LAPACK  3.6.1
LAPACK: Linear Algebra PACKage
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.

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

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 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]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 .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 CLA_HERFSX_EXTENDED had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
September 2012

Definition at line 400 of file cla_herfsx_extended.f.

400 *
401 * -- LAPACK computational routine (version 3.4.2) --
402 * -- LAPACK is a software package provided by Univ. of Tennessee, --
403 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
404 * September 2012
405 *
406 * .. Scalar Arguments ..
407  INTEGER info, lda, ldaf, ldb, ldy, n, nrhs, prec_type,
408  $ n_norms, ithresh
409  CHARACTER uplo
410  LOGICAL colequ, ignore_cwise
411  REAL rthresh, dz_ub
412 * ..
413 * .. Array Arguments ..
414  INTEGER ipiv( * )
415  COMPLEX a( lda, * ), af( ldaf, * ), b( ldb, * ),
416  $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
417  REAL c( * ), ayb( * ), rcond, berr_out( * ),
418  $ err_bnds_norm( nrhs, * ),
419  $ err_bnds_comp( nrhs, * )
420 * ..
421 *
422 * =====================================================================
423 *
424 * .. Local Scalars ..
425  INTEGER uplo2, cnt, i, j, x_state, z_state,
426  $ y_prec_state
427  REAL yk, dyk, ymin, normy, normx, normdx, dxrat,
428  $ dzrat, prevnormdx, prev_dz_z, dxratmax,
429  $ dzratmax, dx_x, dz_z, final_dx_x, final_dz_z,
430  $ eps, hugeval, incr_thresh
431  LOGICAL incr_prec, upper
432  COMPLEX zdum
433 * ..
434 * .. Parameters ..
435  INTEGER unstable_state, working_state, conv_state,
436  $ noprog_state, base_residual, extra_residual,
437  $ extra_y
438  parameter ( unstable_state = 0, working_state = 1,
439  $ conv_state = 2, noprog_state = 3 )
440  parameter ( base_residual = 0, extra_residual = 1,
441  $ extra_y = 2 )
442  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
443  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
444  INTEGER cmp_err_i, piv_growth_i
445  parameter ( final_nrm_err_i = 1, final_cmp_err_i = 2,
446  $ berr_i = 3 )
447  parameter ( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
448  parameter ( cmp_rcond_i = 7, cmp_err_i = 8,
449  $ piv_growth_i = 9 )
450  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
451  $ la_linrx_cwise_i
452  parameter ( la_linrx_itref_i = 1,
453  $ la_linrx_ithresh_i = 2 )
454  parameter ( la_linrx_cwise_i = 3 )
455  INTEGER la_linrx_trust_i, la_linrx_err_i,
456  $ la_linrx_rcond_i
457  parameter ( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
458  parameter ( la_linrx_rcond_i = 3 )
459 * ..
460 * .. External Functions ..
461  LOGICAL lsame
462  EXTERNAL ilauplo
463  INTEGER ilauplo
464 * ..
465 * .. External Subroutines ..
466  EXTERNAL caxpy, ccopy, chetrs, chemv, blas_chemv_x,
467  $ blas_chemv2_x, cla_heamv, cla_wwaddw,
468  $ cla_lin_berr
469  REAL slamch
470 * ..
471 * .. Intrinsic Functions ..
472  INTRINSIC abs, REAL, aimag, max, min
473 * ..
474 * .. Statement Functions ..
475  REAL cabs1
476 * ..
477 * .. Statement Function Definitions ..
478  cabs1( zdum ) = abs( REAL( ZDUM ) ) + abs( aimag( zdum ) )
479 * ..
480 * .. Executable Statements ..
481 *
482  info = 0
483  upper = lsame( uplo, 'U' )
484  IF( .NOT.upper .AND. .NOT.lsame( uplo, 'L' ) ) THEN
485  info = -2
486  ELSE IF( n.LT.0 ) THEN
487  info = -3
488  ELSE IF( nrhs.LT.0 ) THEN
489  info = -4
490  ELSE IF( lda.LT.max( 1, n ) ) THEN
491  info = -6
492  ELSE IF( ldaf.LT.max( 1, n ) ) THEN
493  info = -8
494  ELSE IF( ldb.LT.max( 1, n ) ) THEN
495  info = -13
496  ELSE IF( ldy.LT.max( 1, n ) ) THEN
497  info = -15
498  END IF
499  IF( info.NE.0 ) THEN
500  CALL xerbla( 'CLA_HERFSX_EXTENDED', -info )
501  RETURN
502  END IF
503  eps = slamch( 'Epsilon' )
504  hugeval = slamch( 'Overflow' )
505 * Force HUGEVAL to Inf
506  hugeval = hugeval * hugeval
507 * Using HUGEVAL may lead to spurious underflows.
508  incr_thresh = REAL( N ) * eps
509 
510  IF ( lsame( uplo, 'L' ) ) THEN
511  uplo2 = ilauplo( 'L' )
512  ELSE
513  uplo2 = ilauplo( 'U' )
514  ENDIF
515 
516  DO j = 1, nrhs
517  y_prec_state = extra_residual
518  IF ( y_prec_state .EQ. extra_y ) THEN
519  DO i = 1, n
520  y_tail( i ) = 0.0
521  END DO
522  END IF
523 
524  dxrat = 0.0
525  dxratmax = 0.0
526  dzrat = 0.0
527  dzratmax = 0.0
528  final_dx_x = hugeval
529  final_dz_z = hugeval
530  prevnormdx = hugeval
531  prev_dz_z = hugeval
532  dz_z = hugeval
533  dx_x = hugeval
534 
535  x_state = working_state
536  z_state = unstable_state
537  incr_prec = .false.
538 
539  DO cnt = 1, ithresh
540 *
541 * Compute residual RES = B_s - op(A_s) * Y,
542 * op(A) = A, A**T, or A**H depending on TRANS (and type).
543 *
544  CALL ccopy( n, b( 1, j ), 1, res, 1 )
545  IF ( y_prec_state .EQ. base_residual ) THEN
546  CALL chemv( uplo, n, cmplx(-1.0), a, lda, y( 1, j ), 1,
547  $ cmplx(1.0), res, 1 )
548  ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
549  CALL blas_chemv_x( uplo2, n, cmplx(-1.0), a, lda,
550  $ y( 1, j ), 1, cmplx(1.0), res, 1, prec_type)
551  ELSE
552  CALL blas_chemv2_x(uplo2, n, cmplx(-1.0), a, lda,
553  $ y(1, j), y_tail, 1, cmplx(1.0), res, 1, prec_type)
554  END IF
555 
556 ! XXX: RES is no longer needed.
557  CALL ccopy( n, res, 1, dy, 1 )
558  CALL chetrs( uplo, n, 1, af, ldaf, ipiv, dy, n, info )
559 *
560 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
561 *
562  normx = 0.0
563  normy = 0.0
564  normdx = 0.0
565  dz_z = 0.0
566  ymin = hugeval
567 
568  DO i = 1, n
569  yk = cabs1( y( i, j ) )
570  dyk = cabs1( dy( i ) )
571 
572  IF (yk .NE. 0.0) THEN
573  dz_z = max( dz_z, dyk / yk )
574  ELSE IF ( dyk .NE. 0.0 ) THEN
575  dz_z = hugeval
576  END IF
577 
578  ymin = min( ymin, yk )
579 
580  normy = max( normy, yk )
581 
582  IF ( colequ ) THEN
583  normx = max( normx, yk * c( i ) )
584  normdx = max( normdx, dyk * c( i ) )
585  ELSE
586  normx = normy
587  normdx = max( normdx, dyk )
588  END IF
589  END DO
590 
591  IF ( normx .NE. 0.0 ) THEN
592  dx_x = normdx / normx
593  ELSE IF ( normdx .EQ. 0.0 ) THEN
594  dx_x = 0.0
595  ELSE
596  dx_x = hugeval
597  END IF
598 
599  dxrat = normdx / prevnormdx
600  dzrat = dz_z / prev_dz_z
601 *
602 * Check termination criteria.
603 *
604  IF ( ymin*rcond .LT. incr_thresh*normy
605  $ .AND. y_prec_state .LT. extra_y )
606  $ incr_prec = .true.
607 
608  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
609  $ x_state = working_state
610  IF ( x_state .EQ. working_state ) THEN
611  IF ( dx_x .LE. eps ) THEN
612  x_state = conv_state
613  ELSE IF ( dxrat .GT. rthresh ) THEN
614  IF ( y_prec_state .NE. extra_y ) THEN
615  incr_prec = .true.
