LAPACK  3.9.1
LAPACK: Linear Algebra PACKage

◆ zla_gbrfsx_extended()

subroutine zla_gbrfsx_extended ( integer  PREC_TYPE,
integer  TRANS_TYPE,
integer  N,
integer  KL,
integer  KU,
integer  NRHS,
complex*16, dimension( ldab, * )  AB,
integer  LDAB,
complex*16, dimension( ldafb, * )  AFB,
integer  LDAFB,
integer, dimension( * )  IPIV,
logical  COLEQU,
double precision, dimension( * )  C,
complex*16, dimension( ldb, * )  B,
integer  LDB,
complex*16, dimension( ldy, * )  Y,
integer  LDY,
double precision, dimension( * )  BERR_OUT,
integer  N_NORMS,
double precision, dimension( nrhs, * )  ERR_BNDS_NORM,
double precision, dimension( nrhs, * )  ERR_BNDS_COMP,
complex*16, dimension( * )  RES,
double precision, dimension(*)  AYB,
complex*16, dimension( * )  DY,
complex*16, dimension( * )  Y_TAIL,
double precision  RCOND,
integer  ITHRESH,
double precision  RTHRESH,
double precision  DZ_UB,
logical  IGNORE_CWISE,
integer  INFO 
)

ZLA_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:
 ZLA_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 ZGBRFSX 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]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*16 array, dimension (LDAB,N)
     On entry, the N-by-N matrix A.
[in]LDAB
          LDAB is INTEGER
     The leading dimension of the array A.  LDAB >= max(1,N).
[in]AFB
          AFB is COMPLEX*16 array, dimension (LDAF,N)
     The factors L and U from the factorization
     A = P*L*U as computed by ZGBTRF.
[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 ZGBTRF; 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 DOUBLE PRECISION 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*16 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*16 array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by ZGBTRS.
     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 DOUBLE PRECISION 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 ZLA_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 DOUBLE PRECISION 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 DOUBLE PRECISION 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*16 array, dimension (N)
     Workspace to hold the intermediate residual.
[in]AYB
          AYB is DOUBLE PRECISION array, dimension (N)
     Workspace.
[in]DY
          DY is COMPLEX*16 array, dimension (N)
     Workspace to hold the intermediate solution.
[in]Y_TAIL
          Y_TAIL is COMPLEX*16 array, dimension (N)
     Workspace to hold the trailing bits of the intermediate solution.
[in]RCOND
          RCOND is DOUBLE PRECISION
     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 DOUBLE PRECISION
     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 DOUBLE PRECISION
     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 ZGBTRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 403 of file zla_gbrfsx_extended.f.

410 *
411 * -- LAPACK computational routine --
412 * -- LAPACK is a software package provided by Univ. of Tennessee, --
413 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
414 *
415 * .. Scalar Arguments ..
416  INTEGER INFO, LDAB, LDAFB, LDB, LDY, N, KL, KU, NRHS,
417  $ PREC_TYPE, TRANS_TYPE, N_NORMS, ITHRESH
418  LOGICAL COLEQU, IGNORE_CWISE
419  DOUBLE PRECISION RTHRESH, DZ_UB
420 * ..
421 * .. Array Arguments ..
422  INTEGER IPIV( * )
423  COMPLEX*16 AB( LDAB, * ), AFB( LDAFB, * ), B( LDB, * ),
424  $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
425  DOUBLE PRECISION C( * ), AYB(*), RCOND, BERR_OUT( * ),
426  $ ERR_BNDS_NORM( NRHS, * ),
427  $ ERR_BNDS_COMP( NRHS, * )
428 * ..
429 *
430 * =====================================================================
431 *
432 * .. Local Scalars ..
433  CHARACTER TRANS
434  INTEGER CNT, I, J, M, X_STATE, Z_STATE, Y_PREC_STATE
435  DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
436  $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
437  $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
438  $ EPS, HUGEVAL, INCR_THRESH
439  LOGICAL INCR_PREC
440  COMPLEX*16 ZDUM
441 * ..
442 * .. Parameters ..
443  INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
444  $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
445  $ EXTRA_Y
446  parameter( unstable_state = 0, working_state = 1,
447  $ conv_state = 2, noprog_state = 3 )
448  parameter( base_residual = 0, extra_residual = 1,
449  $ extra_y = 2 )
450  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
451  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
452  INTEGER CMP_ERR_I, PIV_GROWTH_I
453  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
454  $ berr_i = 3 )
455  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
456  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
457  $ piv_growth_i = 9 )
458  INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
459  $ LA_LINRX_CWISE_I
460  parameter( la_linrx_itref_i = 1,
461  $ la_linrx_ithresh_i = 2 )
462  parameter( la_linrx_cwise_i = 3 )
463  INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
464  $ LA_LINRX_RCOND_I
465  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
466  parameter( la_linrx_rcond_i = 3 )
467 * ..
468 * .. External Subroutines ..
469  EXTERNAL zaxpy, zcopy, zgbtrs, zgbmv, blas_zgbmv_x,
470  $ blas_zgbmv2_x, zla_gbamv, zla_wwaddw, dlamch,
472  DOUBLE PRECISION DLAMCH
473  CHARACTER CHLA_TRANSTYPE
474 * ..
475 * .. Intrinsic Functions..
476  INTRINSIC abs, max, min
477 * ..
478 * .. Statement Functions ..
479  DOUBLE PRECISION CABS1
480 * ..
481 * .. Statement Function Definitions ..
482  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
483 * ..
484 * .. Executable Statements ..
485 *
486  IF (info.NE.0) RETURN
487  trans = chla_transtype(trans_type)
488  eps = dlamch( 'Epsilon' )
489  hugeval = dlamch( 'Overflow' )
490 * Force HUGEVAL to Inf
491  hugeval = hugeval * hugeval
492 * Using HUGEVAL may lead to spurious underflows.
493  incr_thresh = dble( n ) * eps
494  m = kl+ku+1
495 
496  DO j = 1, nrhs
497  y_prec_state = extra_residual
498  IF ( y_prec_state .EQ. extra_y ) THEN
499  DO i = 1, n
500  y_tail( i ) = 0.0d+0
501  END DO
502  END IF
503 
504  dxrat = 0.0d+0
505  dxratmax = 0.0d+0
506  dzrat = 0.0d+0
507  dzratmax = 0.0d+0
508  final_dx_x = hugeval
509  final_dz_z = hugeval
510  prevnormdx = hugeval
511  prev_dz_z = hugeval
512  dz_z = hugeval
513  dx_x = hugeval
514 
515  x_state = working_state
516  z_state = unstable_state
517  incr_prec = .false.
518 
519  DO cnt = 1, ithresh
520 *
521 * Compute residual RES = B_s - op(A_s) * Y,
522 * op(A) = A, A**T, or A**H depending on TRANS (and type).
523 *
524  CALL zcopy( n, b( 1, j ), 1, res, 1 )
525  IF ( y_prec_state .EQ. base_residual ) THEN
526  CALL zgbmv( trans, m, n, kl, ku, (-1.0d+0,0.0d+0), ab,
527  $ ldab, y( 1, j ), 1, (1.0d+0,0.0d+0), res, 1 )
528  ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
529  CALL blas_zgbmv_x( trans_type, n, n, kl, ku,
530  $ (-1.0d+0,0.0d+0), ab, ldab, y( 1, j ), 1,
531  $ (1.0d+0,0.0d+0), res, 1, prec_type )
532  ELSE
533  CALL blas_zgbmv2_x( trans_type, n, n, kl, ku,
534  $ (-1.0d+0,0.0d+0), ab, ldab, y( 1, j ), y_tail, 1,
535  $ (1.0d+0,0.0d+0), res, 1, prec_type )
536  END IF
537 
538 ! XXX: RES is no longer needed.
539  CALL zcopy( n, res, 1, dy, 1 )
540  CALL zgbtrs( trans, n, kl, ku, 1, afb, ldafb, ipiv, dy, n,
541  $ info )
542 *
543 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
544 *
545  normx = 0.0d+0
546  normy = 0.0d+0
547  normdx = 0.0d+0
548  dz_z = 0.0d+0
549  ymin = hugeval
550 
551  DO i = 1, n
552  yk = cabs1( y( i, j ) )
553  dyk = cabs1( dy( i ) )
554 
555  IF (yk .NE. 0.0d+0) THEN
556  dz_z = max( dz_z, dyk / yk )
557  ELSE IF ( dyk .NE. 0.0d+0 ) THEN
558  dz_z = hugeval
559  END IF
560 
561  ymin = min( ymin, yk )
562 
563  normy = max( normy, yk )
564 
565  IF ( colequ ) THEN
566  normx = max( normx, yk * c( i ) )
567  normdx = max(normdx, dyk * c(i))
568  ELSE
569  normx = normy
570  normdx = max( normdx, dyk )
571  END IF
572  END DO
573 
574  IF ( normx .NE. 0.0d+0 ) THEN
575  dx_x = normdx / normx
576  ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
577  dx_x = 0.0d+0
578  ELSE
579  dx_x = hugeval
580  END IF
581 
582  dxrat = normdx / prevnormdx
583  dzrat = dz_z / prev_dz_z
584 *
585 * Check termination criteria.
586 *
587  IF (.NOT.ignore_cwise
588  $ .AND. ymin*rcond .LT. incr_thresh*normy
589  $ .AND. y_prec_state .LT. extra_y )
590  $ incr_prec = .true.
