LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine zla_gerfsx_extended ( integer PREC_TYPE, integer TRANS_TYPE, integer N, integer NRHS, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldaf, * ) AF, integer LDAF, integer, dimension( * ) IPIV, logical COLEQU, double precision, dimension( * ) C, complex*16, dimension( ldb, * ) B, integer LDB, complex*16, 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, complex*16, dimension( * ) RES, double precision, dimension( * ) AYB, complex*16, dimension( * ) DY, complex*16, dimension( * ) Y_TAIL, double precision RCOND, integer ITHRESH, double precision RTHRESH, double precision DZ_UB, logical IGNORE_CWISE, integer INFO )

ZLA_GERFSX_EXTENDED

Purpose:
``` ZLA_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 ZGERFSX 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDAF,N) The factors L and U from the factorization A = P*L*U as computed by ZGETRF.``` [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 ZGETRF; 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 COMPLEX*16 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 COMPLEX*16 array, dimension (LDY,NRHS) On entry, the solution matrix X, as computed by ZGETRS. 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 ZLA_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 COMPLEX*16 array, dimension (N) Workspace to hold the intermediate residual.``` [in] AYB ``` AYB is DOUBLE PRECISION array, dimension (N) Workspace.``` [in] DY ``` DY is COMPLEX*16 array, dimension (N) Workspace to hold the intermediate solution.``` [in] Y_TAIL ``` Y_TAIL is COMPLEX*16 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 ZGETRS had an illegal value```
Date
November 2011

Definition at line 400 of file zla_gerfsx_extended.f.

400 *
401 * -- LAPACK computational routine (version 3.4.0) --
402 * -- LAPACK is a software package provided by Univ. of Tennessee, --
403 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
404 * November 2011
405 *
406 * .. Scalar Arguments ..
407  INTEGER info, lda, ldaf, ldb, ldy, n, nrhs, prec_type,
408  \$ trans_type, n_norms
409  LOGICAL colequ, ignore_cwise
410  INTEGER ithresh
411  DOUBLE PRECISION rthresh, dz_ub
412 * ..
413 * .. Array Arguments
414  INTEGER ipiv( * )
415  COMPLEX*16 a( lda, * ), af( ldaf, * ), b( ldb, * ),
416  \$ y( ldy, * ), res( * ), dy( * ), y_tail( * )
417  DOUBLE PRECISION c( * ), ayb( * ), rcond, berr_out( * ),
418  \$ errs_n( nrhs, * ), 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  DOUBLE PRECISION 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  COMPLEX*16 zdum
432 * ..
433 * .. Parameters ..
434  INTEGER unstable_state, working_state, conv_state,
435  \$ noprog_state, base_residual, extra_residual,
436  \$ extra_y
437  parameter ( unstable_state = 0, working_state = 1,
438  \$ conv_state = 2,
439  \$ noprog_state = 3 )
440  parameter ( base_residual = 0, extra_residual = 1,
441  \$ extra_y = 2 )
442  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
443  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
444  INTEGER cmp_err_i, piv_growth_i
445  parameter ( final_nrm_err_i = 1, final_cmp_err_i = 2,
446  \$ berr_i = 3 )
447  parameter ( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
448  parameter ( cmp_rcond_i = 7, cmp_err_i = 8,
449  \$ piv_growth_i = 9 )
450  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
451  \$ la_linrx_cwise_i
452  parameter ( la_linrx_itref_i = 1,
453  \$ la_linrx_ithresh_i = 2 )
454  parameter ( la_linrx_cwise_i = 3 )
455  INTEGER la_linrx_trust_i, la_linrx_err_i,
456  \$ la_linrx_rcond_i
457  parameter ( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
458  parameter ( la_linrx_rcond_i = 3 )
459 * ..
460 * .. External Subroutines ..
461  EXTERNAL zaxpy, zcopy, zgetrs, zgemv, blas_zgemv_x,
462  \$ blas_zgemv2_x, zla_geamv, zla_wwaddw, dlamch,
464  DOUBLE PRECISION dlamch
465  CHARACTER chla_transtype
466 * ..
467 * .. Intrinsic Functions ..
468  INTRINSIC abs, max, min
469 * ..
470 * .. Statement Functions ..
471  DOUBLE PRECISION cabs1
472 * ..
473 * .. Statement Function Definitions ..
474  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
475 * ..
476 * .. Executable Statements ..
477 *
478  IF ( info.NE.0 ) RETURN
479  trans = chla_transtype(trans_type)
480  eps = dlamch( 'Epsilon' )
481  hugeval = dlamch( 'Overflow' )
482 * Force HUGEVAL to Inf
483  hugeval = hugeval * hugeval
484 * Using HUGEVAL may lead to spurious underflows.
485  incr_thresh = dble( n ) * eps
486 *
487  DO j = 1, nrhs
488  y_prec_state = extra_residual
489  IF ( y_prec_state .EQ. extra_y ) THEN
490  DO i = 1, n
491  y_tail( i ) = 0.0d+0
492  END DO
493  END IF
494
495  dxrat = 0.0d+0
496  dxratmax = 0.0d+0
497  dzrat = 0.0d+0
498  dzratmax = 0.0d+0
499  final_dx_x = hugeval
500  final_dz_z = hugeval
501  prevnormdx = hugeval
502  prev_dz_z = hugeval
503  dz_z = hugeval
504  dx_x = hugeval
505
506  x_state = working_state
507  z_state = unstable_state
508  incr_prec = .false.
509
510  DO cnt = 1, ithresh
511 *
512 * Compute residual RES = B_s - op(A_s) * Y,
513 * op(A) = A, A**T, or A**H depending on TRANS (and type).
514 *
515  CALL zcopy( n, b( 1, j ), 1, res, 1 )
516  IF ( y_prec_state .EQ. base_residual ) THEN
517  CALL zgemv( trans, n, n, (-1.0d+0,0.0d+0), a, lda,
518  \$ y( 1, j ), 1, (1.0d+0,0.0d+0), res, 1)
519  ELSE IF (y_prec_state .EQ. extra_residual) THEN
520  CALL blas_zgemv_x( trans_type, n, n, (-1.0d+0,0.0d+0), a,
521  \$ lda, y( 1, j ), 1, (1.0d+0,0.0d+0),
522  \$ res, 1, prec_type )
523  ELSE
524  CALL blas_zgemv2_x( trans_type, n, n, (-1.0d+0,0.0d+0),
525  \$ a, lda, y(1, j), y_tail, 1, (1.0d+0,0.0d+0), res, 1,
526  \$ prec_type)
527  END IF
528
529 ! XXX: RES is no longer needed.
530  CALL zcopy( n, res, 1, dy, 1 )
531  CALL zgetrs( trans, n, 1, af, ldaf, ipiv, dy, n, info )
532 *
533 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
534 *
535  normx = 0.0d+0
536  normy = 0.0d+0
537  normdx = 0.0d+0
538  dz_z = 0.0d+0
539  ymin = hugeval
540 *
541  DO i = 1, n
542  yk = cabs1( y( i, j ) )
543  dyk = cabs1( dy( i ) )
544
545  IF ( yk .NE. 0.0d+0 ) THEN
546  dz_z = max( dz_z, dyk / yk )
547  ELSE IF ( dyk .NE. 0.0d+0 ) THEN
548  dz_z = hugeval
549  END IF
550
551  ymin = min( ymin, yk )
552
553  normy = max( normy, yk )
554
555  IF ( colequ ) THEN
556  normx = max( normx, yk * c( i ) )
557  normdx = max( normdx, dyk * c( i ) )
558  ELSE
559  normx = normy
560  normdx = max(normdx, dyk)
561  END IF
562  END DO
563
564  IF ( normx .NE. 0.0d+0 ) THEN
565  dx_x = normdx / normx
566  ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
567  dx_x = 0.0d+0
568  ELSE
569  dx_x = hugeval
570  END IF
571
572  dxrat = normdx / prevnormdx
573  dzrat = dz_z / prev_dz_z
574 *
575 * Check termination criteria
576 *
577  IF (.NOT.ignore_cwise
578  \$ .AND. ymin*rcond .LT. incr_thresh*normy
579  \$ .AND. y_prec_state .LT. extra_y )
580  \$ incr_prec = .true.