616  ELSE
617  x_state = noprog_state
618  END IF
619  ELSE
620  IF (dxrat .GT. dxratmax) dxratmax = dxrat
621  END IF
622  IF ( x_state .GT. working_state ) final_dx_x = dx_x
623  END IF
624 
625  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
626  $ z_state = working_state
627  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
628  $ z_state = working_state
629  IF ( z_state .EQ. working_state ) THEN
630  IF ( dz_z .LE. eps ) THEN
631  z_state = conv_state
632  ELSE IF ( dz_z .GT. dz_ub ) THEN
633  z_state = unstable_state
634  dzratmax = 0.0
635  final_dz_z = hugeval
636  ELSE IF ( dzrat .GT. rthresh ) THEN
637  IF ( y_prec_state .NE. extra_y ) THEN
638  incr_prec = .true.
639  ELSE
640  z_state = noprog_state
641  END IF
642  ELSE
643  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
644  END IF
645  IF ( z_state .GT. working_state ) final_dz_z = dz_z
646  END IF
647 
648  IF ( x_state.NE.working_state.AND.
649  $ ( ignore_cwise.OR.z_state.NE.working_state ) )
650  $ GOTO 666
651 
652  IF ( incr_prec ) THEN
653  incr_prec = .false.
654  y_prec_state = y_prec_state + 1
655  DO i = 1, n
656  y_tail( i ) = 0.0
657  END DO
658  END IF
659 
660  prevnormdx = normdx
661  prev_dz_z = dz_z
662 *
663 * Update soluton.
664 *
665  IF ( y_prec_state .LT. extra_y ) THEN
666  CALL caxpy( n, cmplx(1.0), dy, 1, y(1,j), 1 )
667  ELSE
668  CALL cla_wwaddw( n, y(1,j), y_tail, dy )
669  END IF
670 
671  END DO
672 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
673  666 CONTINUE
674 *
675 * Set final_* when cnt hits ithresh.
676 *
677  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
678  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
679 *
680 * Compute error bounds.
681 *
682  IF ( n_norms .GE. 1 ) THEN
683  err_bnds_norm( j, la_linrx_err_i ) =
684  $ final_dx_x / (1 - dxratmax)
685  END IF
686  IF (n_norms .GE. 2) THEN
687  err_bnds_comp( j, la_linrx_err_i ) =
688  $ final_dz_z / (1 - dzratmax)
689  END IF
690 *
691 * Compute componentwise relative backward error from formula
692 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
693 * where abs(Z) is the componentwise absolute value of the matrix
694 * or vector Z.
695 *
696 * Compute residual RES = B_s - op(A_s) * Y,
697 * op(A) = A, A**T, or A**H depending on TRANS (and type).
698 *
699  CALL ccopy( n, b( 1, j ), 1, res, 1 )
700  CALL chemv( uplo, n, cmplx(-1.0), a, lda, y(1,j), 1,
701  $ cmplx(1.0), res, 1 )
702 
703  DO i = 1, n
704  ayb( i ) = cabs1( b( i, j ) )
705  END DO
706 *
707 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
708 *
709  CALL cla_heamv( uplo2, n, 1.0,
710  $ a, lda, y(1, j), 1, 1.0, ayb, 1 )
711 
712  CALL cla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
713 *
714 * End of loop for each RHS.
715 *
716  END DO
717 *
718  RETURN
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
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:180
integer function ilauplo(UPLO)
ILAUPLO
Definition: ilauplo.f:60
subroutine chetrs(UPLO, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
CHETRS
Definition: chetrs.f:122
subroutine chemv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
CHEMV
Definition: chemv.f:156
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
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

Here is the call graph for this function:

Here is the caller graph for this function: