LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ 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.

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

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 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 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 < 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 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 SGETRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 391 of file sla_gerfsx_extended.f.

398*
399* -- LAPACK computational routine --
400* -- LAPACK is a software package provided by Univ. of Tennessee, --
401* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
402*
403* .. Scalar Arguments ..
404 INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
405 $ TRANS_TYPE, N_NORMS, ITHRESH
406 LOGICAL COLEQU, IGNORE_CWISE
407 REAL RTHRESH, DZ_UB
408* ..
409* .. Array Arguments ..
410 INTEGER IPIV( * )
411 REAL A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
412 $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
413 REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ),
414 $ ERRS_N( NRHS, * ),
415 $ 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 REAL 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 saxpy, scopy, sgetrs, sgemv, blas_sgemv_x,
457 $ blas_sgemv2_x, sla_geamv, sla_wwaddw, slamch,
459 REAL SLAMCH
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 = slamch( 'Epsilon' )
470 hugeval = slamch( 'Overflow' )
471* Force HUGEVAL to Inf
472 hugeval = hugeval * hugeval
473* Using HUGEVAL may lead to spurious underflows.
474 incr_thresh = real( 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 ) = 0.0
481 END DO
482 END IF
483
484 dxrat = 0.0
485 dxratmax = 0.0
486 dzrat = 0.0
487 dzratmax = 0.0
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 scopy( n, b( 1, j ), 1, res, 1 )
505 IF ( y_prec_state .EQ. base_residual ) THEN
506 CALL sgemv( trans, n, n, -1.0, a, lda, y( 1, j ), 1,
507 $ 1.0, res, 1 )
508 ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
509 CALL blas_sgemv_x( trans_type, n, n, -1.0, a, lda,
510 $ y( 1, j ), 1, 1.0, res, 1, prec_type )
511 ELSE
512 CALL blas_sgemv2_x( trans_type, n, n, -1.0, a, lda,
513 $ y( 1, j ), y_tail, 1, 1.0, res, 1, prec_type )
514 END IF
515
516! XXX: RES is no longer needed.
517 CALL scopy( n, res, 1, dy, 1 )
518 CALL sgetrs( 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 = 0.0
523 normy = 0.0
524 normdx = 0.0
525 dz_z = 0.0
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. 0.0 ) THEN
533 dz_z = max( dz_z, dyk / yk )
534 ELSE IF ( dyk .NE. 0.0 ) 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. 0.0 ) THEN
552 dx_x = normdx / normx
553 ELSE IF ( normdx .EQ. 0.0 ) THEN
554 dx_x = 0.0
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 = 0.0
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 ) = 0.0
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 saxpy( n, 1.0, dy, 1, y( 1, j ), 1 )
635 ELSE
636 CALL sla_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 ) =
652 $ final_dx_x / (1 - dxratmax)
653 END IF
654 IF ( n_norms .GE. 2 ) THEN
655 errs_c( j, la_linrx_err_i ) =
656 $ final_dz_z / (1 - dzratmax)
657 END IF
658*
659* Compute componentwise relative backward error from formula
660* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
661* where abs(Z) is the componentwise absolute value of the matrix
662* or vector Z.
663*
664* Compute residual RES = B_s - op(A_s) * Y,
665* op(A) = A, A**T, or A**H depending on TRANS (and type).
666*
667 CALL scopy( n, b( 1, j ), 1, res, 1 )
668 CALL sgemv( trans, n, n, -1.0, a, lda, y(1,j), 1, 1.0, res, 1 )
669
670 DO i = 1, n
671 ayb( i ) = abs( b( i, j ) )
672 END DO
673*
674* Compute abs(op(A_s))*abs(Y) + abs(B_s).
675*
676 CALL sla_geamv ( trans_type, n, n, 1.0,
677 $ a, lda, y(1, j), 1, 1.0, ayb, 1 )
678
679 CALL sla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
680*
681* End of loop for each RHS.
682*
683 END DO
684*
685 RETURN
686*
687* End of SLA_GERFSX_EXTENDED
688*
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
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:174
subroutine sgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
SGETRS
Definition: sgetrs.f:121
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 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 sgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGEMV
Definition: sgemv.f:156
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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