LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
subroutine cla_gbrfsx_extended ( integer  PREC_TYPE,
integer  TRANS_TYPE,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
complex, dimension( ldab, * )  AB,
integer  LDAB,
complex, dimension( ldafb, * )  AFB,
integer  LDAFB,
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_GBRFSX_EXTENDED improves the computed solution to a system of linear equations for general banded matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

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Purpose:
 CLA_GBRFSX_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 CGBRFSX 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]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]KL
          KL is INTEGER
     The number of subdiagonals within the band of A.  KL >= 0.
[in]KU
          KU is INTEGER
     The number of superdiagonals within the band of A.  KU >= 0
[in]NRHS
          NRHS is INTEGER
     The number of right-hand-sides, i.e., the number of columns of the
     matrix B.
[in]AB
          AB is COMPLEX array, dimension (LDAB,N)
     On entry, the N-by-N matrix AB.
[in]LDAB
          LDAB is INTEGER
     The leading dimension of the array AB.  LDAB >= max(1,N).
[in]AFB
          AFB is COMPLEX array, dimension (LDAF,N)
     The factors L and U from the factorization
     A = P*L*U as computed by CGBTRF.
[in]LDAFB
          LDAFB 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 CGBTRF; 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 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 CGBTRS.
     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 CGBTRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
September 2012

Definition at line 414 of file cla_gbrfsx_extended.f.

414 *
415 * -- LAPACK computational routine (version 3.4.2) --
416 * -- LAPACK is a software package provided by Univ. of Tennessee, --
417 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
418 * September 2012
419 *
420 * .. Scalar Arguments ..
421  INTEGER info, ldab, ldafb, ldb, ldy, n, kl, ku, nrhs,
422  $ prec_type, trans_type, n_norms, ithresh
423  LOGICAL colequ, ignore_cwise
424  REAL rthresh, dz_ub
425 * ..
426 * .. Array Arguments ..
427  INTEGER ipiv( * )
428  COMPLEX ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
429  $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
430  REAL c( * ), ayb(*), rcond, berr_out( * ),
431  $ err_bnds_norm( nrhs, * ),
432  $ err_bnds_comp( nrhs, * )
433 * ..
434 *
435 * =====================================================================
436 *
437 * .. Local Scalars ..
438  CHARACTER trans
439  INTEGER cnt, i, j, m, x_state, z_state, y_prec_state
440  REAL yk, dyk, ymin, normy, normx, normdx, dxrat,
441  $ dzrat, prevnormdx, prev_dz_z, dxratmax,
442  $ dzratmax, dx_x, dz_z, final_dx_x, final_dz_z,
443  $ eps, hugeval, incr_thresh
444  LOGICAL incr_prec
445  COMPLEX zdum
446 * ..
447 * .. Parameters ..
448  INTEGER unstable_state, working_state, conv_state,
449  $ noprog_state, base_residual, extra_residual,
450  $ extra_y
451  parameter ( unstable_state = 0, working_state = 1,
452  $ conv_state = 2, noprog_state = 3 )
453  parameter ( base_residual = 0, extra_residual = 1,
454  $ extra_y = 2 )
455  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
456  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
457  INTEGER cmp_err_i, piv_growth_i
458  parameter ( final_nrm_err_i = 1, final_cmp_err_i = 2,
459  $ berr_i = 3 )
460  parameter ( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
461  parameter ( cmp_rcond_i = 7, cmp_err_i = 8,
462  $ piv_growth_i = 9 )
463  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
464  $ la_linrx_cwise_i
465  parameter ( la_linrx_itref_i = 1,
466  $ la_linrx_ithresh_i = 2 )
467  parameter ( la_linrx_cwise_i = 3 )
468  INTEGER la_linrx_trust_i, la_linrx_err_i,
469  $ la_linrx_rcond_i
470  parameter ( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
471  parameter ( la_linrx_rcond_i = 3 )
472 * ..
473 * .. External Subroutines ..
474  EXTERNAL caxpy, ccopy, cgbtrs, cgbmv, blas_cgbmv_x,
475  $ blas_cgbmv2_x, cla_gbamv, cla_wwaddw, slamch,
477  REAL slamch
478  CHARACTER chla_transtype
479 * ..
480 * .. Intrinsic Functions..
481  INTRINSIC abs, max, min
482 * ..
483 * .. Statement Functions ..
484  REAL cabs1
485 * ..
