LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ 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.

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

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 responsible 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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