL blas_ssymv2_x(uplo2, n, -1.0, a, lda,
540  $ y(1, j), y_tail, 1, 1.0, res, 1, prec_type)
541  END IF
542 
543 ! XXX: RES is no longer needed.
544  CALL scopy( n, res, 1, dy, 1 )
545  CALL ssytrs( uplo, n, 1, af, ldaf, ipiv, dy, n, info )
546 *
547 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
548 *
549  normx = 0.0
550  normy = 0.0
551  normdx = 0.0
552  dz_z = 0.0
553  ymin = hugeval
554 
555  DO i = 1, n
556  yk = abs( y( i, j ) )
557  dyk = abs( dy( i ) )
558 
559  IF ( yk .NE. 0.0 ) THEN
560  dz_z = max( dz_z, dyk / yk )
561  ELSE IF ( dyk .NE. 0.0 ) THEN
562  dz_z = hugeval
563  END IF
564 
565  ymin = min( ymin, yk )
566 
567  normy = max( normy, yk )
568 
569  IF ( colequ ) THEN
570  normx = max( normx, yk * c( i ) )
571  normdx = max( normdx, dyk * c( i ) )
572  ELSE
573  normx = normy
574  normdx = max(normdx, dyk)
575  END IF
576  END DO
577 
578  IF ( normx .NE. 0.0 ) THEN
579  dx_x = normdx / normx
580  ELSE IF ( normdx .EQ. 0.0 ) THEN
581  dx_x = 0.0
582  ELSE
583  dx_x = hugeval
584  END IF
585 
586  dxrat = normdx / prevnormdx
587  dzrat = dz_z / prev_dz_z
588 *
589 * Check termination criteria.
590 *
591  IF ( ymin*rcond .LT. incr_thresh*normy
592  $ .AND. y_prec_state .LT. extra_y )
593  $ incr_prec = .true.
594 
595  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
596  $ x_state = working_state
597  IF ( x_state .EQ. working_state ) THEN
598  IF ( dx_x .LE. eps ) THEN
599  x_state = conv_state
600  ELSE IF ( dxrat .GT. rthresh ) THEN
601  IF ( y_prec_state .NE. extra_y ) THEN
602  incr_prec = .true.
603  ELSE
604  x_state = noprog_state
605  END IF
606  ELSE
607  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
608  END IF
609  IF ( x_state .GT. working_state ) final_dx_x = dx_x
610  END IF
611 
612  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
613  $ z_state = working_state
614  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
615  $ z_state = working_state
616  IF ( z_state .EQ. working_state ) THEN
617  IF ( dz_z .LE. eps ) THEN
618  z_state = conv_state
619  ELSE IF ( dz_z .GT. dz_ub ) THEN
620  z_state = unstable_state
621  dzratmax = 0.0
622  final_dz_z = hugeval
623  ELSE IF ( dzrat .GT. rthresh ) THEN
624  IF ( y_prec_state .NE. extra_y ) THEN
625  incr_prec = .true.
626  ELSE
627  z_state = noprog_state
628  END IF
629  ELSE
630  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
631  END IF
632  IF ( z_state .GT. working_state ) final_dz_z = dz_z
633  END IF
634 
635  IF ( x_state.NE.working_state.AND.
636  $ ( ignore_cwise.OR.z_state.NE.working_state ) )
637  $ GOTO 666
638 
639  IF ( incr_prec ) THEN
640  incr_prec = .false.
641  y_prec_state = y_prec_state + 1
642  DO i = 1, n
643  y_tail( i ) = 0.0
644  END DO
645  END IF
646 
647  prevnormdx = normdx
648  prev_dz_z = dz_z
649 *
650 * Update soluton.
651 *
652  IF (y_prec_state .LT. extra_y) THEN
653  CALL saxpy( n, 1.0, dy, 1, y(1,j), 1 )
654  ELSE
655  CALL sla_wwaddw( n, y(1,j), y_tail, dy )
656  END IF
657 
658  END DO
659 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
660  666 CONTINUE
661 *
662 * Set final_* when cnt hits ithresh.
663 *
664  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
665  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
666 *
667 * Compute error bounds.
668 *
669  IF ( n_norms .GE. 1 ) THEN
670  err_bnds_norm( j, la_linrx_err_i ) =
671  $ final_dx_x / (1 - dxratmax)
672  END IF
673  IF ( n_norms .GE. 2 ) THEN
674  err_bnds_comp( j, la_linrx_err_i ) =
675  $ final_dz_z / (1 - dzratmax)
676  END IF
677 *
678 * Compute componentwise relative backward error from formula
679 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
680 * where abs(Z) is the componentwise absolute value of the matrix
681 * or vector Z.
682 *
683 * Compute residual RES = B_s - op(A_s) * Y,
684 * op(A) = A, A**T, or A**H depending on TRANS (and type).
685  CALL scopy( n, b( 1, j ), 1, res, 1 )
686  CALL ssymv( uplo, n, -1.0, a, lda, y(1,j), 1, 1.0, res, 1 )
687 
688  DO i = 1, n
689  ayb( i ) = abs( b( i, j ) )
690  END DO
691 *
692 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
693 *
694  CALL sla_syamv( uplo2, n, 1.0,
695  $ a, lda, y(1, j), 1, 1.0, ayb, 1 )
696 
697  CALL sla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
698 *
699 * End of loop for each RHS.
700 *
701  END DO
702 *
703  RETURN
704 *
705 * End of SLA_SYRFSX_EXTENDED
706 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
integer function ilauplo(UPLO)
ILAUPLO
Definition: ilauplo.f:58
subroutine sla_wwaddw(N, X, Y, W)
SLA_WWADDW adds a vector into a doubled-single vector.
Definition: sla_wwaddw.f:81
subroutine sla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
SLA_LIN_BERR computes a component-wise relative backward error.
Definition: sla_lin_berr.f:101
subroutine sla_syamv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bou...
Definition: sla_syamv.f:177
subroutine ssytrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SSYTRS
Definition: ssytrs.f:120
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:82
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:89
subroutine ssymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SSYMV
Definition: ssymv.f:152
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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