LAPACK  3.8.0
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', '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 .LT. 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 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 CPOTRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
June 2017

Definition at line 389 of file cla_porfsx_extended.f.

389 *
390 * -- LAPACK computational routine (version 3.7.1) --
391 * -- LAPACK is a software package provided by Univ. of Tennessee, --
392 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
393 * June 2017
394 *
395 * .. Scalar Arguments ..
396  INTEGER info, lda, ldaf, ldb, ldy, n, nrhs, prec_type,
397  $ n_norms, ithresh
398  CHARACTER uplo
399  LOGICAL colequ, ignore_cwise
400  REAL rthresh, dz_ub
401 * ..
402 * .. Array Arguments ..
403  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
404  $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
405  REAL c( * ), ayb( * ), rcond, berr_out( * ),
406  $ err_bnds_norm( nrhs, * ),
407  $ err_bnds_comp( nrhs, * )
408 * ..
409 *
410 * =====================================================================
411 *
412 * .. Local Scalars ..
413  INTEGER uplo2, cnt, i, j, x_state, z_state,
414  $ y_prec_state
415  REAL yk, dyk, ymin, normy, normx, normdx, dxrat,
416  $ dzrat, prevnormdx, prev_dz_z, dxratmax,
417  $ dzratmax, dx_x, dz_z, final_dx_x, final_dz_z,
418  $ eps, hugeval, incr_thresh
419  LOGICAL incr_prec
420  COMPLEX zdum
421 * ..
422 * .. Parameters ..
423  INTEGER unstable_state, working_state, conv_state,
424  $ noprog_state, base_residual, extra_residual,
425  $ extra_y
426  parameter( unstable_state = 0, working_state = 1,
427  $ conv_state = 2, noprog_state = 3 )
428  parameter( base_residual = 0, extra_residual = 1,
429  $ extra_y = 2 )
430  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
431  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
432  INTEGER cmp_err_i, piv_growth_i
433  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
434  $ berr_i = 3 )
435  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
436  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
437  $ piv_growth_i = 9 )
438  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
439  $ la_linrx_cwise_i
440  parameter( la_linrx_itref_i = 1,
441  $ la_linrx_ithresh_i = 2 )
442  parameter( la_linrx_cwise_i = 3 )
443  INTEGER la_linrx_trust_i, la_linrx_err_i,
444  $ la_linrx_rcond_i
445  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
446  parameter( la_linrx_rcond_i = 3 )
447 * ..
448 * .. External Functions ..
449  LOGICAL lsame
450  EXTERNAL ilauplo
451  INTEGER ilauplo
452 * ..
453 * .. External Subroutines ..
454  EXTERNAL caxpy, ccopy, cpotrs, chemv, blas_chemv_x,
455  $ blas_chemv2_x, cla_heamv, cla_wwaddw,
457  REAL slamch
458 * ..
459 * .. Intrinsic Functions ..
460  INTRINSIC abs, REAL, aimag, max, min
461 * ..
462 * .. Statement Functions ..
463  REAL cabs1
464 * ..
465 * .. Statement Function Definitions ..
466  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( aimag( zdum ) )
467 * ..
468 * .. Executable Statements ..
469 *
470  IF (info.NE.0) RETURN
471  eps = slamch( 'Epsilon' )
472  hugeval = slamch( 'Overflow' )
473 * Force HUGEVAL to Inf
474  hugeval = hugeval * hugeval
475 * Using HUGEVAL may lead to spurious underflows.
476  incr_thresh = REAL(N) * eps
477 
478  IF (lsame(uplo, 'L')) THEN
479  uplo2 = ilauplo( 'L' )
480  ELSE
481  uplo2 = ilauplo( 'U' )
482  ENDIF
483 
484  DO j = 1, nrhs
485  y_prec_state = extra_residual
486  IF (y_prec_state .EQ. extra_y) THEN
487  DO i = 1, n
488  y_tail( i ) = 0.0
489  END DO
490  END IF
491 
492  dxrat = 0.0
493  dxratmax = 0.0
494  dzrat = 0.0
495  dzratmax = 0.0
496  final_dx_x = hugeval
497  final_dz_z = hugeval
498  prevnormdx = hugeval
499  prev_dz_z = hugeval
500  dz_z = hugeval
501  dx_x = hugeval
502 
503  x_state = working_state
504  z_state = unstable_state
505  incr_prec = .false.
506 
507  DO cnt = 1, ithresh
508 *
509 * Compute residual RES = B_s - op(A_s) * Y,
510 * op(A) = A, A**T, or A**H depending on TRANS (and type).
511 *
512  CALL ccopy( n, b( 1, j ), 1, res, 1 )
513  IF (y_prec_state .EQ. base_residual) THEN
514  CALL chemv(uplo, n, cmplx(-1.0), a, lda, y(1,j), 1,
515  $ cmplx(1.0), res, 1)
516  ELSE IF (y_prec_state .EQ. extra_residual) THEN
517  CALL blas_chemv_x(uplo2, n, cmplx(-1.0), a, lda,
518  $ y( 1, j ), 1, cmplx(1.0), res, 1, prec_type)
519  ELSE
520  CALL blas_chemv2_x(uplo2, n, cmplx(-1.0), a, lda,
521  $ y(1, j), y_tail, 1, cmplx(1.0), res, 1, prec_type)
522  END IF
523 
524 ! XXX: RES is no longer needed.
525  CALL ccopy( n, res, 1, dy, 1 )
526  CALL cpotrs( uplo, n, 1, af, ldaf, dy, n, info)
527 *
528 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
529 *
530  normx = 0.0
531  normy = 0.0
532  normdx = 0.0
533  dz_z = 0.0
534  ymin = hugeval
535 
536  DO i = 1, n
537  yk = cabs1(y(i, j))
538  dyk = cabs1(dy(i))
539 
540  IF (yk .NE. 0.0) THEN
541  dz_z = max( dz_z, dyk / yk )
542  ELSE IF (dyk .NE. 0.0) THEN
543  dz_z = hugeval
544  END IF
545 
546  ymin = min( ymin, yk )
547 
548  normy = max( normy, yk )
549 
550  IF ( colequ ) THEN
551  normx = max(normx, yk * c(i))
552  normdx = max(normdx, dyk * c(i))
553  ELSE
554  normx = normy
555  normdx = max(normdx, dyk)
556  END IF
557  END DO
558 
559  IF (normx .NE. 0.0) THEN
560  dx_x = normdx / normx
561  ELSE IF (normdx .EQ. 0.0) THEN
562  dx_x = 0.0
563  ELSE
564  dx_x = hugeval
565  END IF
566 
567  dxrat = normdx / prevnormdx
568  dzrat = dz_z / prev_dz_z
569 *
570 * Check termination criteria.
571 *
572  IF (ymin*rcond .LT. incr_thresh*normy
573  $ .AND. y_prec_state .LT. extra_y)
574  $ incr_prec = .true.
575 
576  IF (x_state .EQ. noprog_state .AND. dxrat .LE. rthresh)
577  $ x_state = working_state
578  IF (x_state .EQ. working_state) THEN
579  IF (dx_x .LE. eps) THEN
580  x_state = conv_state
581  ELSE IF (dxrat .GT. rthresh) THEN
582  IF (y_prec_state .NE. extra_y) THEN
583  incr_prec = .true.
584  ELSE
585  x_state = noprog_state
586  END IF
587  ELSE
588  IF (dxrat .GT. dxratmax) dxratmax = dxrat
589  END IF
590  IF (x_state .GT. working_state) final_dx_x = dx_x
591  END IF
592 
593  IF (z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub)
594  $ z_state = working_state
595  IF (z_state .EQ. noprog_state .AND. dzrat .LE. rthresh)
596  $ z_state = working_state
597  IF (z_state .EQ. working_state) THEN
598  IF (dz_z .LE. eps) THEN
599  z_state = conv_state
600  ELSE IF (dz_z .GT. dz_ub) THEN
601  z_state = unstable_state
602  dzratmax = 0.0
603  final_dz_z = hugeval
604  ELSE IF (dzrat .GT. rthresh) THEN
605  IF (y_prec_state .NE. extra_y) THEN
606  incr_prec = .true.
607  ELSE
608  z_state = noprog_state
609  END IF
610  ELSE
611  IF (dzrat .GT. dzratmax) dzratmax = dzrat
612  END IF
613  IF (z_state .GT. working_state) final_dz_z = dz_z
614  END IF
615 
616  IF ( x_state.NE.working_state.AND.
617  $ (ignore_cwise.OR.z_state.NE.working_state) )
618  $ GOTO 666
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 caxpy( n, cmplx(1.0), dy, 1, y(1,j), 1 )
635  ELSE
636  CALL cla_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  err_bnds_norm( j, la_linrx_err_i ) =
652  $ final_dx_x / (1 - dxratmax)
653  END IF
654  IF (n_norms .GE. 2) THEN
655  err_bnds_comp( 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 ccopy( n, b( 1, j ), 1, res, 1 )
668  CALL chemv(uplo, n, cmplx(-1.0), a, lda, y(1,j), 1, cmplx(1.0),
669  $ res, 1)
670 
671  DO i = 1, n
672  ayb( i ) = cabs1( b( i, j ) )
673  END DO
674 *
675 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
676 *
677  CALL cla_heamv (uplo2, n, 1.0,
678  $ a, lda, y(1, j), 1, 1.0, ayb, 1)
679 
680  CALL cla_lin_berr (n, n, 1, res, ayb, berr_out(j))
681 *
682 * End of loop for each RHS.
683 *
684  END DO
685 *
686  RETURN
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:180
subroutine cla_wwaddw(N, X, Y, W)
CLA_WWADDW adds a vector into a doubled-single vector.
Definition: cla_wwaddw.f:83
subroutine cpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
CPOTRS
Definition: cpotrs.f:112
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:103
subroutine chemv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CHEMV
Definition: chemv.f:156
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
integer function ilauplo(UPLO)
ILAUPLO
Definition: ilauplo.f:60
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:83
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:90
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