LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
Loading...
Searching...
No Matches

◆ dla_gerfsx_extended()

subroutine dla_gerfsx_extended ( integer  PREC_TYPE,
integer  TRANS_TYPE,
integer  N,
integer  NRHS,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( ldaf, * )  AF,
integer  LDAF,
integer, dimension( * )  IPIV,
logical  COLEQU,
double precision, dimension( * )  C,
double precision, dimension( ldb, * )  B,
integer  LDB,
double precision, dimension( ldy, * )  Y,
integer  LDY,
double precision, dimension( * )  BERR_OUT,
integer  N_NORMS,
double precision, dimension( nrhs, * )  ERRS_N,
double precision, dimension( nrhs, * )  ERRS_C,
double precision, dimension( * )  RES,
double precision, dimension( * )  AYB,
double precision, dimension( * )  DY,
double precision, dimension( * )  Y_TAIL,
double precision  RCOND,
integer  ITHRESH,
double precision  RTHRESH,
double precision  DZ_UB,
logical  IGNORE_CWISE,
integer  INFO 
)

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

Download DLA_GERFSX_EXTENDED + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 DLA_GERFSX_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 DGERFSX to perform iterative refinement.
 In addition to normwise error bound, the code provides maximum
 componentwise error bound if possible. See comments for ERRS_N
 and ERRS_C for details of the error bounds. Note that this
 subroutine is only responsible for setting the second fields of
 ERRS_N and ERRS_C.
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]TRANS_TYPE
          TRANS_TYPE is INTEGER
     Specifies the transposition operation on A.
     The value is defined by ILATRANS(T) where T is a CHARACTER and T
          = 'N':  No transpose
          = 'T':  Transpose
          = 'C':  Conjugate transpose
[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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDAF,N)
     The factors L and U from the factorization
     A = P*L*U as computed by DGETRF.
[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 the factorization A = P*L*U
     as computed by DGETRF; row i of the matrix was interchanged
     with row IPIV(i).
[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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by DGETRS.
     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 DOUBLE PRECISION 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 DLA_LIN_BERR.
[in]N_NORMS
          N_NORMS is INTEGER
     Determines which error bounds to return (see ERRS_N
     and ERRS_C).
     If N_NORMS >= 1 return normwise error bounds.
     If N_NORMS >= 2 return componentwise error bounds.
[in,out]ERRS_N
          ERRS_N is DOUBLE PRECISION 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 ERRS_N(i,:) corresponds to the ith
     right-hand side.

     The second index in ERRS_N(:,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]ERRS_C
          ERRS_C is DOUBLE PRECISION 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
     ERRS_C is not accessed.  If N_ERR_BNDS < 3, then at most
     the first (:,N_ERR_BNDS) entries are returned.

     The first index in ERRS_C(i,:) corresponds to the ith
     right-hand side.

