LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ sla_gbrfsx_extended()

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

SLA_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.

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

Purpose:
 SLA_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 SGBRFSX 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 REAL 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 REAL array, dimension (LDAFB,N)
     The factors L and U from the factorization
     A = P*L*U as computed by SGBTRF.
[in]LDAFB
          LDAFB is INTEGER
     The leading dimension of the array AF.  LDAFB >= max(1,N).
[in]IPIV
          IPIV is INTEGER array, dimension (N)
     The pivot indices from the factorization A = P*L*U
     as computed by SGBTRF; 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 REAL 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 REAL array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by SGBTRS.
     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 SLA_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 REAL array, dimension (N)
     Workspace to hold the intermediate residual.
[in]AYB
          AYB is REAL array, dimension (N)
     Workspace. This can be the same workspace passed for Y_TAIL.
[in]DY
          DY is REAL array, dimension (N)
     Workspace to hold the intermediate solution.
[in]Y_TAIL
          Y_TAIL is REAL 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 SGBTRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
June 2017

Definition at line 412 of file sla_gbrfsx_extended.f.

412 *
413 * -- LAPACK computational routine (version 3.7.1) --
414 * -- LAPACK is a software package provided by Univ. of Tennessee, --
415 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
416 * June 2017
417 *
418 * .. Scalar Arguments ..
419  INTEGER info, ldab, ldafb, ldb, ldy, n, kl, ku, nrhs,
420  $ prec_type, trans_type, n_norms, ithresh
421  LOGICAL colequ, ignore_cwise
422  REAL rthresh, dz_ub
423 * ..
424 * .. Array Arguments ..
425  INTEGER ipiv( * )
426  REAL ab( ldab, * ), afb( ldafb, * ), b( ldb, * ),
427  $ y( ldy, * ), res(*), dy(*), y_tail(*)
428  REAL c( * ), ayb(*), rcond, berr_out(*),
429  $ err_bnds_norm( nrhs, * ),
430  $ err_bnds_comp( nrhs, * )
431 * ..
432 *
433 * =====================================================================
434 *
435 * .. Local Scalars ..
436  CHARACTER trans
437  INTEGER cnt, i, j, m, x_state, z_state, y_prec_state
438  REAL yk, dyk, ymin, normy, normx, normdx, dxrat,
439  $ dzrat, prevnormdx, prev_dz_z, dxratmax,
440  $ dzratmax, dx_x, dz_z, final_dx_x, final_dz_z,
441  $ eps, hugeval, incr_thresh
442  LOGICAL incr_prec
443 * ..
444 * .. Parameters ..
445  INTEGER unstable_state, working_state, conv_state,
446  $ noprog_state, base_residual, extra_residual,
447  $ extra_y
448  parameter( unstable_state = 0, working_state = 1,
449  $ conv_state = 2, noprog_state = 3 )
450  parameter( base_residual = 0, extra_residual = 1,
451  $ extra_y = 2 )
452  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
453  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
454  INTEGER cmp_err_i, piv_growth_i
455  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
456  $ berr_i = 3 )
457  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
458  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
459  $ piv_growth_i = 9 )
460  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
461  $ la_linrx_cwise_i
462  parameter( la_linrx_itref_i = 1,
463  $ la_linrx_ithresh_i = 2 )
464  parameter( la_linrx_cwise_i = 3 )
465  INTEGER la_linrx_trust_i, la_linrx_err_i,
466  $ la_linrx_rcond_i
467  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
468  parameter( la_linrx_rcond_i = 3 )
469 * ..
470 * .. External Subroutines ..
471  EXTERNAL saxpy, scopy, sgbtrs, sgbmv, blas_sgbmv_x,
472  $ blas_sgbmv2_x, sla_gbamv, sla_wwaddw, slamch,
474  REAL slamch
475  CHARACTER chla_transtype
476 * ..
477 * .. Intrinsic Functions ..
478  INTRINSIC abs, max, min
479 * ..
480 * .. Executable Statements ..
481 *
482  IF (info.NE.0) RETURN
483  trans = chla_transtype(trans_type)
484  eps = slamch( 'Epsilon' )
485  hugeval = slamch( 'Overflow' )
486 * Force HUGEVAL to Inf
487  hugeval = hugeval * hugeval
488 * Using HUGEVAL may lead to spurious underflows.
489  incr_thresh = REAL( N ) * eps
490  m = kl+ku+1
491 
492  DO j = 1, nrhs
493  y_prec_state = extra_residual
494  IF ( y_prec_state .EQ. extra_y ) THEN
495  DO i = 1, n
496  y_tail( i ) = 0.0
497  END DO
498  END IF
499 
500  dxrat = 0.0
501  dxratmax = 0.0
502  dzrat = 0.0
503  dzratmax = 0.0
504  final_dx_x = hugeval
505  final_dz_z = hugeval
506  prevnormdx = hugeval
507  prev_dz_z = hugeval
508  dz_z = hugeval
509  dx_x = hugeval
510 
511  x_state = working_state
512  z_state = unstable_state
513  incr_prec = .false.
514 
515  DO cnt = 1, ithresh
516 *
517 * Compute residual RES = B_s - op(A_s) * Y,
518 * op(A) = A, A**T, or A**H depending on TRANS (and type).
519 *
520  CALL scopy( n, b( 1, j ), 1, res, 1 )
521  IF ( y_prec_state .EQ. base_residual ) THEN
522  CALL sgbmv( trans, m, n, kl, ku, -1.0, ab, ldab,
523  $ y( 1, j ), 1, 1.0, res, 1 )
524  ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
525  CALL blas_sgbmv_x( trans_type, n, n, kl, ku,
526  $ -1.0, ab, ldab, y( 1, j ), 1, 1.0, res, 1,
527  $ prec_type )
528  ELSE
529  CALL blas_sgbmv2_x( trans_type, n, n, kl, ku, -1.0,
530  $ ab, ldab, y( 1, j ), y_tail, 1, 1.0, res, 1,
531  $ prec_type )
532  END IF
533 
534 ! XXX: RES is no longer needed.
535  CALL scopy( n, res, 1, dy, 1 )
536  CALL sgbtrs( trans, n, kl, ku, 1, afb, ldafb, ipiv, dy, n,
537  $ info )
538 *
539 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
540 *
541  normx = 0.0
542  normy = 0.0
543  normdx = 0.0
544  dz_z = 0.0
545  ymin = hugeval
546 
547  DO i = 1, n
548  yk = abs( y( i, j ) )
549  dyk = abs( dy( i ) )
550 
551  IF ( yk .NE. 0.0 ) THEN
552  dz_z = max( dz_z, dyk / yk )
553  ELSE IF ( dyk .NE. 0.0 ) THEN
554  dz_z = hugeval
555  END IF
556 
557  ymin = min( ymin, yk )
558 
559  normy = max( normy, yk )
560 
561  IF ( colequ ) THEN
562  normx = max( normx, yk * c( i ) )
563  normdx = max( normdx, dyk * c( i ) )
564  ELSE
565  normx = normy
566  normdx = max( normdx, dyk )
567  END IF
568  END DO
569 
570  IF ( normx .NE. 0.0 ) THEN
571  dx_x = normdx / normx
572  ELSE IF ( normdx .EQ. 0.0 ) THEN
573  dx_x = 0.0
574  ELSE
575  dx_x = hugeval
576  END IF
577 
578  dxrat = normdx / prevnormdx
579  dzrat = dz_z / prev_dz_z
580 *
581 * Check termination criteria.
582 *
583  IF ( .NOT.ignore_cwise
584  $ .AND. ymin*rcond .LT. incr_thresh*normy
585  $ .AND. y_prec_state .LT. extra_y )
586  $ incr_prec = .true.
587 
588  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
589  $ x_state = working_state
590  IF ( x_state .EQ. working_state ) THEN
591  IF ( dx_x .LE. eps ) THEN
592  x_state = conv_state
593  ELSE IF ( dxrat .GT. rthresh ) THEN
594  IF ( y_prec_state .NE. extra_y ) THEN
595  incr_prec = .true.
596  ELSE
597  x_state = noprog_state
598  END IF
599  ELSE
600  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
601  END IF
602  IF ( x_state .GT. working_state ) final_dx_x = dx_x
603  END IF
604 
605  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
606  $ z_state = working_state
607  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
