LAPACK  3.10.1
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 responsible 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' 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]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 < 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 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 ZGETRS had an illegal
             value
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.

Definition at line 390 of file zla_gerfsx_extended.f.

396 *
397 * -- LAPACK computational routine --
398 * -- LAPACK is a software package provided by Univ. of Tennessee, --
399 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
400 *
401 * .. Scalar Arguments ..
402  INTEGER INFO, LDA, LDAF, LDB, LDY, N, NRHS, PREC_TYPE,
403  $ TRANS_TYPE, N_NORMS
404  LOGICAL COLEQU, IGNORE_CWISE
405  INTEGER ITHRESH
406  DOUBLE PRECISION RTHRESH, DZ_UB
407 * ..
408 * .. Array Arguments
409  INTEGER IPIV( * )
410  COMPLEX*16 A( LDA, * ), AF( LDAF, * ), B( LDB, * ),
411  $ Y( LDY, * ), RES( * ), DY( * ), Y_TAIL( * )
412  DOUBLE PRECISION C( * ), AYB( * ), RCOND, BERR_OUT( * ),
413  $ ERRS_N( NRHS, * ), ERRS_C( NRHS, * )
414 * ..
415 *
416 * =====================================================================
417 *
418 * .. Local Scalars ..
419  CHARACTER TRANS
420  INTEGER CNT, I, J, X_STATE, Z_STATE, Y_PREC_STATE
421  DOUBLE PRECISION YK, DYK, YMIN, NORMY, NORMX, NORMDX, DXRAT,
422  $ DZRAT, PREVNORMDX, PREV_DZ_Z, DXRATMAX,
423  $ DZRATMAX, DX_X, DZ_Z, FINAL_DX_X, FINAL_DZ_Z,
424  $ EPS, HUGEVAL, INCR_THRESH
425  LOGICAL INCR_PREC
426  COMPLEX*16 ZDUM
427 * ..
428 * .. Parameters ..
429  INTEGER UNSTABLE_STATE, WORKING_STATE, CONV_STATE,
430  $ NOPROG_STATE, BASE_RESIDUAL, EXTRA_RESIDUAL,
431  $ EXTRA_Y
432  parameter( unstable_state = 0, working_state = 1,
433  $ conv_state = 2,
434  $ noprog_state = 3 )
435  parameter( base_residual = 0, extra_residual = 1,
436  $ extra_y = 2 )
437  INTEGER FINAL_NRM_ERR_I, FINAL_CMP_ERR_I, BERR_I
438  INTEGER RCOND_I, NRM_RCOND_I, NRM_ERR_I, CMP_RCOND_I
439  INTEGER CMP_ERR_I, PIV_GROWTH_I
440  parameter( final_nrm_err_i = 1, final_cmp_err_i = 2,
441  $ berr_i = 3 )
442  parameter( rcond_i = 4, nrm_rcond_i = 5, nrm_err_i = 6 )
443  parameter( cmp_rcond_i = 7, cmp_err_i = 8,
444  $ piv_growth_i = 9 )
445  INTEGER LA_LINRX_ITREF_I, LA_LINRX_ITHRESH_I,
446  $ LA_LINRX_CWISE_I
447  parameter( la_linrx_itref_i = 1,
448  $ la_linrx_ithresh_i = 2 )
449  parameter( la_linrx_cwise_i = 3 )
450  INTEGER LA_LINRX_TRUST_I, LA_LINRX_ERR_I,
451  $ LA_LINRX_RCOND_I
452  parameter( la_linrx_trust_i = 1, la_linrx_err_i = 2 )
453  parameter( la_linrx_rcond_i = 3 )
454 * ..
455 * .. External Subroutines ..
456  EXTERNAL zaxpy, zcopy, zgetrs, zgemv, blas_zgemv_x,
457  $ blas_zgemv2_x, zla_geamv, zla_wwaddw, dlamch,
459  DOUBLE PRECISION DLAMCH
460  CHARACTER CHLA_TRANSTYPE
461 * ..
462 * .. Intrinsic Functions ..
463  INTRINSIC abs, max, min
464 * ..
465 * .. Statement Functions ..
466  DOUBLE PRECISION CABS1
467 * ..
468 * .. Statement Function Definitions ..
469  cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
470 * ..
471 * .. Executable Statements ..
472 *
473  IF ( info.NE.0 ) RETURN
474  trans = chla_transtype(trans_type)
475  eps = dlamch( 'Epsilon' )
476  hugeval = dlamch( 'Overflow' )
477 * Force HUGEVAL to Inf
478  hugeval = hugeval * hugeval
479 * Using HUGEVAL may lead to spurious underflows.
480  incr_thresh = dble( n ) * eps
481 *
482  DO j = 1, nrhs
483  y_prec_state = extra_residual
484  IF ( y_prec_state .EQ. extra_y ) THEN
485  DO i = 1, n
486  y_tail( i ) = 0.0d+0
487  END DO
488  END IF
489 
490  dxrat = 0.0d+0
491  dxratmax = 0.0d+0
492  dzrat = 0.0d+0
493  dzratmax = 0.0d+0
494  final_dx_x = hugeval
495  final_dz_z = hugeval
496  prevnormdx = hugeval
497  prev_dz_z = hugeval
498  dz_z = hugeval
499  dx_x = hugeval
500 
501  x_state = working_state
502  z_state = unstable_state
503  incr_prec = .false.
504 
505  DO cnt = 1, ithresh
506 *
507 * Compute residual RES = B_s - op(A_s) * Y,
508 * op(A) = A, A**T, or A**H depending on TRANS (and type).
509 *
510  CALL zcopy( n, b( 1, j ), 1, res, 1 )
511  IF ( y_prec_state .EQ. base_residual ) THEN
512  CALL zgemv( trans, n, n, (-1.0d+0,0.0d+0), a, lda,
513  $ y( 1, j ), 1, (1.0d+0,0.0d+0), res, 1)
514  ELSE IF (y_prec_state .EQ. extra_residual) THEN
515  CALL blas_zgemv_x( trans_type, n, n, (-1.0d+0,0.0d+0), a,
516  $ lda, y( 1, j ), 1, (1.0d+0,0.0d+0),
517  $ res, 1, prec_type )
518  ELSE
519  CALL blas_zgemv2_x( trans_type, n, n, (-1.0d+0,0.0d+0),
520  $ a, lda, y(1, j), y_tail, 1, (1.0d+0,0.0d+0), res, 1,
521  $ prec_type)
522  END IF
523 
524 ! XXX: RES is no longer needed.
525  CALL zcopy( n, res, 1, dy, 1 )
526  CALL zgetrs( trans, n, 1, af, ldaf, ipiv, dy, n, info )
527 *
528 * Calculate relative changes DX_X, DZ_Z and ratios DXRAT, DZRAT.
529 *
530  normx = 0.0d+0
531  normy = 0.0d+0
532  normdx = 0.0d+0
533  dz_z = 0.0d+0
534  ymin = hugeval
535 *
536  DO i = 1, n
537  yk = cabs1( y( i, j ) )
538  dyk = cabs1( dy( i ) )
539 
540  IF ( yk .NE. 0.0d+0 ) THEN
541  dz_z = max( dz_z, dyk / yk )
542  ELSE IF ( dyk .NE. 0.0d+0 ) THEN
543  dz_z = hugeval
544  END IF
545 
546  ymin = min( ymin, yk )
547 
548  normy = max( normy, yk )
549 
550  IF ( colequ ) THEN
551  normx = max( normx, yk * c( i ) )
552  normdx = max( normdx, dyk * c( i ) )
553  ELSE
554  normx = normy
555  normdx = max(normdx, dyk)
556  END IF
557  END DO
558 
559  IF ( normx .NE. 0.0d+0 ) THEN
560  dx_x = normdx / normx
561  ELSE IF ( normdx .EQ. 0.0d+0 ) THEN
562  dx_x = 0.0d+0
563  ELSE
564  dx_x = hugeval
565  END IF
566 
567  dxrat = normdx / prevnormdx
568  dzrat = dz_z / prev_dz_z
569 *
570 * Check termination criteria
571 *
572  IF (.NOT.ignore_cwise
573  $ .AND. ymin*rcond .LT. incr_thresh*normy
574  $ .AND. y_prec_state .LT. extra_y )
575  $ incr_prec = .true.
576 
577  IF ( x_state .EQ. noprog_state .AND. dxrat .LE. rthresh )
578  $ x_state = working_state
579  IF ( x_state .EQ. working_state ) THEN
580  IF (dx_x .LE. eps) THEN
581  x_state = conv_state
582  ELSE IF ( dxrat .GT. rthresh ) THEN
583  IF ( y_prec_state .NE. extra_y ) THEN
584  incr_prec = .true.
585  ELSE
586  x_state = noprog_state
587  END IF
588  ELSE
589  IF ( dxrat .GT. dxratmax ) dxratmax = dxrat
590  END IF
591  IF ( x_state .GT. working_state ) final_dx_x = dx_x
592  END IF
593 
594  IF ( z_state .EQ. unstable_state .AND. dz_z .LE. dz_ub )
595  $ z_state = working_state
596  IF ( z_state .EQ. noprog_state .AND. dzrat .LE. rthresh )
597  $ z_state = working_state
598  IF ( z_state .EQ. working_state ) THEN
599  IF ( dz_z .LE. eps ) THEN
600  z_state = conv_state
601  ELSE IF ( dz_z .GT. dz_ub ) THEN
602  z_state = unstable_state
603  dzratmax = 0.0d+0
604  final_dz_z = hugeval
605  ELSE IF ( dzrat .GT. rthresh ) THEN
606  IF ( y_prec_state .NE. extra_y ) THEN
607  incr_prec = .true.
608  ELSE
609  z_state = noprog_state
610  END IF
611  ELSE
612  IF ( dzrat .GT. dzratmax ) dzratmax = dzrat
613  END IF
614  IF ( z_state .GT. working_state ) final_dz_z = dz_z
615  END IF
616 *
617 * Exit if both normwise and componentwise stopped working,
618 * but if componentwise is unstable, let it go at least two
619 * iterations.
620 *
621  IF ( x_state.NE.working_state ) THEN
622  IF ( ignore_cwise ) GOTO 666
623  IF ( z_state.EQ.noprog_state .OR. z_state.EQ.conv_state )
624  $ GOTO 666
625  IF ( z_state.EQ.unstable_state .AND. cnt.GT.1 ) GOTO 666
626  END IF
627 
628  IF ( incr_prec ) THEN
629  incr_prec = .false.
630  y_prec_state = y_prec_state + 1
631  DO i = 1, n
632  y_tail( i ) = 0.0d+0
633  END DO
634  END IF
635 
636  prevnormdx = normdx
637  prev_dz_z = dz_z
638 *
639 * Update soluton.
640 *
641  IF ( y_prec_state .LT. extra_y ) THEN
642  CALL zaxpy( n, (1.0d+0,0.0d+0), dy, 1, y(1,j), 1 )
643  ELSE
644  CALL zla_wwaddw( n, y( 1, j ), y_tail, dy )
645  END IF
646 
647  END DO
648 * Target of "IF (Z_STOP .AND. X_STOP)". Sun's f77 won't EXIT.
649  666 CONTINUE
650 *
651 * Set final_* when cnt hits ithresh
652 *
653  IF ( x_state .EQ. working_state ) final_dx_x = dx_x
654  IF ( z_state .EQ. working_state ) final_dz_z = dz_z
655 *
656 * Compute error bounds
657 *
658  IF (n_norms .GE. 1) THEN
659  errs_n( j, la_linrx_err_i ) = final_dx_x / (1 - dxratmax)
660 
661  END IF
662  IF ( n_norms .GE. 2 ) THEN
663  errs_c( j, la_linrx_err_i ) = final_dz_z / (1 - dzratmax)
664  END IF
665 *
666 * Compute componentwise relative backward error from formula
667 * max(i) ( abs(R(i)) / ( abs(op(A_s))*abs(Y) + abs(B_s) )(i) )
668 * where abs(Z) is the componentwise absolute value of the matrix
669 * or vector Z.
670 *
671 * Compute residual RES = B_s - op(A_s) * Y,
672 * op(A) = A, A**T, or A**H depending on TRANS (and type).
673 *
674  CALL zcopy( n, b( 1, j ), 1, res, 1 )
675  CALL zgemv( trans, n, n, (-1.0d+0,0.0d+0), a, lda, y(1,j), 1,
676  $ (1.0d+0,0.0d+0), res, 1 )
677 
678  DO i = 1, n
679  ayb( i ) = cabs1( b( i, j ) )
680  END DO
681 *
682 * Compute abs(op(A_s))*abs(Y) + abs(B_s).
683 *
684  CALL zla_geamv ( trans_type, n, n, 1.0d+0,
685  $ a, lda, y(1, j), 1, 1.0d+0, ayb, 1 )
686 
687  CALL zla_lin_berr ( n, n, 1, res, ayb, berr_out( j ) )
688 *
689 * End of loop for each RHS.
690 *
691  END DO
692 *
693  RETURN
694 *
695 * End of ZLA_GERFSX_EXTENDED
696 *
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 zgemv(TRANS, M, N, ALPHA, A, LDA, X, INCX, BETA, Y, INCY)
ZGEMV
Definition: zgemv.f:158
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:175
subroutine zgetrs(TRANS, N, NRHS, A, LDA, IPIV, B, LDB, INFO)
ZGETRS
Definition: zgetrs.f:121
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
Here is the call graph for this function:
Here is the caller graph for this function: