LAPACK  3.10.0
LAPACK: Linear Algebra PACKage

◆ cla_porfsx_extended()

subroutine cla_porfsx_extended ( integer  PREC_TYPE,
character  UPLO,
integer  N,
integer  NRHS,
complex, dimension( lda, * )  A,
integer  LDA,
complex, dimension( ldaf, * )  AF,
integer  LDAF,
logical  COLEQU,
real, dimension( * )  C,
complex, dimension( ldb, * )  B,
integer  LDB,
complex, dimension( ldy, * )  Y,
integer  LDY,
real, dimension( * )  BERR_OUT,
integer  N_NORMS,
real, dimension( nrhs, * )  ERR_BNDS_NORM,
real, dimension( nrhs, * )  ERR_BNDS_COMP,
complex, dimension( * )  RES,
real, dimension( * )  AYB,
complex, dimension( * )  DY,
complex, dimension( * )  Y_TAIL,
real  RCOND,
integer  ITHRESH,
real  RTHRESH,
real  DZ_UB,
logical  IGNORE_CWISE,
integer  INFO 
)

CLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

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Purpose:
 CLA_PORFSX_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 CPORFSX to perform iterative refinement.
 In addition to normwise error bound, the code provides maximum
 componentwise error bound if possible. See comments for ERR_BNDS_NORM
 and ERR_BNDS_COMP for details of the error bounds. Note that this
 subroutine is only resonsible for setting the second fields of
 ERR_BNDS_NORM and ERR_BNDS_COMP.
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]UPLO
          UPLO is CHARACTER*1
       = 'U':  Upper triangle of A is stored;
       = 'L':  Lower triangle of A is stored.
[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 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 array, dimension (LDAF,N)
     The triangular factor U or L from the Cholesky factorization
     A = U**T*U or A = L*L**T, as computed by CPOTRF.
[in]LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
[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 COMPLEX 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 array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by CPOTRS.
     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 CLA_LIN_BERR.
[in]N_NORMS
          N_NORMS is INTEGER
     Determines which error bounds to return (see ERR_BNDS_NORM
     and ERR_BNDS_COMP).
     If N_NORMS >= 1 return normwise error bounds.
     If N_NORMS >= 2 return componentwise error bounds.
[in,out]ERR_BNDS_NORM
          ERR_BNDS_NORM 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 ERR_BNDS_NORM(i,:) corresponds to the ith
     right-hand side.

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

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

     The second index in ERR_BNDS_COMP(:,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 array, dimension (N)
     Workspace to hold the intermediate residual.
[in]AYB
          AYB is REAL array, dimension (N)
     Workspace.
[in]DY
          DY is COMPLEX array, dimension (N)
     Workspace to hold the intermediate solution.
[in]Y_TAIL
          Y_TAIL is COMPLEX 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
     ERR_BNDS_NORM and ERR_BNDS_COMP 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 CPOTRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 380 of file cla_porfsx_extended.f.

387 *
388 * -- LAPACK computational routine --
389 * -- LAPACK is a software package provided by Univ. of Tennessee, --
390 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
391 *
392 * .. Scalar Arguments ..
393  INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
394  $ N_NORMS, ITHRESH
395  CHARACTER UPLO
396  LOGICAL COLEQU, IGNORE_CWISE
397  REAL RTHRESH, DZ_UB
398 * ..
399 * .. Array Arguments ..
400  COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
401  $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
402  REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ),
403  $ ERR_BNDS_NORM( NRHS, * ),
404  $ ERR_BNDS_COMP( NRHS, * )
405 * ..
406 *
407 * =====================================================================
408 *
409 * .. Local Scalars ..
410  INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE,
411  $ Y_PREC_STATE
412  REAL YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
413  $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
414  $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
415  $ EPS, HUGEVAL, INCR_THRESH
416  LOGICAL INCR_PREC
417  COMPLEX ZDUM
418 * ..
419 * .. Parameters ..
420  INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
421  $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
422  $ EXTRA_Y
423  parameter( unstable_state = 0, working_state = 1,
424  $ conv_state = 2, noprog_state = 3 )
425  parameter( base_residual = 0, extra_residual = 1,
426  $ extra_y = 2 )
427  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
428  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
429  INTEGER CMP_ERR_I, PIV_GROWTH_I
430  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
431  $ berr_i = 3 )
432  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
433  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
434  $ piv_growth_i = 9 )
435  INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
436  $ LA_LINRX_CWISE_I
437  parameter( la_linrx_itref_i = 1,
438  $ la_linrx_ithresh_i = 2 )
439  parameter( la_linrx_cwise_i = 3 )
440  INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
441  $ LA_LINRX_RCOND_I
442  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
443  parameter( la_linrx_rcond_i = 3 )
444 * ..
445 * .. External Functions ..
446  LOGICAL LSAME
447  EXTERNAL ilauplo
448  INTEGER ILAUPLO
449 * ..
450 * .. External Subroutines ..
451  EXTERNAL caxpy, ccopy, cpotrs, chemv, blas_chemv_x,
452  $ blas_chemv2_x, cla_heamv, cla_wwaddw,
454  REAL SLAMCH
455 * ..
456 * .. Intrinsic Functions ..
457  INTRINSIC abs, real, aimag, max, min
458 * ..
459 * .. Statement Functions ..
460  REAL CABS1
461 * ..
462 * .. Statement Function Definitions ..
463  cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
464 * ..
465 * .. Executable Statements ..
466 *
467  IF (info.NE.0) RETURN
468  eps = slamch( 'Epsilon' )
469  hugeval = slamch( 'Overflow' )
470 * Force HUGEVAL to Inf
471  hugeval = hugeval * hugeval
472 * Using HUGEVAL may lead to spurious underflows.
473  incr_thresh = real(n) * eps
474 
475  IF (lsame(uplo, 'L')) THEN
476  uplo2 = ilauplo( 'L' )
477  ELSE
478  uplo2 = ilauplo( 'U' )
479  ENDIF
480 
481  DO j = 1, nrhs
482  y_prec_state = extra_residual
483  IF (y_prec_state .EQ. extra_y) THEN
484  DO i = 1, n
485  y_tail( i ) = 0.0
486  END DO
487  END IF
488 
489  dxrat = 0.0
490  dxratmax = 0.0
491  dzrat = 0.0
492  dzratmax = 0.0
493  final_dx_x = hugeval
494  final_dz_z = hugeval
495  prevnormdx = hugeval
496  prev_dz_z = hugeval
497  dz_z = hugeval
498  dx_x = hugeval
499 
500  x_state = working_state
501  z_state = unstable_state
502  incr_prec = .false.
503 
504  DO cnt = 1, ithresh
505 *
506 * Compute residual RES = B_s - op(A_s) * Y,
507 * op(A) = A, A**T, or A**H depending on TRANS (and type).
508 *
509  CALL ccopy( n, b( 1, j ), 1, res, 1 )
510  IF (y_prec_state .EQ. base_residual) THEN
511  CALL chemv(uplo, n, cmplx(-1.0), a, lda, y(1,j), 1,
512  $ cmplx(1.0), res, 1)
513  ELSE IF (y_prec_state .EQ. extra_residual) THEN
514  CALL blas_chemv_x(uplo2, n, cmplx(-1.0), a, lda,
515  $ y( 1, j ), 1, cmplx(1.0), res, 1, prec_type)
516  ELSE
517  CALL blas_chemv2_x(uplo2, n, cmplx(-1.0), a, lda,
518  $ y(1, j), y_tail, 1, cmplx(1.0), res, 1, prec_type)
519  END IF
520 
521 ! XXX: RES is no longer needed.
522  CALL ccopy( n, res, 1, dy, 1 )
523  CALL cpotrs( uplo, n, 1, af, ldaf, dy, n, info)
524 *
525 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
526 *
527  normx = 0.0