608  $ z_state = working_state
609  IF ( z_state .EQ. working_state ) THEN
610  IF ( dz_z .LE. eps ) THEN
611  z_state = conv_state
612  ELSE IF ( dz_z .GT. dz_ub ) THEN
613  z_state = unstable_state
614  dzratmax = 0.0
615  final_dz_z = hugeval
616  ELSE IF ( dzrat .GT. rthresh ) THEN
617  IF ( y_prec_state .NE. extra_y ) THEN
618  incr_prec = .true.
619  ELSE
620  z_state = noprog_state
621  END IF
622  ELSE
623  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
624  END IF
625  IF ( z_state .GT. working_state ) final_dz_z = dz_z
626  END IF
627 *
628 * Exit if both normwise and componentwise stopped working,
629 * but if componentwise is unstable, let it go at least two
630 * iterations.
631 *
632  IF ( x_state.NE.working_state ) THEN
633  IF ( ignore_cwise ) GOTO 666
634  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
635  $ GOTO 666
636  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
637  END IF
638 
639  IF ( incr_prec ) THEN
640  incr_prec = .false.
641  y_prec_state = y_prec_state + 1
642  DO i = 1, n
643  y_tail( i ) = 0.0
644  END DO
645  END IF
646 
647  prevnormdx = normdx
648  prev_dz_z = dz_z
649 *
650 * Update soluton.
651 *
652  IF (y_prec_state .LT. extra_y) THEN
653  CALL saxpy( n, 1.0, dy, 1, y(1,j), 1 )
654  ELSE
655  CALL sla_wwaddw( n, y(1,j), y_tail, dy )
656  END IF
657 
658  END DO
659 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
660  666 CONTINUE
661 *
662 * Set final_* when cnt hits ithresh.
663 *
664  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
665  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
666 *
667 * Compute error bounds.
668 *
669  IF ( n_norms .GE. 1 ) THEN
670  err_bnds_norm( j, la_linrx_err_i ) =
671  $ final_dx_x / (1 - dxratmax)
672  END IF
673  IF (n_norms .GE. 2) THEN
674  err_bnds_comp( j, la_linrx_err_i ) =
675  $ final_dz_z / (1 - dzratmax)
676  END IF
677 *
678 * Compute componentwise relative backward error from formula
679 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
680 * where abs(Z) is the componentwise absolute value of the matrix
681 * or vector Z.
682 *
683 * Compute residual RES = B_s - op(A_s) * Y,
684 * op(A) = A, A**T, or A**H depending on TRANS (and type).
685 *
686  CALL scopy( n, b( 1, j ), 1, res, 1 )
687  CALL sgbmv(trans, n, n, kl, ku, -1.0, ab, ldab, y(1,j),
688  $ 1, 1.0, res, 1 )
689 
690  DO i = 1, n
691  ayb( i ) = abs( b( i, j ) )
692  END DO
693 *
694 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
695 *
696  CALL sla_gbamv( trans_type, n, n, kl, ku, 1.0,
697  $ ab, ldab, y(1, j), 1, 1.0, ayb, 1 )
698 
699  CALL sla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
700 *
701 * End of loop for each RHS
702 *
703  END DO
704 *
705  RETURN
subroutine sla_wwaddw(N, X, Y, W)
SLA_WWADDW adds a vector into a doubled-single vector.
Definition: sla_wwaddw.f:83
subroutine sgbmv(TRANS, M, N, KL, KU, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
SGBMV
Definition: sgbmv.f:187
subroutine sla_gbamv(TRANS, M, N, KL, KU, ALPHA, AB, LDAB, X, INCX, BETA, Y, INCY)
SLA_GBAMV performs a matrix-vector operation to calculate error bounds.
Definition: sla_gbamv.f:187
subroutine sgbtrs(TRANS, N, KL, KU, NRHS, AB, LDAB, IPIV, B, LDB, INFO)
SGBTRS
Definition: sgbtrs.f:140
subroutine sla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
SLA_LIN_BERR computes a component-wise relative backward error.
Definition: sla_lin_berr.f:103
subroutine saxpy(N, SA, SX, INCX, SY, INCY)
SAXPY
Definition: saxpy.f:91
real function slamch(CMACH)
SLAMCH
Definition: slamch.f:69
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
subroutine scopy(N, SX, INCX, SY, INCY)
SCOPY
Definition: scopy.f:84
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