581
582  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
583  \$ x_state = working_state
584  IF ( x_state .EQ. working_state ) THEN
585  IF (dx_x .LE. eps) THEN
586  x_state = conv_state
587  ELSE IF ( dxrat .GT. rthresh ) THEN
588  IF ( y_prec_state .NE. extra_y ) THEN
589  incr_prec = .true.
590  ELSE
591  x_state = noprog_state
592  END IF
593  ELSE
594  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
595  END IF
596  IF ( x_state .GT. working_state ) final_dx_x = dx_x
597  END IF
598
599  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
600  \$ z_state = working_state
601  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
602  \$ z_state = working_state
603  IF ( z_state .EQ. working_state ) THEN
604  IF ( dz_z .LE. eps ) THEN
605  z_state = conv_state
606  ELSE IF ( dz_z .GT. dz_ub ) THEN
607  z_state = unstable_state
608  dzratmax = 0.0d+0
609  final_dz_z = hugeval
610  ELSE IF ( dzrat .GT. rthresh ) THEN
611  IF ( y_prec_state .NE. extra_y ) THEN
612  incr_prec = .true.
613  ELSE
614  z_state = noprog_state
615  END IF
616  ELSE
617  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
618  END IF
619  IF ( z_state .GT. working_state ) final_dz_z = dz_z
620  END IF
621 *
622 * Exit if both normwise and componentwise stopped working,
623 * but if componentwise is unstable, let it go at least two
624 * iterations.
625 *
626  IF ( x_state.NE.working_state ) THEN
627  IF ( ignore_cwise ) GOTO 666
628  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
629  \$ GOTO 666
630  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
631  END IF
632
633  IF ( incr_prec ) THEN
634  incr_prec = .false.
635  y_prec_state = y_prec_state + 1
636  DO i = 1, n
637  y_tail( i ) = 0.0d+0
638  END DO
639  END IF
640
641  prevnormdx = normdx
642  prev_dz_z = dz_z
643 *
644 * Update soluton.
645 *
646  IF ( y_prec_state .LT. extra_y ) THEN
647  CALL zaxpy( n, (1.0d+0,0.0d+0), dy, 1, y(1,j), 1 )
648  ELSE
649  CALL zla_wwaddw( n, y( 1, j ), y_tail, dy )
650  END IF
651
652  END DO
653 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
654  666 CONTINUE
655 *
656 * Set final_* when cnt hits ithresh
657 *
658  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
659  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
660 *
661 * Compute error bounds
662 *
663  IF (n_norms .GE. 1) THEN
664  errs_n( j, la_linrx_err_i ) = final_dx_x / (1 - dxratmax)
665
666  END IF
667  IF ( n_norms .GE. 2 ) THEN
668  errs_c( j, la_linrx_err_i ) = final_dz_z / (1 - dzratmax)
669  END IF
670 *
671 * Compute componentwise relative backward error from formula
672 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
673 * where abs(Z) is the componentwise absolute value of the matrix
674 * or vector Z.
675 *
676 * Compute residual RES = B_s - op(A_s) * Y,
677 * op(A) = A, A**T, or A**H depending on TRANS (and type).
678 *
679  CALL zcopy( n, b( 1, j ), 1, res, 1 )
680  CALL zgemv( trans, n, n, (-1.0d+0,0.0d+0), a, lda, y(1,j), 1,
681  \$ (1.0d+0,0.0d+0), res, 1 )
682
683  DO i = 1, n
684  ayb( i ) = cabs1( b( i, j ) )
685  END DO
686 *
687 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
688 *
689  CALL zla_geamv ( trans_type, n, n, 1.0d+0,
690  \$ a, lda, y(1, j), 1, 1.0d+0, ayb, 1 )
691
692  CALL zla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
693 *
694 * End of loop for each RHS.
695 *
696  END DO
697 *
698  RETURN
subroutine zgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZGETRS
Definition: zgetrs.f:123
subroutine zla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
ZLA_LIN_BERR computes a component-wise relative backward error.
Definition: zla_lin_berr.f:103
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:52
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160