LAPACK  3.8.0
LAPACK: Linear Algebra PACKage

◆ zla_gerfsx_extended()

subroutine zla_gerfsx_extended ( integer  PREC_TYPE,
integer  TRANS_TYPE,
integer  N,
integer  NRHS,
complex*16, dimension( lda, * )  A,
integer  LDA,
complex*16, dimension( ldaf, * )  AF,
integer  LDAF,
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, * )  ERRS_N,
double precision, dimension( nrhs, * )  ERRS_C,
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_GERFSX_EXTENDED

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

Purpose:
 ZLA_GERFSX_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 ZGERFSX to perform iterative refinement.
 In addition to normwise error bound, the code provides maximum
 componentwise error bound if possible. See comments for ERRS_N
 and ERRS_C for details of the error bounds. Note that this
 subroutine is only resonsible for setting the second fields of
 ERRS_N and ERRS_C.
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]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*16 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*16 array, dimension (LDAF,N)
     The factors L and U from the factorization
     A = P*L*U as computed by ZGETRF.
[in]LDAF
          LDAF 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 ZGETRF; 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 ZGETRS.
     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 ERRS_N
     and ERRS_C).
     If N_NORMS >= 1 return normwise error bounds.
     If N_NORMS >= 2 return componentwise error bounds.
[in,out]ERRS_N
          ERRS_N 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 ERRS_N(i,:) corresponds to the ith
     right-hand side.

     The second index in ERRS_N(:,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]ERRS_C
          ERRS_C 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
     ERRS_C is not accessed.  If N_ERR_BNDS .LT. 3, then at most
     the first (:,N_ERR_BNDS) entries are returned.

     The first index in ERRS_C(i,:) corresponds to the ith
     right-hand side.

     The second index in ERRS_C(:,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
     ERRS_N and ERRS_C 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 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 ZGETRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date
June 2017

Definition at line 398 of file zla_gerfsx_extended.f.

398 *
399 * -- LAPACK computational routine (version 3.7.1) --
400 * -- LAPACK is a software package provided by Univ. of Tennessee, --
401 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
402 * June 2017
403 *
404 * .. Scalar Arguments ..
405  INTEGER info, lda, ldaf, ldb, ldy, n, nrhs, prec_type,
406  $ trans_type, n_norms
407  LOGICAL colequ, ignore_cwise
408  INTEGER ithresh
409  DOUBLE PRECISION rthresh, dz_ub
410 * ..
411 * .. Array Arguments
412  INTEGER ipiv( * )
413  COMPLEX*16 a( lda, * ), af( ldaf, * ), b( ldb, * ),
414  $ y( ldy, * ), res( * ), dy( * ), y_tail( * )
415  DOUBLE PRECISION c( * ), ayb( * ), rcond, berr_out( * ),
416  $ errs_n( nrhs, * ), errs_c( nrhs, * )
417 * ..
418 *
419 * =====================================================================
420 *
421 * .. Local Scalars ..
422  CHARACTER trans
423  INTEGER cnt, i, j, x_state, z_state, y_prec_state
424  DOUBLE PRECISION yk, dyk, ymin, normy, normx, normdx, dxrat,
425  $ dzrat, prevnormdx, prev_dz_z, dxratmax,
426  $ dzratmax, dx_x, dz_z, final_dx_x, final_dz_z,
427  $ eps, hugeval, incr_thresh
428  LOGICAL incr_prec
429  COMPLEX*16 zdum
430 * ..
431 * .. Parameters ..
432  INTEGER unstable_state, working_state, conv_state,
433  $ noprog_state, base_residual, extra_residual,
434  $ extra_y
435  parameter( unstable_state = 0, working_state = 1,
436  $ conv_state = 2,
437  $ noprog_state = 3 )
438  parameter( base_residual = 0, extra_residual = 1,
439  $ extra_y = 2 )
440  INTEGER final_nrm_err_i, final_cmp_err_i, berr_i
441  INTEGER rcond_i, nrm_rcond_i, nrm_err_i, cmp_rcond_i
442  INTEGER cmp_err_i, piv_growth_i
443  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
444  $ berr_i = 3 )
445  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
446  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
447  $ piv_growth_i = 9 )
448  INTEGER la_linrx_itref_i, la_linrx_ithresh_i,
449  $ la_linrx_cwise_i
450  parameter( la_linrx_itref_i = 1,
451  $ la_linrx_ithresh_i = 2 )
452  parameter( la_linrx_cwise_i = 3 )
453  INTEGER la_linrx_trust_i, la_linrx_err_i,
454  $ la_linrx_rcond_i
455  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
456  parameter( la_linrx_rcond_i = 3 )
457 * ..
458 * .. External Subroutines ..
459  EXTERNAL zaxpy, zcopy, zgetrs, zgemv, blas_zgemv_x,
460  $ blas_zgemv2_x, zla_geamv, zla_wwaddw, dlamch,
462  DOUBLE PRECISION dlamch
463  CHARACTER chla_transtype
464 * ..
465 * .. Intrinsic Functions ..
466  INTRINSIC abs, max, min
467 * ..
468 * .. Statement Functions ..
469  DOUBLE PRECISION cabs1
470 * ..
471 * .. Statement Function Definitions ..
472  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
473 * ..
474 * .. Executable Statements ..
475 *
476  IF ( info.NE.0 ) RETURN
477  trans = chla_transtype(trans_type)
478  eps = dlamch( 'Epsilon' )
479  hugeval = dlamch( 'Overflow' )
480 * Force HUGEVAL to Inf
481  hugeval = hugeval * hugeval
482 * Using HUGEVAL may lead to spurious underflows.
483  incr_thresh = dble( n ) * eps
484 *
485  DO j = 1, nrhs
486  y_prec_state = extra_residual
487  IF ( y_prec_state .EQ. extra_y ) THEN
488  DO i = 1, n
489  y_tail( i ) = 0.0d+0
490  END DO
491  END IF
492 
493  dxrat = 0.0d+0
494  dxratmax = 0.0d+0
495  dzrat = 0.0d+0
496  dzratmax = 0.0d+0
497  final_dx_x = hugeval
498  final_dz_z = hugeval
499  prevnormdx = hugeval
500  prev_dz_z = hugeval
501  dz_z = hugeval
502  dx_x = hugeval
503 
504  x_state = working_state
505  z_state = unstable_state
506  incr_prec = .false.
507 
508  DO cnt = 1, ithresh
509 *
510 * Compute residual RES = B_s - op(A_s) * Y,
511 * op(A) = A, A**T, or A**H depending on TRANS (and type).
512 *
513  CALL zcopy( n, b( 1, j ), 1, res, 1 )
514  IF ( y_prec_state .EQ. base_residual ) THEN
515  CALL zgemv( trans, n, n, (-1.0d+0,0.0d+0), a, lda,
516  $ y( 1, j ), 1, (1.0d+0,0.0d+0), res, 1)
517  ELSE IF (y_prec_state .EQ. extra_residual) THEN
518  CALL blas_zgemv_x( trans_type, n, n, (-1.0d+0,0.0d+0), a,
519  $ lda, y( 1, j ), 1, (1.0d+0,0.0d+0),
520  $ res, 1, prec_type )
521  ELSE
522  CALL blas_zgemv2_x( trans_type, n, n, (-1.0d+0,0.0d+0),
523  $ a, lda, y(1, j), y_tail, 1, (1.0d+0,0.0d+0), res, 1,
524  $ prec_type)
525  END IF
526 
527 ! XXX: RES is no longer needed.
528  CALL zcopy( n, res, 1, dy, 1 )
529  CALL zgetrs( trans, n, 1, af, ldaf, ipiv, dy, n, info )
530 *
531 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
532 *
533  normx = 0.0d+0
534  normy = 0.0d+0
535  normdx = 0.0d+0
536  dz_z = 0.0d+0
537  ymin = hugeval
538 *
539  DO i = 1, n
540  yk = cabs1( y( i, j ) )
541  dyk = cabs1( dy( i ) )
542 
543  IF ( yk .NE. 0.0d+0 ) THEN
544  dz_z = max( dz_z, dyk / yk )
545  ELSE IF ( dyk .NE. 0.0d+0 ) THEN
546  dz_z = hugeval
547  END IF
548 
549  ymin = min( ymin, yk )
550 
551  normy = max( normy, yk )
552 
553  IF ( colequ ) THEN
554  normx = max( normx, yk * c( i ) )
555  normdx = max( normdx, dyk * c( i ) )
556  ELSE
557  normx = normy
558  normdx = max(normdx, dyk)
559  END IF
560  END DO
561 
562  IF ( normx .NE. 0.0d+0 ) THEN
563  dx_x = normdx / normx
564  ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
565  dx_x = 0.0d+0
566  ELSE
567  dx_x = hugeval
568  END IF
569 
570  dxrat = normdx / prevnormdx
571  dzrat = dz_z / prev_dz_z
572 *
573 * Check termination criteria
574 *
575  IF (.NOT.ignore_cwise
576  $ .AND. ymin*rcond .LT. incr_thresh*normy
577  $ .AND. y_prec_state .LT. extra_y )
578  $ incr_prec = .true.
579 
580  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
581  $ x_state = working_state
582  IF ( x_state .EQ. working_state ) THEN
583  IF (dx_x .LE. eps) THEN
584  x_state = conv_state
585  ELSE IF ( dxrat .GT. rthresh ) THEN
586  IF ( y_prec_state .NE. extra_y ) THEN
587  incr_prec = .true.
588  ELSE
589  x_state = noprog_state
590  END IF
591  ELSE
592  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
593  END IF
594  IF ( x_state .GT. working_state ) final_dx_x = dx_x
595  END IF
596 
597  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
598  $ z_state = working_state
599  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
600  $ z_state = working_state
601  IF ( z_state .EQ. working_state ) THEN
602  IF ( dz_z .LE. eps ) THEN
603  z_state = conv_state
604  ELSE IF ( dz_z .GT. dz_ub ) THEN
605  z_state = unstable_state
606  dzratmax = 0.0d+0
607  final_dz_z = hugeval
608  ELSE IF ( dzrat .GT. rthresh ) THEN
609  IF ( y_prec_state .NE. extra_y ) THEN
610  incr_prec = .true.
611  ELSE
612  z_state = noprog_state
613  END IF
614  ELSE
615  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
616  END IF
617  IF ( z_state .GT. working_state ) final_dz_z = dz_z
618  END IF
619 *
620 * Exit if both normwise and componentwise stopped working,
621 * but if componentwise is unstable, let it go at least two
622 * iterations.
623 *
624  IF ( x_state.NE.working_state ) THEN
625  IF ( ignore_cwise ) GOTO 666
626  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
627  $ GOTO 666
628  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
629  END IF
630 
631  IF ( incr_prec ) THEN
632  incr_prec = .false.
633  y_prec_state = y_prec_state + 1
634  DO i = 1, n
635  y_tail( i ) = 0.0d+0
636  END DO
637  END IF
638 
639  prevnormdx = normdx
640  prev_dz_z = dz_z
641 *
642 * Update soluton.
643 *
644  IF ( y_prec_state .LT. extra_y ) THEN
645  CALL zaxpy( n, (1.0d+0,0.0d+0), dy, 1, y(1,j), 1 )
646  ELSE
647  CALL zla_wwaddw( n, y( 1, j ), y_tail, dy )
648  END IF
649 
650  END DO
651 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
652  666 CONTINUE
653 *
654 * Set final_* when cnt hits ithresh
655 *
656  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
657  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
658 *
659 * Compute error bounds
660 *
661  IF (n_norms .GE. 1) THEN
662  errs_n( j, la_linrx_err_i ) = final_dx_x / (1 - dxratmax)
663 
664  END IF
665  IF ( n_norms .GE. 2 ) THEN
666  errs_c( j, la_linrx_err_i ) = final_dz_z / (1 - dzratmax)
667  END IF
668 *
669 * Compute componentwise relative backward error from formula
670 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
671 * where abs(Z) is the componentwise absolute value of the matrix
672 * or vector Z.
673 *
674 * Compute residual RES = B_s - op(A_s) * Y,
675 * op(A) = A, A**T, or A**H depending on TRANS (and type).
676 *
677  CALL zcopy( n, b( 1, j ), 1, res, 1 )
678  CALL zgemv( trans, n, n, (-1.0d+0,0.0d+0), a, lda, y(1,j), 1,
679  $ (1.0d+0,0.0d+0), res, 1 )
680 
681  DO i = 1, n
682  ayb( i ) = cabs1( b( i, j ) )
683  END DO
684 *
685 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
686 *
687  CALL zla_geamv ( trans_type, n, n, 1.0d+0,
688  $ a, lda, y(1, j), 1, 1.0d+0, ayb, 1 )
689 
690  CALL zla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
691 *
692 * End of loop for each RHS.
693 *
694  END DO
695 *
696  RETURN
subroutine zgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZGETRS
Definition: zgetrs.f:123
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:65
subroutine zcopy(N, ZX, INCX, ZY, INCY)
ZCOPY
Definition: zcopy.f:83
subroutine zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:160
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:103
subroutine zla_wwaddw(N, X, Y, W)
ZLA_WWADDW adds a vector into a doubled-single vector.
Definition: zla_wwaddw.f:83
character *1 function chla_transtype(TRANS)
CHLA_TRANSTYPE
subroutine zaxpy(N, ZA, ZX, INCX, ZY, INCY)
ZAXPY
Definition: zaxpy.f:90
subroutine zla_geamv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZLA_GEAMV computes a matrix-vector product using a general matrix to calculate error bounds...
Definition: zla_geamv.f:177
Here is the call graph for this function:
Here is the caller graph for this function: