LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ 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.

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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 resonsible 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', '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 .LT. 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 definte 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.
Date
June 2017

Definition at line 398 of file dla_gerfsx_extended.f.

398 *
399 * -- LAPACK computational routine (version 3.7.1) --
400 * -- LAPACK is a software package provided by Univ. of Tennessee, --
401 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
402 * June 2017
403 *
404 * .. Scalar Arguments ..
405  INTEGER info, lda, ldaf, ldb, ldy, n, nrhs, prec_type,
406  $ trans_type, n_norms, ithresh
407  LOGICAL colequ, ignore_cwise
408  DOUBLE PRECISION rthresh, dz_ub
409 * ..
410 * .. Array Arguments ..
411  INTEGER ipiv( * )
412  DOUBLE PRECISION a( lda, * ), af( ldaf, * ), b( ldb, * ),
413  $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
414  DOUBLE PRECISION c( * ), ayb( * ), rcond, berr_out( * ),
415  $ errs_n( nrhs, * ), errs_c( nrhs, * )
416 * ..
417 *
418 * =====================================================================
419 *
420 * .. Local Scalars ..
421  CHARACTER trans
422  INTEGER cnt, i, j, x_state, z_state, y_prec_state
423  DOUBLE PRECISION yk, dyk, ymin, normy, normx, normdx, dxrat,
424  $ dzrat, prevnormdx, prev_dz_z, dxratmax,
425  $ dzratmax, dx_x, dz_z, final_dx_x, final_dz_z,
426  $ eps, hugeval, incr_thresh
427  LOGICAL incr_prec
428 * ..
429 * .. Parameters ..
430  INTEGER unstable_state, working_state, conv_state,
431  $ noprog_state, base_residual, extra_residual,
432  $ extra_y
433  parameter( unstable_state = 0, working_state = 1,
434  $ conv_state = 2, noprog_state = 3 )
435  parameter( base_residual = 0, extra_residual = 1,
436  $ extra_y = 2 )
437  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
438  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
439  INTEGER cmp_err_i, piv_growth_i
440  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
441  $ berr_i = 3 )
442  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
443  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
444  $ piv_growth_i = 9 )
445  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
446  $ la_linrx_cwise_i
447  parameter( la_linrx_itref_i = 1,
448  $ la_linrx_ithresh_i = 2 )
449  parameter( la_linrx_cwise_i = 3 )
450  INTEGER la_linrx_trust_i, la_linrx_err_i,
451  $ la_linrx_rcond_i
452  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
453  parameter( la_linrx_rcond_i = 3 )
454 * ..
455 * .. External Subroutines ..
456  EXTERNAL daxpy, dcopy, dgetrs, dgemv, blas_dgemv_x,
457  $ blas_dgemv2_x, dla_geamv, dla_wwaddw, dlamch,
459  DOUBLE PRECISION dlamch
460  CHARACTER chla_transtype
461 * ..
462 * .. Intrinsic Functions ..
463  INTRINSIC abs, max, min
464 * ..
465 * .. Executable Statements ..
466 *
467  IF ( info.NE.0 ) RETURN
468  trans = chla_transtype(trans_type)
469  eps = dlamch( 'Epsilon' )
470  hugeval = dlamch( 'Overflow' )
471 * Force HUGEVAL to Inf
472  hugeval = hugeval * hugeval
473 * Using HUGEVAL may lead to spurious underflows.
474  incr_thresh = dble( n ) * eps
475 *
476  DO j = 1, nrhs
477  y_prec_state = extra_residual
478  IF ( y_prec_state .EQ. extra_y ) THEN
479  DO i = 1, n
480  y_tail( i ) = 0.0d+0
481  END DO
482  END IF
483 
484  dxrat = 0.0d+0
485  dxratmax = 0.0d+0
486  dzrat = 0.0d+0
487  dzratmax = 0.0d+0
488  final_dx_x = hugeval
489  final_dz_z = hugeval
490  prevnormdx = hugeval
491  prev_dz_z = hugeval
492  dz_z = hugeval
493  dx_x = hugeval
494 
495  x_state = working_state
496  z_state = unstable_state
497  incr_prec = .false.
498 
499  DO cnt = 1, ithresh
500 *
501 * Compute residual RES = B_s - op(A_s) * Y,
502 * op(A) = A, A**T, or A**H depending on TRANS (and type).
503 *
504  CALL dcopy( n, b( 1, j ), 1, res, 1 )
505  IF ( y_prec_state .EQ. base_residual ) THEN
506  CALL dgemv( trans, n, n, -1.0d+0, a, lda, y( 1, j ), 1,
507  $ 1.0d+0, res, 1 )
508  ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
509  CALL blas_dgemv_x( trans_type, n, n, -1.0d+0, a, lda,
510  $ y( 1, j ), 1, 1.0d+0, res, 1, prec_type )
511  ELSE
512  CALL blas_dgemv2_x( trans_type, n, n, -1.0d+0, a, lda,
513  $ y( 1, j ), y_tail, 1, 1.0d+0, res, 1, prec_type )
514  END IF
515 
516 ! XXX: RES is no longer needed.
517  CALL dcopy( n, res, 1, dy, 1 )
518  CALL dgetrs( trans, n, 1, af, ldaf, ipiv, dy, n, info )
519 *
520 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
521 *
522  normx = 0.0d+0
523  normy = 0.0d+0
524  normdx = 0.0d+0
525  dz_z = 0.0d+0
526  ymin = hugeval
527 *
528  DO i = 1, n
529  yk = abs( y( i, j ) )
530  dyk = abs( dy( i ) )
531 
532  IF ( yk .NE. 0.0d+0 ) THEN
533  dz_z = max( dz_z, dyk / yk )
534  ELSE IF ( dyk .NE. 0.0d+0 ) THEN
535  dz_z = hugeval
536  END IF
537 
538  ymin = min( ymin, yk )
539 
540  normy = max( normy, yk )
541 
542  IF ( colequ ) THEN
543  normx = max( normx, yk * c( i ) )
544  normdx = max( normdx, dyk * c( i ) )
545  ELSE
546  normx = normy
547  normdx = max( normdx, dyk )
548  END IF
549  END DO
550 
551  IF ( normx .NE. 0.0d+0 ) THEN
552  dx_x = normdx / normx
553  ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
554  dx_x = 0.0d+0
555  ELSE
556  dx_x = hugeval
557  END IF
558 
559  dxrat = normdx / prevnormdx
560  dzrat = dz_z / prev_dz_z
561 *
562 * Check termination criteria
563 *
564  IF (.NOT.ignore_cwise
565  $ .AND. ymin*rcond .LT. incr_thresh*normy
566  $ .AND. y_prec_state .LT. extra_y)
567  $ incr_prec = .true.
568 
569  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
570  $ x_state = working_state
571  IF ( x_state .EQ. working_state ) THEN
572  IF ( dx_x .LE. eps ) THEN
573  x_state = conv_state
574  ELSE IF ( dxrat .GT. rthresh ) THEN
575  IF ( y_prec_state .NE. extra_y ) THEN
576  incr_prec = .true.
577  ELSE
578  x_state = noprog_state
579  END IF
580  ELSE
581  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
582  END IF
583  IF ( x_state .GT. working_state ) final_dx_x = dx_x
584  END IF
585 
586  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
587  $ z_state = working_state
588  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
589  $ z_state = working_state
590  IF ( z_state .EQ. working_state ) THEN
591  IF ( dz_z .LE. eps ) THEN
592  z_state = conv_state
593  ELSE IF ( dz_z .GT. dz_ub ) THEN
594  z_state = unstable_state
595  dzratmax = 0.0d+0
596  final_dz_z = hugeval
597  ELSE IF ( dzrat .GT. rthresh ) THEN
598  IF ( y_prec_state .NE. extra_y ) THEN
599  incr_prec = .true.
600  ELSE
601  z_state = noprog_state
602  END IF
603  ELSE
604  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
605  END IF
606  IF ( z_state .GT. working_state ) final_dz_z = dz_z
607  END IF
608 *
609 * Exit if both normwise and componentwise stopped working,
610 * but if componentwise is unstable, let it go at least two
611 * iterations.
612 *
613  IF ( x_state.NE.working_state ) THEN
614  IF ( ignore_cwise) GOTO 666
615  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
616  $ GOTO 666
617  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
618  END IF
619 
620  IF ( incr_prec ) THEN
621  incr_prec = .false.
622  y_prec_state = y_prec_state + 1
623  DO i = 1, n
624  y_tail( i ) = 0.0d+0
625  END DO
626  END IF
627 
628  prevnormdx = normdx
629  prev_dz_z = dz_z
630 *
631 * Update soluton.
632 *
633  IF ( y_prec_state .LT. extra_y ) THEN
634  CALL daxpy( n, 1.0d+0, dy, 1, y( 1, j ), 1 )
635  ELSE
636  CALL dla_wwaddw( n, y( 1, j ), y_tail, dy )
637  END IF
638 
639  END DO
640 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
641  666 CONTINUE
642 *
643 * Set final_* when cnt hits ithresh.
644 *
645  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
646  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
647 *
648 * Compute error bounds
649 *
650  IF (n_norms .GE. 1) THEN
651  errs_n( j, la_linrx_err_i ) = final_dx_x / (1 - dxratmax)
652  END IF
653  IF ( n_norms .GE. 2 ) THEN
654  errs_c( j, la_linrx_err_i ) = final_dz_z / (1 - dzratmax)
655  END IF
656 *
657 * Compute componentwise relative backward error from formula
658 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
659 * where abs(Z) is the componentwise absolute value of the matrix
660 * or vector Z.
661 *
662 * Compute residual RES = B_s - op(A_s) * Y,
663 * op(A) = A, A**T, or A**H depending on TRANS (and type).
664 *
665  CALL dcopy( n, b( 1, j ), 1, res, 1 )
666  CALL dgemv( trans, n, n, -1.0d+0, a, lda, y(1,j), 1, 1.0d+0,
667  $ res, 1 )
668 
669  DO i = 1, n
670  ayb( i ) = abs( b( i, j ) )
671  END DO
672 *
673 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
674 *
675  CALL dla_geamv ( trans_type, n, n, 1.0d+0,
676  $ a, lda, y(1, j), 1, 1.0d+0, ayb, 1 )
677 
678  CALL dla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
679 *
680 * End of loop for each RHS.
681 *
682  END DO
683 *
684  RETURN
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:84
subroutine dgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DGEMV
Definition: dgemv.f:158
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:91
subroutine dgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
DGETRS
Definition: dgetrs.f:123
subroutine dla_wwaddw(N, X, Y, W)
DLA_WWADDW adds a vector into a doubled-single vector.
Definition: dla_wwaddw.f:83
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:176
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:103
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
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