     The second index in ERRS_C(:,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 DOUBLE PRECISION array, dimension (N)
     Workspace to hold the intermediate residual.
[in]AYB
          AYB is DOUBLE PRECISION array, dimension (N)
     Workspace. This can be the same workspace passed for Y_TAIL.
[in]DY
          DY is DOUBLE PRECISION array, dimension (N)
     Workspace to hold the intermediate solution.
[in]Y_TAIL
          Y_TAIL is DOUBLE PRECISION array, dimension (N)
     Workspace to hold the trailing bits of the intermediate solution.
[in]RCOND
          RCOND is DOUBLE PRECISION
     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
     ERRS_N and ERRS_C may no longer be trustworthy.
[in]RTHRESH
          RTHRESH is DOUBLE PRECISION
     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 DOUBLE PRECISION
     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 DGETRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 390 of file dla_gerfsx_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 $ TRANS_TYPE, N_NORMS, ITHRESH
404 LOGICAL COLEQU, IGNORE_CWISE
405 DOUBLE PRECISION RTHRESH, DZ_UB
406* ..
407* .. Array Arguments ..
408 INTEGER IPIV( * )
409 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
410 $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
411 DOUBLE PRECISION C( * ), AYB( * ), RCOND, BERR_OUT( * ),
412 $ ERRS_N( NRHS, * ), ERRS_C( NRHS, * )
413* ..
414*
415* =====================================================================
416*
417* .. Local Scalars ..
418 CHARACTER TRANS
419 INTEGER CNT, I, J, X_STATE, Z_STATE, Y_PREC_STATE
420 DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
421 $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
422 $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
423 $ EPS, HUGEVAL, INCR_THRESH
424 LOGICAL INCR_PREC
425* ..
426* .. Parameters ..
427 INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
428 $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
429 $ EXTRA_Y
430 parameter( unstable_state = 0, working_state = 1,
431 $ conv_state = 2, noprog_state = 3 )
432 parameter( base_residual = 0, extra_residual = 1,
433 $ extra_y = 2 )
434 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
435 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
436 INTEGER CMP_ERR_I, PIV_GROWTH_I
437 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
438 $ berr_i = 3 )
439 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
440 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
441 $ piv_growth_i = 9 )
442 INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
443 $ LA_LINRX_CWISE_I
444 parameter( la_linrx_itref_i = 1,
445 $ la_linrx_ithresh_i = 2 )
446 parameter( la_linrx_cwise_i = 3 )
447 INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
448 $ LA_LINRX_RCOND_I
449 parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
450 parameter( la_linrx_rcond_i = 3 )
451* ..
452* .. External Subroutines ..
453 EXTERNAL daxpy, dcopy, dgetrs, dgemv, blas_dgemv_x,
454 $ blas_dgemv2_x, dla_geamv, dla_wwaddw, dlamch,
456 DOUBLE PRECISION DLAMCH
457 CHARACTER CHLA_TRANSTYPE
458* ..
459* .. Intrinsic Functions ..
460 INTRINSIC abs, max, min
461* ..
462* .. Executable Statements ..
463*
464 IF ( info.NE.0 ) RETURN
465 trans = chla_transtype(trans_type)
466 eps = dlamch( 'Epsilon' )
467 hugeval = dlamch( 'Overflow' )
468* Force HUGEVAL to Inf
469 hugeval = hugeval * hugeval
470* Using HUGEVAL may lead to spurious underflows.
471 incr_thresh = dble( n ) * eps
472*
473 DO j = 1, nrhs
474 y_prec_state = extra_residual
475 IF ( y_prec_state .EQ. extra_y ) THEN
476 DO i = 1, n
477 y_tail( i ) = 0.0d+0
478 END DO
479 END IF
480
481 dxrat = 0.0d+0
482 dxratmax = 0.0d+0
483 dzrat = 0.0d+0
484 dzratmax = 0.0d+0
485 final_dx_x = hugeval
486 final_dz_z = hugeval
487 prevnormdx = hugeval
488 prev_dz_z = hugeval
489 dz_z = hugeval
490 dx_x = hugeval
491
492 x_state = working_state
493 z_state = unstable_state
494 incr_prec = .false.
495
496 DO cnt = 1, ithresh
497*
498* Compute residual RES = B_s - op(A_s) * Y,
499* op(A) = A, A**T, or A**H depending on TRANS (and type).
500*
501 CALL dcopy( n, b( 1, j ), 1, res, 1 )
502 IF ( y_prec_state .EQ. base_residual ) THEN
503 CALL dgemv( trans, n, n, -1.0d+0, a, lda, y( 1, j ), 1,
504 $ 1.0d+0, res, 1 )
505 ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
506 CALL blas_dgemv_x( trans_type, n, n, -1.0d+0, a, lda,
507 $ y( 1, j ), 1, 1.0d+0, res, 1, prec_type )
508 ELSE
509 CALL blas_dgemv2_x( trans_type, n, n, -1.0d+0, a, lda,
510 $ y( 1, j ), y_tail, 1, 1.0d+0, res, 1, prec_type )
511 END IF
512
513! XXX: RES is no longer needed.
514 CALL dcopy( n, res, 1, dy, 1 )
515 CALL dgetrs( trans, n, 1, af, ldaf, ipiv, dy, n, info )
516*
517* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
518*
519 normx = 0.0d+0
520 normy = 0.0d+0
