LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ sla_gerfsx_extended()

subroutine sla_gerfsx_extended ( integer  PREC_TYPE,
integer  TRANS_TYPE,
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, * )  ERRS_N,
real, dimension( nrhs, * )  ERRS_C,
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_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:
 SLA_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 SGERFSX 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 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 factors L and U from the factorization
     A = P*L*U as computed by SGETRF.
[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 SGETRF; 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 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 SGETRS.
     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 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 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 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 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
     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 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
     ERRS_N and ERRS_C 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 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 SGETRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
December 2016

Definition at line 400 of file sla_gerfsx_extended.f.

400 *
401 * -- LAPACK computational routine (version 3.7.0) --
402 * -- LAPACK is a software package provided by Univ. of Tennessee, --
403 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
404 * December 2016
405 *
406 * .. Scalar Arguments ..
407  INTEGER info, lda, ldaf, ldb, ldy, n, nrhs, prec_type,
408  $ trans_type, n_norms, ithresh
409  LOGICAL colequ, ignore_cwise
410  REAL rthresh, dz_ub
411 * ..
412 * .. Array Arguments ..
413  INTEGER ipiv( * )
414  REAL a( lda, * ), af( ldaf, * ), b( ldb, * ),
415  $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
416  REAL c( * ), ayb( * ), rcond, berr_out( * ),
417  $ errs_n( nrhs, * ),
418  $ errs_c( nrhs, * )
419 * ..
420 *
421 * =====================================================================
422 *
423 * .. Local Scalars ..
424  CHARACTER trans
425  INTEGER cnt, i, j, x_state, z_state, y_prec_state
426  REAL yk, dyk, ymin, normy, normx, normdx, dxrat,
427  $ dzrat, prevnormdx, prev_dz_z, dxratmax,
428  $ dzratmax, dx_x, dz_z, final_dx_x, final_dz_z,
429  $ eps, hugeval, incr_thresh
430  LOGICAL incr_prec
431 * ..
432 * .. Parameters ..
433  INTEGER unstable_state, working_state, conv_state,
434  $ noprog_state, base_residual, extra_residual,
435  $ extra_y
436  parameter( unstable_state = 0, working_state = 1,
437  $ conv_state = 2, noprog_state = 3 )
438  parameter( base_residual = 0, extra_residual = 1,
439  $ extra_y = 2 )
440  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
441  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
442  INTEGER cmp_err_i, piv_growth_i
443  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
444  $ berr_i = 3 )
445  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
446  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
447  $ piv_growth_i = 9 )
448  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
449  $ la_linrx_cwise_i
450  parameter( la_linrx_itref_i = 1,
451  $ la_linrx_ithresh_i = 2 )
452  parameter( la_linrx_cwise_i = 3 )
453  INTEGER la_linrx_trust_i, la_linrx_err_i,
454  $ la_linrx_rcond_i
455  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
456  parameter( la_linrx_rcond_i = 3 )
457 * ..
458 * .. External Subroutines ..
459  EXTERNAL saxpy, scopy, sgetrs, sgemv, blas_sgemv_x,
460  $ blas_sgemv2_x, sla_geamv, sla_wwaddw, slamch,
462  REAL slamch
463  CHARACTER chla_transtype
464 * ..
465 * .. Intrinsic Functions ..
466  INTRINSIC abs, max, min
467 * ..
468 * .. Executable Statements ..
469 *
470  IF ( info.NE.0 ) RETURN
471  trans = chla_transtype(trans_type)
472  eps = slamch( 'Epsilon' )
473  hugeval = slamch( 'Overflow' )
474 * Force HUGEVAL to Inf
475  hugeval = hugeval * hugeval
476 * Using HUGEVAL may lead to spurious underflows.
477  incr_thresh = REAL( N ) * eps
478 *
479  DO j = 1, nrhs
480  y_prec_state = extra_residual
481  IF ( y_prec_state .EQ. extra_y ) THEN
482  DO i = 1, n
483  y_tail( i ) = 0.0
484  END DO
485  END IF
486 
487  dxrat = 0.0
488  dxratmax = 0.0
489  dzrat = 0.0
490  dzratmax = 0.0
491  final_dx_x = hugeval
492  final_dz_z = hugeval
493  prevnormdx = hugeval
494  prev_dz_z = hugeval
495  dz_z = hugeval
496  dx_x = hugeval
497 
498  x_state = working_state
499  z_state = unstable_state
500  incr_prec = .false.
501 
502  DO cnt = 1, ithresh
503 *
504 * Compute residual RES = B_s - op(A_s) * Y,
505 * op(A) = A, A**T, or A**H depending on TRANS (and type).
506 *
507  CALL scopy( n, b( 1, j ), 1, res, 1 )
508  IF ( y_prec_state .EQ. base_residual ) THEN
509  CALL sgemv( trans, n, n, -1.0, a, lda, y( 1, j ), 1,
510  $ 1.0, res, 1 )
511  ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
512  CALL blas_sgemv_x( trans_type, n, n, -1.0, a, lda,
513  $ y( 1, j ), 1, 1.0, res, 1, prec_type )
514  ELSE
515  CALL blas_sgemv2_x( trans_type, n, n, -1.0, a, lda,
516  $ y( 1, j ), y_tail, 1, 1.0, res, 1, prec_type )
517  END IF
518 
519 ! XXX: RES is no longer needed.
520  CALL scopy( n, res, 1, dy, 1 )
521  CALL sgetrs( trans, n, 1, af, ldaf, ipiv, dy, n, info )
522 *
523 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
524 *
525  normx = 0.0
526  normy = 0.0
527  normdx = 0.0
528  dz_z = 0.0
529  ymin = hugeval
530 *
531  DO i = 1, n
532  yk = abs( y( i, j ) )
533  dyk = abs( dy( i ) )
534 
535  IF ( yk .NE. 0.0 ) THEN
536  dz_z = max( dz_z, dyk / yk )
537  ELSE IF ( dyk .NE. 0.0 ) THEN
538  dz_z = hugeval
539  END IF
540 
541  ymin = min( ymin, yk )
542 
543  normy = max( normy, yk )
544 
545  IF ( colequ ) THEN
546  normx = max( normx, yk * c( i ) )
547  normdx = max( normdx, dyk * c( i ) )
548  ELSE
549  normx = normy
550  normdx = max( normdx, dyk )
551  END IF
552  END DO
553 
554  IF ( normx .NE. 0.0 ) THEN
555  dx_x = normdx / normx
556  ELSE IF ( normdx .EQ. 0.0 ) THEN
557  dx_x = 0.0
558  ELSE
559  dx_x = hugeval
560  END IF
561 
562  dxrat = normdx / prevnormdx
563  dzrat = dz_z / prev_dz_z
564 *
565 * Check termination criteria
566 *
567  IF (.NOT.ignore_cwise
568  $ .AND. ymin*rcond .LT. incr_thresh*normy
569  $ .AND. y_prec_state .LT. extra_y)
570  $ incr_prec = .true.
571 
572  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
573  $ x_state = working_state
574  IF ( x_state .EQ. working_state ) THEN
575  IF ( dx_x .LE. eps ) THEN
576  x_state = conv_state
577  ELSE IF ( dxrat .GT. rthresh ) THEN
578  IF ( y_prec_state .NE. extra_y ) THEN
579  incr_prec = .true.
580  ELSE
581  x_state = noprog_state
582  END IF
583  ELSE
584  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
585  END IF
586  IF ( x_state .GT. working_state ) final_dx_x = dx_x
587  END IF
588 
589  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
590  $ z_state = working_state
591  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
592  $ z_state = working_state
593  IF ( z_state .EQ. working_state ) THEN
594  IF ( dz_z .LE. eps ) THEN
595  z_state = conv_state
596  ELSE IF ( dz_z .GT. dz_ub ) THEN
597  z_state = unstable_state
598  dzratmax = 0.0
599  final_dz_z = hugeval
600  ELSE IF ( dzrat .GT. rthresh ) THEN
601  IF ( y_prec_state .NE. extra_y ) THEN
602  incr_prec = .true.
603  ELSE
604  z_state = noprog_state
605  END IF
606  ELSE
607  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
608  END IF
609  IF ( z_state .GT. working_state ) final_dz_z = dz_z
610  END IF
611 *
612 * Exit if both normwise and componentwise stopped working,
613 * but if componentwise is unstable, let it go at least two
614 * iterations.
615 *
616  IF ( x_state.NE.working_state ) THEN
617  IF ( ignore_cwise) GOTO 666
618  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
619  $ GOTO 666
620  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
621  END IF
622 
623  IF ( incr_prec ) THEN
624  incr_prec = .false.
625  y_prec_state = y_prec_state + 1
626  DO i = 1, n
627  y_tail( i ) = 0.0
628  END DO
629  END IF
630 
631  prevnormdx = normdx
632  prev_dz_z = dz_z
633 *
634 * Update soluton.
635 *
636  IF ( y_prec_state .LT. extra_y ) THEN
637  CALL saxpy( n, 1.0, dy, 1, y( 1, j ), 1 )
638  ELSE
639  CALL sla_wwaddw( n, y( 1, j ), y_tail, dy )
640  END IF
641 
642  END DO
643 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
644  666 CONTINUE
645 *
646 * Set final_* when cnt hits ithresh.
647 *
648  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
649  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
650 *
651 * Compute error bounds
652 *
653  IF (n_norms .GE. 1) THEN
654  errs_n( j, la_linrx_err_i ) =
655  $ final_dx_x / (1 - dxratmax)
656  END IF
657  IF ( n_norms .GE. 2 ) THEN
658  errs_c( j, la_linrx_err_i ) =
659  $ final_dz_z / (1 - dzratmax)
660  END IF
661 *
662 * Compute componentwise relative backward error from formula
663 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
664 * where abs(Z) is the componentwise absolute value of the matrix
665 * or vector Z.
666 *
667 * Compute residual RES = B_s - op(A_s) * Y,
668 * op(A) = A, A**T, or A**H depending on TRANS (and type).
669 *
670  CALL scopy( n, b( 1, j ), 1, res, 1 )
671  CALL sgemv( trans, n, n, -1.0, a, lda, y(1,j), 1, 1.0, res, 1 )
672 
673  DO i = 1, n
674  ayb( i ) = abs( b( i, j ) )
675  END DO
676 *
677 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
678 *
679  CALL sla_geamv ( trans_type, n, n, 1.0,
680  $ a, lda, y(1, j), 1, 1.0, ayb, 1 )
681 
682  CALL sla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
683 *
684 * End of loop for each RHS.
685 *
686  END DO
687 *
688  RETURN
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:123
subroutine sla_wwaddw(N, X, Y, W)
SLA_WWADDW adds a vector into a doubled-single vector.
Definition: sla_wwaddw.f:83
subroutine sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:158
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:103
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:91
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84
subroutine sla_geamv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds...
Definition: sla_geamv.f:176
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