528  normy = 0.0
529  normdx = 0.0
530  dz_z = 0.0
531  ymin = hugeval
532 
533  DO i = 1, n
534  yk = cabs1(y(i, j))
535  dyk = cabs1(dy(i))
536 
537  IF (yk .NE. 0.0) THEN
538  dz_z = max( dz_z, dyk / yk )
539  ELSE IF (dyk .NE. 0.0) THEN
540  dz_z = hugeval
541  END IF
542 
543  ymin = min( ymin, yk )
544 
545  normy = max( normy, yk )
546 
547  IF ( colequ ) THEN
548  normx = max(normx, yk * c(i))
549  normdx = max(normdx, dyk * c(i))
550  ELSE
551  normx = normy
552  normdx = max(normdx, dyk)
553  END IF
554  END DO
555 
556  IF (normx .NE. 0.0) THEN
557  dx_x = normdx / normx
558  ELSE IF (normdx .EQ. 0.0) THEN
559  dx_x = 0.0
560  ELSE
561  dx_x = hugeval
562  END IF
563 
564  dxrat = normdx / prevnormdx
565  dzrat = dz_z / prev_dz_z
566 *
567 * Check termination criteria.
568 *
569  IF (ymin*rcond .LT. incr_thresh*normy
570  $ .AND. y_prec_state .LT. extra_y)
571  $ incr_prec = .true.
572 
573  IF (x_state .EQ. noprog_state .AND. dxrat .LE. rthresh)
574  $ x_state = working_state
575  IF (x_state .EQ. working_state) THEN
576  IF (dx_x .LE. eps) THEN
577  x_state = conv_state
578  ELSE IF (dxrat .GT. rthresh) THEN
579  IF (y_prec_state .NE. extra_y) THEN
580  incr_prec = .true.
581  ELSE
582  x_state = noprog_state
583  END IF
584  ELSE
585  IF (dxrat .GT. dxratmax) dxratmax = dxrat
586  END IF
587  IF (x_state .GT. working_state) final_dx_x = dx_x
588  END IF
589 
590  IF (z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub)
591  $ z_state = working_state
592  IF (z_state .EQ. noprog_state .AND. dzrat .LE. rthresh)
593  $ z_state = working_state
594  IF (z_state .EQ. working_state) THEN
595  IF (dz_z .LE. eps) THEN
596  z_state = conv_state
597  ELSE IF (dz_z .GT. dz_ub) THEN
598  z_state = unstable_state
599  dzratmax = 0.0
600  final_dz_z = hugeval
601  ELSE IF (dzrat .GT. rthresh) THEN
602  IF (y_prec_state .NE. extra_y) THEN
603  incr_prec = .true.
604  ELSE
605  z_state = noprog_state
606  END IF
607  ELSE
608  IF (dzrat .GT. dzratmax) dzratmax = dzrat
609  END IF
610  IF (z_state .GT. working_state) final_dz_z = dz_z
611  END IF
612 
613  IF ( x_state.NE.working_state.AND.
614  $ (ignore_cwise.OR.z_state.NE.working_state) )
615  $ GOTO 666
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 ) = 0.0
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 caxpy( n, cmplx(1.0), dy, 1, y(1,j), 1 )
632  ELSE
633  CALL cla_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  err_bnds_norm( j, la_linrx_err_i ) =
649  $ final_dx_x / (1 - dxratmax)
650  END IF
651  IF (n_norms .GE. 2) THEN
652  err_bnds_comp( j, la_linrx_err_i ) =
653  $ final_dz_z / (1 - dzratmax)
654  END IF
655 *
656 * Compute componentwise relative backward error from formula
657 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
658 * where abs(Z) is the componentwise absolute value of the matrix
659 * or vector Z.
660 *
661 * Compute residual RES = B_s - op(A_s) * Y,
662 * op(A) = A, A**T, or A**H depending on TRANS (and type).
663 *
664  CALL ccopy( n, b( 1, j ), 1, res, 1 )
665  CALL chemv(uplo, n, cmplx(-1.0), a, lda, y(1,j), 1, cmplx(1.0),
666  $ res, 1)
667 
668  DO i = 1, n
669  ayb( i ) = cabs1( b( i, j ) )
670  END DO
671 *
672 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
673 *
674  CALL cla_heamv (uplo2, n, 1.0,
675  $ a, lda, y(1, j), 1, 1.0, ayb, 1)
676 
677  CALL cla_lin_berr (n, n, 1, res, ayb, berr_out(j))
678 *
679 * End of loop for each RHS.
680 *
681  END DO
682 *
683  RETURN
684 *
685 * End of CLA_PORFSX_EXTENDED
686 *
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
integer function ilauplo(UPLO)
ILAUPLO
Definition: ilauplo.f:58
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:88
subroutine chemv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CHEMV
Definition: chemv.f:154
subroutine cla_heamv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CLA_HEAMV computes a matrix-vector product using a Hermitian indefinite matrix to calculate error bou...
Definition: cla_heamv.f:178
subroutine cla_wwaddw(N, X, Y, W)
CLA_WWADDW adds a vector into a doubled-single vector.
Definition: cla_wwaddw.f:81
subroutine cla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
CLA_LIN_BERR computes a component-wise relative backward error.
Definition: cla_lin_berr.f:101
subroutine cpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
CPOTRS
Definition: cpotrs.f:110
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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