LAPACK  3.10.1
LAPACK: Linear Algebra PACKage

◆ cla_herfsx_extended()

subroutine cla_herfsx_extended ( integer  PREC_TYPE,
character  UPLO,
integer  N,
integer  NRHS,
complex, dimension( lda, * )  A,
integer  LDA,
complex, dimension( ldaf, * )  AF,
integer  LDAF,
integer, dimension( * )  IPIV,
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_HERFSX_EXTENDED improves the computed solution to a system of linear equations for Hermitian indefinite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

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Purpose:
 CLA_HERFSX_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 CHERFSX 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 block diagonal matrix D and the multipliers used to
     obtain the factor U or L as computed by CHETRF.
[in]LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
[in]IPIV
          IPIV is INTEGER array, dimension (N)
     Details of the interchanges and the block structure of D
     as determined by CHETRF.
[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 CHETRS.
     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 CLA_HERFSX_EXTENDED had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 388 of file cla_herfsx_extended.f.

395 *
396 * -- LAPACK computational routine --
397 * -- LAPACK is a software package provided by Univ. of Tennessee, --
398 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
399 *
400 * .. Scalar Arguments ..
401  INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
402  $ N_NORMS, ITHRESH
403  CHARACTER UPLO
404  LOGICAL COLEQU, IGNORE_CWISE
405  REAL RTHRESH, DZ_UB
406 * ..
407 * .. Array Arguments ..
408  INTEGER IPIV( * )
409  COMPLEX A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
410  $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
411  REAL C( * ), AYB( * ), RCOND, BERR_OUT( * ),
412  $ ERR_BNDS_NORM( NRHS, * ),
413  $ ERR_BNDS_COMP( NRHS, * )
414 * ..
415 *
416 * =====================================================================
417 *
418 * .. Local Scalars ..
419  INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE,
420  $ Y_PREC_STATE
421  REAL YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
422  $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
423  $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
424  $ EPS, HUGEVAL, INCR_THRESH
425  LOGICAL INCR_PREC, UPPER
426  COMPLEX ZDUM
427 * ..
428 * .. Parameters ..
429  INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
430  $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
431  $ EXTRA_Y
432  parameter( unstable_state = 0, working_state = 1,
433  $ conv_state = 2, noprog_state = 3 )
434  parameter( base_residual = 0, extra_residual = 1,
435  $ extra_y = 2 )
436  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
437  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
438  INTEGER CMP_ERR_I, PIV_GROWTH_I
439  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
440  $ berr_i = 3 )
441  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
442  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
443  $ piv_growth_i = 9 )
444  INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
445  $ LA_LINRX_CWISE_I
446  parameter( la_linrx_itref_i = 1,
447  $ la_linrx_ithresh_i = 2 )
448  parameter( la_linrx_cwise_i = 3 )
449  INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
450  $ LA_LINRX_RCOND_I
451  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
452  parameter( la_linrx_rcond_i = 3 )
453 * ..
454 * .. External Functions ..
455  LOGICAL LSAME
456  EXTERNAL ilauplo
457  INTEGER ILAUPLO
458 * ..
459 * .. External Subroutines ..
460  EXTERNAL caxpy, ccopy, chetrs, chemv, blas_chemv_x,
461  $ blas_chemv2_x, cla_heamv, cla_wwaddw,
462  $ cla_lin_berr
463  REAL SLAMCH
464 * ..
465 * .. Intrinsic Functions ..
466  INTRINSIC abs, real, aimag, max, min
467 * ..
468 * .. Statement Functions ..
469  REAL CABS1
470 * ..
471 * .. Statement Function Definitions ..
472  cabs1( zdum ) = abs( real( zdum ) ) + abs( aimag( zdum ) )
473 * ..
474 * .. Executable Statements ..
475 *
476  info = 0
477  upper = lsame( uplo, 'U' )
478  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
479  info = -2
480  ELSE IF( n.LT.0 ) THEN
481  info = -3
482  ELSE IF( nrhs.LT.0 ) THEN
483  info = -4
484  ELSE IF( lda.LT.max( 1, n ) ) THEN
485  info = -6
486  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
487  info = -8
488  ELSE IF( ldb.LT.max( 1, n ) ) THEN
489  info = -13
490  ELSE IF( ldy.LT.max( 1, n ) ) THEN
491  info = -15
492  END IF
493  IF( info.NE.0 ) THEN
494  CALL xerbla( 'CLA_HERFSX_EXTENDED', -info )
495  RETURN
496  END IF
497  eps = slamch( 'Epsilon' )
498  hugeval = slamch( 'Overflow' )
499 * Force HUGEVAL to Inf
500  hugeval = hugeval * hugeval
501 * Using HUGEVAL may lead to spurious underflows.
502  incr_thresh = real( n ) * eps
503 
504  IF ( lsame( uplo, 'L' ) ) THEN
505  uplo2 = ilauplo( 'L' )
506  ELSE
507  uplo2 = ilauplo( 'U' )
508  ENDIF
509 
510  DO j = 1, nrhs
511  y_prec_state = extra_residual
512  IF ( y_prec_state .EQ. extra_y ) THEN
513  DO i = 1, n
514  y_tail( i ) = 0.0
515  END DO
516  END IF
517 
518  dxrat = 0.0
519  dxratmax = 0.0
520  dzrat = 0.0
521  dzratmax = 0.0
522  final_dx_x = hugeval
523  final_dz_z = hugeval
524  prevnormdx = hugeval
525  prev_dz_z = hugeval
526  dz_z = hugeval
527  dx_x = hugeval
528 
529  x_state = working_state
530  z_state = unstable_state
531  incr_prec = .false.
532 
533  DO cnt = 1, ithresh
534 *
535 * Compute residual RES = B_s - op(A_s) * Y,
536 * op(A) = A, A**T, or A**H depending on TRANS (and type).
537 *
538  CALL ccopy( n, b( 1, j ), 1, res, 1 )
539  IF ( y_prec_state .EQ. base_residual ) THEN
540  CALL chemv( uplo, n, cmplx(-1.0), a, lda, y( 1, j ), 1,
541  $ cmplx(1.0), res, 1 )
542  ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
543  CALL blas_chemv_x( uplo2, n, cmplx(-1.0), a, lda,
544  $ y( 1, j ), 1, cmplx(1.0), res, 1, prec_type)
545  ELSE
546  CALL blas_chemv2_x(uplo2, n, cmplx(-1.0), a, lda,
547  $ y(1, j), y_tail, 1, cmplx(1.0), res, 1, prec_type)
548  END IF
549 
550 ! XXX: RES is no longer needed.
551  CALL ccopy( n, res, 1, dy, 1 )
552  CALL chetrs( uplo, n, 1, af, ldaf, ipiv, dy, n, info )
553 *
554 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
555 *
556  normx = 0.0
557  normy = 0.0
558  normdx = 0.0
559  dz_z = 0.0
560  ymin = hugeval
561 
562  DO i = 1, n
563  yk = cabs1( y( i, j ) )
564  dyk = cabs1( dy( i ) )
565 
566  IF (yk .NE. 0.0) THEN
567  dz_z = max( dz_z, dyk / yk )
568  ELSE IF ( dyk .NE. 0.0 ) THEN
569  dz_z = hugeval
570  END IF
571 
572  ymin = min( ymin, yk )
573 
574  normy = max( normy, yk )
575 
576  IF ( colequ ) THEN
577  normx = max( normx, yk * c( i ) )
578  normdx = max( normdx, dyk * c( i ) )
579  ELSE
580  normx = normy
581  normdx = max( normdx, dyk )
582  END IF
583  END DO
584 
585  IF ( normx .NE. 0.0 ) THEN
586  dx_x = normdx / normx
587  ELSE IF ( normdx .EQ. 0.0 ) THEN
588  dx_x = 0.0
589  ELSE
590  dx_x = hugeval
591  END IF
592 
593  dxrat = normdx / prevnormdx
594  dzrat = dz_z / prev_dz_z
595 *
596 * Check termination criteria.
