LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ dla_porfsx_extended()

subroutine dla_porfsx_extended ( integer  PREC_TYPE,
character  UPLO,
integer  N,
integer  NRHS,
double precision, dimension( lda, * )  A,
integer  LDA,
double precision, dimension( ldaf, * )  AF,
integer  LDAF,
logical  COLEQU,
double precision, dimension( * )  C,
double precision, dimension( ldb, * )  B,
integer  LDB,
double precision, 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,
double precision, dimension( * )  RES,
double precision, dimension(*)  AYB,
double precision, dimension( * )  DY,
double precision, dimension( * )  Y_TAIL,
double precision  RCOND,
integer  ITHRESH,
double precision  RTHRESH,
double precision  DZ_UB,
logical  IGNORE_CWISE,
integer  INFO 
)

DLA_PORFSX_EXTENDED improves the computed solution to a system of linear equations for symmetric or Hermitian positive-definite matrices by performing extra-precise iterative refinement and provides error bounds and backward error estimates for the solution.

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

Purpose:
 DLA_PORFSX_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 DPORFSX 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 responsible 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]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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDAF,N)
     The triangular factor U or L from the Cholesky factorization
     A = U**T*U or A = L*L**T, as computed by DPOTRF.
[in]LDAF
          LDAF is INTEGER
     The leading dimension of the array AF.  LDAF >= max(1,N).
[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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDY,NRHS)
     On entry, the solution matrix X, as computed by DPOTRS.
     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 DLA_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 DOUBLE PRECISION array, dimension (N)
     Workspace to hold the intermediate residual.
[in]AYB
          AYB is DOUBLE PRECISION array, dimension (N)
     Workspace. This can be the same workspace passed for Y_TAIL.
[in]DY
          DY is DOUBLE PRECISION array, dimension (N)
     Workspace to hold the intermediate solution.
[in]Y_TAIL
          Y_TAIL is DOUBLE PRECISION 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 DPOTRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 380 of file dla_porfsx_extended.f.

387*
388* -- LAPACK computational routine --
389* -- LAPACK is a software package provided by Univ. of Tennessee, --
390* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
391*
392* .. Scalar Arguments ..
393 INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
394 $ N_NORMS, ITHRESH
395 CHARACTER UPLO
396 LOGICAL COLEQU, IGNORE_CWISE
397 DOUBLE PRECISION RTHRESH, DZ_UB
398* ..
399* .. Array Arguments ..
400 DOUBLE PRECISION A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
401 $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
402 DOUBLE PRECISION C( * ), AYB(*), RCOND, BERR_OUT( * ),
403 $ ERR_BNDS_NORM( NRHS, * ),
404 $ ERR_BNDS_COMP( NRHS, * )
405* ..
406*
407* =====================================================================
408*
409* .. Local Scalars ..
410 INTEGER UPLO2, CNT, I, J, X_STATE, Z_STATE
411 DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
412 $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
413 $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
414 $ EPS, HUGEVAL, INCR_THRESH
415 LOGICAL INCR_PREC
416* ..
417* .. Parameters ..
418 INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
419 $ NOPROG_STATE, Y_PREC_STATE, BASE_RESIDUAL,
420 $ EXTRA_RESIDUAL, EXTRA_Y
421 parameter( unstable_state = 0, working_state = 1,
422 $ conv_state = 2, noprog_state = 3 )
423 parameter( base_residual = 0, extra_residual = 1,
424 $ extra_y = 2 )
425 INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
426 INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
427 INTEGER CMP_ERR_I, PIV_GROWTH_I
428 parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
429 $ berr_i = 3 )
430 parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
431 parameter( cmp_rcond_i = 7, cmp_err_i = 8,
432 $ piv_growth_i = 9 )
433 INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
434 $ LA_LINRX_CWISE_I
435 parameter( la_linrx_itref_i = 1,
436 $ la_linrx_ithresh_i = 2 )
437 parameter( la_linrx_cwise_i = 3 )
438 INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
439 $ LA_LINRX_RCOND_I
440 parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
441 parameter( la_linrx_rcond_i = 3 )
442* ..
443* .. External Functions ..
444 LOGICAL LSAME
445 EXTERNAL ilauplo
446 INTEGER ILAUPLO
447* ..
448* .. External Subroutines ..
449 EXTERNAL daxpy, dcopy, dpotrs, dsymv, blas_dsymv_x,
450 $ blas_dsymv2_x, dla_syamv, dla_wwaddw,
452 DOUBLE PRECISION DLAMCH
453* ..
454* .. Intrinsic Functions ..
455 INTRINSIC abs, max, min
456* ..
457* .. Executable Statements ..
458*
459 IF (info.NE.0) RETURN
460 eps = dlamch( 'Epsilon' )
461 hugeval = dlamch( 'Overflow' )
462* Force HUGEVAL to Inf
463 hugeval = hugeval * hugeval
464* Using HUGEVAL may lead to spurious underflows.
465 incr_thresh = dble( n ) * eps
466
467 IF ( lsame( uplo, 'L' ) ) THEN
468 uplo2 = ilauplo( 'L' )
469 ELSE
470 uplo2 = ilauplo( 'U' )
471 ENDIF
472
473 DO j = 1, nrhs
474 y_prec_state = extra_residual
475 IF ( y_prec_state .EQ. extra_y ) THEN
476 DO i = 1, n
477 y_tail( i ) = 0.0d+0
478 END DO
479 END IF
480
481 dxrat = 0.0d+0
482 dxratmax = 0.0d+0
483 dzrat = 0.0d+0
484 dzratmax = 0.0d+0
485 final_dx_x = hugeval
486 final_dz_z = hugeval
487 prevnormdx = hugeval
488 prev_dz_z = hugeval
489 dz_z = hugeval
490 dx_x = hugeval
491
492 x_state = working_state
493 z_state = unstable_state
494 incr_prec = .false.
495
496 DO cnt = 1, ithresh
497*
498* Compute residual RES = B_s - op(A_s) * Y,
499* op(A) = A, A**T, or A**H depending on TRANS (and type).
500*
501 CALL dcopy( n, b( 1, j ), 1, res, 1 )
502 IF ( y_prec_state .EQ. base_residual ) THEN
503 CALL dsymv( uplo, n, -1.0d+0, a, lda, y(1,j), 1,
504 $ 1.0d+0, res, 1 )
505 ELSE IF ( y_prec_state .EQ. extra_residual ) THEN
506 CALL blas_dsymv_x( uplo2, n, -1.0d+0, a, lda,
507 $ y( 1, j ), 1, 1.0d+0, res, 1, prec_type )
508 ELSE
509 CALL blas_dsymv2_x(uplo2, n, -1.0d+0, a, lda,
510 $ y(1, j), y_tail, 1, 1.0d+0, res, 1, prec_type)
511 END IF
512
513! XXX: RES is no longer needed.
514 CALL dcopy( n, res, 1, dy, 1 )
515 CALL dpotrs( uplo, n, 1, af, ldaf, dy, n, info )
516*
517* Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
518*
519 normx = 0.0d+0
520 normy = 0.0d+0
521 normdx = 0.0d+0
522 dz_z = 0.0d+0
523 ymin = hugeval
524
525 DO i = 1, n
526 yk = abs( y( i, j ) )
527 dyk = abs( dy( i ) )
528
529 IF ( yk .NE. 0.0d+0 ) THEN
530 dz_z = max( dz_z, dyk / yk )
531 ELSE IF ( dyk .NE. 0.0d+0 ) THEN
532 dz_z = hugeval
533 END IF
534
535 ymin = min( ymin, yk )
536
537 normy = max( normy, yk )
538
539 IF ( colequ ) THEN
540 normx = max( normx, yk * c( i ) )
541 normdx = max( normdx, dyk * c( i ) )
542 ELSE
543 normx = normy
544 normdx = max( normdx, dyk )
545 END IF
546 END DO
547
548 IF ( normx .NE. 0.0d+0 ) THEN
549 dx_x = normdx / normx
550 ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
551 dx_x = 0.0d+0
552 ELSE
553 dx_x = hugeval
554 END IF
555
556 dxrat = normdx / prevnormdx
557 dzrat = dz_z / prev_dz_z
558*
559* Check termination criteria.
560*
561 IF ( ymin*rcond .LT. incr_thresh*normy
562 $ .AND. y_prec_state .LT. extra_y )
563 $ incr_prec = .true.
564
565 IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
566 $ x_state = working_state
567 IF ( x_state .EQ. working_state ) THEN
568 IF ( dx_x .LE. eps ) THEN
569 x_state = conv_state
570 ELSE IF ( dxrat .GT. rthresh ) THEN
571 IF ( y_prec_state .NE. extra_y ) THEN
572 incr_prec = .true.
573 ELSE
574 x_state = noprog_state
575 END IF
576 ELSE
577 IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
578 END IF
579 IF ( x_state .GT. working_state ) final_dx_x = dx_x
580 END IF
581
582 IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
583 $ z_state = working_state
584 IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
585 $ z_state = working_state
586 IF ( z_state .EQ. working_state ) THEN
587 IF ( dz_z .LE. eps ) THEN
588 z_state = conv_state
589 ELSE IF ( dz_z .GT. dz_ub ) THEN
590 z_state = unstable_state
591 dzratmax = 0.0d+0
592 final_dz_z = hugeval
593 ELSE IF ( dzrat .GT. rthresh ) THEN
594 IF ( y_prec_state .NE. extra_y ) THEN
595 incr_prec = .true.
596 ELSE
597 z_state = noprog_state
598 END IF
599 ELSE
600 IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
601 END IF
602 IF ( z_state .GT. working_state ) final_dz_z = dz_z
603 END IF
604
605 IF ( x_state.NE.working_state.AND.
606 $ ( ignore_cwise.OR.z_state.NE.working_state ) )
607 $ GOTO 666
608
609 IF ( incr_prec ) THEN
610 incr_prec = .false.
611 y_prec_state = y_prec_state + 1
612 DO i = 1, n
613 y_tail( i ) = 0.0d+0
614 END DO
615 END IF
616
617 prevnormdx = normdx
618 prev_dz_z = dz_z
619*
620* Update soluton.
621*
622 IF (y_prec_state .LT. extra_y) THEN
623 CALL daxpy( n, 1.0d+0, dy, 1, y(1,j), 1 )
624 ELSE
625 CALL dla_wwaddw( n, y( 1, j ), y_tail, dy )
626 END IF
627
628 END DO
629* Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
630 666 CONTINUE
631*
632* Set final_* when cnt hits ithresh.
633*
634 IF ( x_state .EQ. working_state ) final_dx_x = dx_x
635 IF ( z_state .EQ. working_state ) final_dz_z = dz_z
636*
637* Compute error bounds.
638*
639 IF ( n_norms .GE. 1 ) THEN
640 err_bnds_norm( j, la_linrx_err_i ) =
641 $ final_dx_x / (1 - dxratmax)
642 END IF
643 IF ( n_norms .GE. 2 ) THEN
644 err_bnds_comp( j, la_linrx_err_i ) =
645 $ final_dz_z / (1 - dzratmax)
646 END IF
647*
648* Compute componentwise relative backward error from formula
649* max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
650* where abs(Z) is the componentwise absolute value of the matrix
651* or vector Z.
652*
653* Compute residual RES = B_s - op(A_s) * Y,
654* op(A) = A, A**T, or A**H depending on TRANS (and type).
655*
656 CALL dcopy( n, b( 1, j ), 1, res, 1 )
657 CALL dsymv( uplo, n, -1.0d+0, a, lda, y(1,j), 1, 1.0d+0, res,
658 $ 1 )
659
660 DO i = 1, n
661 ayb( i ) = abs( b( i, j ) )
662 END DO
663*
664* Compute abs(op(A_s))*abs(Y) + abs(B_s).
665*
666 CALL dla_syamv( uplo2, n, 1.0d+0,
667 $ a, lda, y(1, j), 1, 1.0d+0, ayb, 1 )
668
669 CALL dla_lin_berr( n, n, 1, res, ayb, berr_out( j ) )
670*
671* End of loop for each RHS.
672*
673 END DO
674*
675 RETURN
676*
677* End of DLA_PORFSX_EXTENDED
678*
double precision function dlamch(CMACH)
DLAMCH
Definition: dlamch.f:69
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
integer function ilauplo(UPLO)
ILAUPLO
Definition: ilauplo.f:58
subroutine dcopy(N, DX, INCX, DY, INCY)
DCOPY
Definition: dcopy.f:82
subroutine daxpy(N, DA, DX, INCX, DY, INCY)
DAXPY
Definition: daxpy.f:89
subroutine dsymv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DSYMV
Definition: dsymv.f:152
subroutine dla_lin_berr(N, NZ, NRHS, RES, AYB, BERR)
DLA_LIN_BERR computes a component-wise relative backward error.
Definition: dla_lin_berr.f:101
subroutine dla_wwaddw(N, X, Y, W)
DLA_WWADDW adds a vector into a doubled-single vector.
Definition: dla_wwaddw.f:81
subroutine dpotrs(UPLO, N, NRHS, A, LDA, B, LDB, INFO)
DPOTRS
Definition: dpotrs.f:110
subroutine dla_syamv(UPLO, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
DLA_SYAMV computes a matrix-vector product using a symmetric indefinite matrix to calculate error bou...
Definition: dla_syamv.f:177
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