486 * .. Statement Function Definitions ..
487  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( aimag( zdum ) )
488 * ..
489 * .. Executable Statements ..
490 *
491  IF (info.NE.0) RETURN
492  trans = chla_transtype(trans_type)
493  eps = slamch( 'Epsilon' )
494  hugeval = slamch( 'Overflow' )
495 * Force HUGEVAL to Inf
496  hugeval = hugeval * hugeval
497 * Using HUGEVAL may lead to spurious underflows.
498  incr_thresh = REAL( N ) * eps
499  m = kl+ku+1
500 
501  DO j = 1, nrhs
502  y_prec_state = extra_residual
503  IF ( y_prec_state .EQ. extra_y ) then
504  DO i = 1, n
505  y_tail( i ) = 0.0
506  END DO
507  END IF
508 
509  dxrat = 0.0e+0
510  dxratmax = 0.0e+0
511  dzrat = 0.0e+0
512  dzratmax = 0.0e+0
513  final_dx_x = hugeval
514  final_dz_z = hugeval
515  prevnormdx = hugeval
516  prev_dz_z = hugeval
517  dz_z = hugeval
518  dx_x = hugeval
519 
520  x_state = working_state
521  z_state = unstable_state
522  incr_prec = .false.
523 
524  DO cnt = 1, ithresh
525 *
526 * Compute residual RES = B_s - op(A_s) * Y,
527 * op(A) = A, A**T, or A**H depending on TRANS (and type).
528 *
529  CALL ccopy( n, b( 1, j ), 1, res, 1 )
530  IF ( y_prec_state .EQ. base_residual ) THEN
531  CALL cgbmv( trans, m, n, kl, ku, (-1.0e+0,0.0e+0), ab,
532  $ ldab, y( 1, j ), 1, (1.0e+0,0.0e+0), res, 1 )
533  ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
534  CALL blas_cgbmv_x( trans_type, n, n, kl, ku,
535  $ (-1.0e+0,0.0e+0), ab, ldab, y( 1, j ), 1,
536  $ (1.0e+0,0.0e+0), res, 1, prec_type )
537  ELSE
538  CALL blas_cgbmv2_x( trans_type, n, n, kl, ku,
539  $ (-1.0e+0,0.0e+0), ab, ldab, y( 1, j ), y_tail, 1,
540  $ (1.0e+0,0.0e+0), res, 1, prec_type )
541  END IF
542 
543 ! XXX: RES is no longer needed.
544  CALL ccopy( n, res, 1, dy, 1 )
545  CALL cgbtrs( trans, n, kl, ku, 1, afb, ldafb, ipiv, dy, n,
546  $ info )
547 *
548 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
549 *
550  normx = 0.0e+0
551  normy = 0.0e+0
552  normdx = 0.0e+0
553  dz_z = 0.0e+0
554  ymin = hugeval
555 
556  DO i = 1, n
557  yk = cabs1( y( i, j ) )
558  dyk = cabs1( dy( i ) )
559 
560  IF (yk .NE. 0.0) THEN
561  dz_z = max( dz_z, dyk / yk )
562  ELSE IF ( dyk .NE. 0.0 ) THEN
563  dz_z = hugeval
564  END IF
565 
566  ymin = min( ymin, yk )
567 
568  normy = max( normy, yk )
569 
570  IF ( colequ ) THEN
571  normx = max( normx, yk * c( i ) )
572  normdx = max(normdx, dyk * c(i))
573  ELSE
574  normx = normy
575  normdx = max( normdx, dyk )
576  END IF
577  END DO
578 
579  IF ( normx .NE. 0.0 ) THEN
580  dx_x = normdx / normx
581  ELSE IF ( normdx .EQ. 0.0 ) THEN
582  dx_x = 0.0
583  ELSE
584  dx_x = hugeval
585  END IF
586 
587  dxrat = normdx / prevnormdx
588  dzrat = dz_z / prev_dz_z
589 *
590 * Check termination criteria.
591 *
592  IF (.NOT.ignore_cwise
593  $ .AND. ymin*rcond .LT. incr_thresh*normy
594  $ .AND. y_prec_state .LT. extra_y )
595  $ incr_prec = .true.
596 
597  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
598  $ x_state = working_state
599  IF ( x_state .EQ. working_state ) THEN
600  IF ( dx_x .LE. eps ) THEN
601  x_state = conv_state
602  ELSE IF ( dxrat .GT. rthresh ) THEN
603  IF ( y_prec_state .NE. extra_y ) THEN
604  incr_prec = .true.
605  ELSE
606  x_state = noprog_state
607  END IF
608  ELSE
609  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
610  END IF
611  IF ( x_state .GT. working_state ) final_dx_x = dx_x
612  END IF
613 
614  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
615  $ z_state = working_state
616  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
617  $ z_state = working_state
618  IF ( z_state .EQ. working_state ) THEN
619  IF ( dz_z .LE. eps ) THEN
620  z_state = conv_state
621  ELSE IF ( dz_z .GT. dz_ub ) THEN
622  z_state = unstable_state
623  dzratmax = 0.0
624  final_dz_z = hugeval
625  ELSE IF ( dzrat .GT. rthresh ) THEN
626  IF ( y_prec_state .NE. extra_y ) THEN
627  incr_prec = .true.
628  ELSE
629  z_state = noprog_state
630  END IF
631  ELSE
632  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
633  END IF
634  IF ( z_state .GT. working_state ) final_dz_z = dz_z
635  END IF
636 *
637 * Exit if both normwise and componentwise stopped working,
638 * but if componentwise is unstable, let it go at least two
639 * iterations.
640 *
641  IF ( x_state.NE.working_state ) THEN
642  IF ( ignore_cwise ) GOTO 666
643  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
644  $ GOTO 666
645  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
646  END IF
647 
648  IF ( incr_prec ) THEN
649  incr_prec = .false.
650  y_prec_state = y_prec_state + 1
651  DO i = 1, n
652  y_tail( i ) = 0.0
653  END DO
654  END IF
655 
656  prevnormdx = normdx
657  prev_dz_z = dz_z
658 *
659 * Update soluton.
660 *
661  IF ( y_prec_state .LT. extra_y ) THEN
662  CALL caxpy( n, (1.0e+0,0.0e+0), dy, 1, y(1,j), 1 )
663  ELSE
664  CALL cla_wwaddw( n, y(1,j), y_tail, dy )
665  END IF
666 
667  END DO
668 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
669  666 CONTINUE
670 *
671 * Set final_* when cnt hits ithresh.
672 *
673  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
674  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
675 *
676 * Compute error bounds.
677 *
678  IF ( n_norms .GE. 1 ) THEN
679  err_bnds_norm( j, la_linrx_err_i ) =
680  $ final_dx_x / (1 - dxratmax)
681  END IF
682  IF ( n_norms .GE. 2 ) THEN
683  err_bnds_comp( j, la_linrx_err_i ) =
684  $ final_dz_z / (1 - dzratmax)
685  END IF
686 *
687 * Compute componentwise relative backward error from formula
688 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
689 * where abs(Z) is the componentwise absolute value of the matrix
690 * or vector Z.
691 *
692 * Compute residual RES = B_s - op(A_s) * Y,
693 * op(A) = A, A**T, or A**H depending on TRANS (and type).
694 *
695  CALL ccopy( n, b( 1, j ), 1, res, 1 )
696  CALL cgbmv( trans, n, n, kl, ku, (-1.0e+0,0.0e+0), ab, ldab,
697  $ y(1,j), 1, (1.0e+0,0.0e+0), res, 1 )
698 
699  DO i = 1, n
700  ayb( i ) = cabs1( b( i, j ) )
701  END DO
702 *
703 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
704 *
705  CALL cla_gbamv( trans_type, n, n, kl, ku, 1.0e+0,
706  $ ab, ldab, y(1, j), 1, 1.0e+0, ayb, 1 )
707 
708  CALL cla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
709 *
710 * End of loop for each RHS.
711 *
712  END DO
713 *
714  RETURN
subroutine cla_gbamv(TRANS, M, N, KL, KU, ALPHA, AB, LDAB, X, INCX, BETA, Y, INCY)
CLA_GBAMV performs a matrix-vector operation to calculate error bounds.
Definition: cla_gbamv.f:188
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
subroutine cgbmv(TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CGBMV
Definition: cgbmv.f:189
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 ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:52
subroutine cla_wwaddw(N, X, Y, W)
CLA_WWADDW adds a vector into a doubled-single vector.
Definition: cla_wwaddw.f:83
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
subroutine caxpy(N, CA, CX, INCX, CY, INCY)
CAXPY
Definition: caxpy.f:53
subroutine cgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
CGBTRS
Definition: cgbtrs.f:140

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