597 *
598  IF ( ymin*rcond .LT. incr_thresh*normy
599  $ .AND. y_prec_state .LT. extra_y )
600  $ incr_prec = .true.
601 
602  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
603  $ x_state = working_state
604  IF ( x_state .EQ. working_state ) THEN
605  IF ( dx_x .LE. eps ) THEN
606  x_state = conv_state
607  ELSE IF ( dxrat .GT. rthresh ) THEN
608  IF ( y_prec_state .NE. extra_y ) THEN
609  incr_prec = .true.
610  ELSE
611  x_state = noprog_state
612  END IF
613  ELSE
614  IF (dxrat .GT. dxratmax) dxratmax = dxrat
615  END IF
616  IF ( x_state .GT. working_state ) final_dx_x = dx_x
617  END IF
618 
619  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
620  $ z_state = working_state
621  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
622  $ z_state = working_state
623  IF ( z_state .EQ. working_state ) THEN
624  IF ( dz_z .LE. eps ) THEN
625  z_state = conv_state
626  ELSE IF ( dz_z .GT. dz_ub ) THEN
627  z_state = unstable_state
628  dzratmax = 0.0
629  final_dz_z = hugeval
630  ELSE IF ( dzrat .GT. rthresh ) THEN
631  IF ( y_prec_state .NE. extra_y ) THEN
632  incr_prec = .true.
633  ELSE
634  z_state = noprog_state
635  END IF
636  ELSE
637  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
638  END IF
639  IF ( z_state .GT. working_state ) final_dz_z = dz_z
640  END IF
641 
642  IF ( x_state.NE.working_state.AND.
643  $ ( ignore_cwise.OR.z_state.NE.working_state ) )
644  $ GOTO 666
645 
646  IF ( incr_prec ) THEN
647  incr_prec = .false.
648  y_prec_state = y_prec_state + 1
649  DO i = 1, n
650  y_tail( i ) = 0.0
651  END DO
652  END IF
653 
654  prevnormdx = normdx
655  prev_dz_z = dz_z
656 *
657 * Update soluton.
658 *
659  IF ( y_prec_state .LT. extra_y ) THEN
660  CALL caxpy( n, cmplx(1.0), dy, 1, y(1,j), 1 )
661  ELSE
662  CALL cla_wwaddw( n, y(1,j), y_tail, dy )
663  END IF
664 
665  END DO
666 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
667  666 CONTINUE
668 *
669 * Set final_* when cnt hits ithresh.
670 *
671  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
672  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
673 *
674 * Compute error bounds.
675 *
676  IF ( n_norms .GE. 1 ) THEN
677  err_bnds_norm( j, la_linrx_err_i ) =
678  $ final_dx_x / (1 - dxratmax)
679  END IF
680  IF (n_norms .GE. 2) THEN
681  err_bnds_comp( j, la_linrx_err_i ) =
682  $ final_dz_z / (1 - dzratmax)
683  END IF
684 *
685 * Compute componentwise relative backward error from formula
686 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
687 * where abs(Z) is the componentwise absolute value of the matrix
688 * or vector Z.
689 *
690 * Compute residual RES = B_s - op(A_s) * Y,
691 * op(A) = A, A**T, or A**H depending on TRANS (and type).
692 *
693  CALL ccopy( n, b( 1, j ), 1, res, 1 )
694  CALL chemv( uplo, n, cmplx(-1.0), a, lda, y(1,j), 1,
695  $ cmplx(1.0), res, 1 )
696 
697  DO i = 1, n
698  ayb( i ) = cabs1( b( i, j ) )
699  END DO
700 *
701 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
702 *
703  CALL cla_heamv( uplo2, n, 1.0,
704  $ a, lda, y(1, j), 1, 1.0, ayb, 1 )
705 
706  CALL cla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
707 *
708 * End of loop for each RHS.
709 *
710  END DO
711 *
712  RETURN
713 *
714 * End of CLA_HERFSX_EXTENDED
715 *
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
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 chetrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CHETRS
Definition: chetrs.f:120
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
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:68
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