591 
592  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
593  $ x_state = working_state
594  IF ( x_state .EQ. working_state ) THEN
595  IF ( dx_x .LE. eps ) THEN
596  x_state = conv_state
597  ELSE IF ( dxrat .GT. rthresh ) THEN
598  IF ( y_prec_state .NE. extra_y ) THEN
599  incr_prec = .true.
600  ELSE
601  x_state = noprog_state
602  END IF
603  ELSE
604  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
605  END IF
606  IF ( x_state .GT. working_state ) final_dx_x = dx_x
607  END IF
608 
609  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
610  $ z_state = working_state
611  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
612  $ z_state = working_state
613  IF ( z_state .EQ. working_state ) THEN
614  IF ( dz_z .LE. eps ) THEN
615  z_state = conv_state
616  ELSE IF ( dz_z .GT. dz_ub ) THEN
617  z_state = unstable_state
618  dzratmax = 0.0d+0
619  final_dz_z = hugeval
620  ELSE IF ( dzrat .GT. rthresh ) THEN
621  IF ( y_prec_state .NE. extra_y ) THEN
622  incr_prec = .true.
623  ELSE
624  z_state = noprog_state
625  END IF
626  ELSE
627  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
628  END IF
629  IF ( z_state .GT. working_state ) final_dz_z = dz_z
630  END IF
631 *
632 * Exit if both normwise and componentwise stopped working,
633 * but if componentwise is unstable, let it go at least two
634 * iterations.
635 *
636  IF ( x_state.NE.working_state ) THEN
637  IF ( ignore_cwise ) GOTO 666
638  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
639  $ GOTO 666
640  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
641  END IF
642 
643  IF ( incr_prec ) THEN
644  incr_prec = .false.
645  y_prec_state = y_prec_state + 1
646  DO i = 1, n
647  y_tail( i ) = 0.0d+0
648  END DO
649  END IF
650 
651  prevnormdx = normdx
652  prev_dz_z = dz_z
653 *
654 * Update soluton.
655 *
656  IF ( y_prec_state .LT. extra_y ) THEN
657  CALL zaxpy( n, (1.0d+0,0.0d+0), dy, 1, y(1,j), 1 )
658  ELSE
659  CALL zla_wwaddw( n, y(1,j), y_tail, dy )
660  END IF
661 
662  END DO
663 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
664  666 CONTINUE
665 *
666 * Set final_* when cnt hits ithresh.
667 *
668  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
669  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
670 *
671 * Compute error bounds.
672 *
673  IF ( n_norms .GE. 1 ) THEN
674  err_bnds_norm( j, la_linrx_err_i ) =
675  $ final_dx_x / (1 - dxratmax)
676  END IF
677  IF ( n_norms .GE. 2 ) THEN
678  err_bnds_comp( j, la_linrx_err_i ) =
679  $ final_dz_z / (1 - dzratmax)
680  END IF
681 *
682 * Compute componentwise relative backward error from formula
683 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
684 * where abs(Z) is the componentwise absolute value of the matrix
685 * or vector Z.
686 *
687 * Compute residual RES = B_s - op(A_s) * Y,
688 * op(A) = A, A**T, or A**H depending on TRANS (and type).
689 *
690  CALL zcopy( n, b( 1, j ), 1, res, 1 )
691  CALL zgbmv( trans, n, n, kl, ku, (-1.0d+0,0.0d+0), ab, ldab,
692  $ y(1,j), 1, (1.0d+0,0.0d+0), res, 1 )
693 
694  DO i = 1, n
695  ayb( i ) = cabs1( b( i, j ) )
696  END DO
697 *
698 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
699 *
700  CALL zla_gbamv( trans_type, n, n, kl, ku, 1.0d+0,
701  $ ab, ldab, y(1, j), 1, 1.0d+0, ayb, 1 )
702 
703  CALL zla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
704 *
705 * End of loop for each RHS.
706 *
707  END DO
708 *
709  RETURN
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:88
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:81
subroutine zgbmv(TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGBMV
Definition: zgbmv.f:187
subroutine zla_gbamv(TRANS, M, N, KL, KU, ALPHA, AB, LDAB, X, INCX, BETA, Y, INCY)
ZLA_GBAMV performs a matrix-vector operation to calculate error bounds.
Definition: zla_gbamv.f:186
subroutine zgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
ZGBTRS
Definition: zgbtrs.f:138
subroutine zla_wwaddw(N, X, Y, W)
ZLA_WWADDW adds a vector into a doubled-single vector.
Definition: zla_wwaddw.f:81
subroutine zla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
ZLA_LIN_BERR computes a component-wise relative backward error.
Definition: zla_lin_berr.f:101
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