521 normdx = 0.0d+0
522 dz_z = 0.0d+0
523 ymin = hugeval
524*
525 DO i = 1, n
526 yk = abs( y( i, j ) )
527 dyk = abs( dy( i ) )
528
529 IF ( yk .NE. 0.0d+0 ) THEN
530 dz_z = max( dz_z, dyk / yk )
531 ELSE IF ( dyk .NE. 0.0d+0 ) THEN
532 dz_z = hugeval
533 END IF
534
535 ymin = min( ymin, yk )
536
537 normy = max( normy, yk )
538
539 IF ( colequ ) THEN
540 normx = max( normx, yk * c( i ) )
541 normdx = max( normdx, dyk * c( i ) )
542 ELSE
543 normx = normy
544 normdx = max( normdx, dyk )
545 END IF
546 END DO
547
548 IF ( normx .NE. 0.0d+0 ) THEN
549 dx_x = normdx / normx
550 ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
551 dx_x = 0.0d+0
552 ELSE
553 dx_x = hugeval
554 END IF
555
556 dxrat = normdx / prevnormdx
557 dzrat = dz_z / prev_dz_z
558*
559* Check termination criteria
560*
561 IF (.NOT.ignore_cwise
562 $ .AND. ymin*rcond .LT. incr_thresh*normy
563 $ .AND. y_prec_state .LT. extra_y)
564 $ incr_prec = .true.
565
566 IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
567 $ x_state = working_state
568 IF ( x_state .EQ. working_state ) THEN
569 IF ( dx_x .LE. eps ) THEN
570 x_state = conv_state
571 ELSE IF ( dxrat .GT. rthresh ) THEN
572 IF ( y_prec_state .NE. extra_y ) THEN
573 incr_prec = .true.
574 ELSE
575 x_state = noprog_state
576 END IF
577 ELSE
578 IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
579 END IF
580 IF ( x_state .GT. working_state ) final_dx_x = dx_x
581 END IF
582
583 IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
584 $ z_state = working_state
585 IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
586 $ z_state = working_state
587 IF ( z_state .EQ. working_state ) THEN
588 IF ( dz_z .LE. eps ) THEN
589 z_state = conv_state
590 ELSE IF ( dz_z .GT. dz_ub ) THEN
591 z_state = unstable_state
592 dzratmax = 0.0d+0
593 final_dz_z = hugeval
594 ELSE IF ( dzrat .GT. rthresh ) THEN
595 IF ( y_prec_state .NE. extra_y ) THEN
596 incr_prec = .true.
597 ELSE
598 z_state = noprog_state
599 END IF
600 ELSE
601 IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
602 END IF
603 IF ( z_state .GT. working_state ) final_dz_z = dz_z
604 END IF
605*
606* Exit if both normwise and componentwise stopped working,
607* but if componentwise is unstable, let it go at least two
608* iterations.
609*
610 IF ( x_state.NE.working_state ) THEN
611 IF ( ignore_cwise) GOTO 666
612 IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
613 $ GOTO 666
614 IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
615 END IF
616
617 IF ( incr_prec ) THEN
618 incr_prec = .false.
619 y_prec_state = y_prec_state + 1
620 DO i = 1, n
621 y_tail( i ) = 0.0d+0
622 END DO
623 END IF
624
625 prevnormdx = normdx
626 prev_dz_z = dz_z
627*
628* Update soluton.
629*
630 IF ( y_prec_state .LT. extra_y ) THEN
631 CALL daxpy( n, 1.0d+0, dy, 1, y( 1, j ), 1 )
632 ELSE
633 CALL dla_wwaddw( n, y( 1, j ), y_tail, dy )
634 END IF
635
636 END DO
637* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
638 666 CONTINUE
639*
640* Set final_* when cnt hits ithresh.
641*
642 IF ( x_state .EQ. working_state ) final_dx_x = dx_x
643 IF ( z_state .EQ. working_state ) final_dz_z = dz_z
644*
645* Compute error bounds
646*
647 IF (n_norms .GE. 1) THEN
648 errs_n( j, la_linrx_err_i ) = final_dx_x / (1 - dxratmax)
649 END IF
650 IF ( n_norms .GE. 2 ) THEN
651 errs_c( j, la_linrx_err_i ) = final_dz_z / (1 - dzratmax)
652 END IF
653*
654* Compute componentwise relative backward error from formula
655* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
656* where abs(Z) is the componentwise absolute value of the matrix
657* or vector Z.
658*
659* Compute residual RES = B_s - op(A_s) * Y,
660* op(A) = A, A**T, or A**H depending on TRANS (and type).
661*
662 CALL dcopy( n, b( 1, j ), 1, res, 1 )
663 CALL dgemv( trans, n, n, -1.0d+0, a, lda, y(1,j), 1, 1.0d+0,
664 $ res, 1 )
665
666 DO i = 1, n
667 ayb( i ) = abs( b( i, j ) )
668 END DO
669*
670* Compute abs(op(A_s))*abs(Y) + abs(B_s).
671*
672 CALL dla_geamv ( trans_type, n, n, 1.0d+0,
673 $ a, lda, y(1, j), 1, 1.0d+0, ayb, 1 )
674
675 CALL dla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
676*
677* End of loop for each RHS.
678*
679 END DO
680*
681 RETURN
682*
683* End of DLA_GERFSX_EXTENDED
684*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:156
subroutine dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
Definition: dgetrs.f:121
subroutine dla_geamv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds.
Definition: dla_geamv.f:174
subroutine dla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
DLA_LIN_BERR computes a component-wise relative backward error.
Definition: dla_lin_berr.f:101
subroutine dla_wwaddw(N, X, Y, W)
DLA_WWADDW adds a vector into a doubled-single vector.
Definition: dla_wwaddw.f:81
Here is the call graph for this function:
Here is the caller graph for this function: