zgtt05.f

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*> \brief \b ZGTT05
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE ZGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
*                          XACT, LDXACT, FERR, BERR, RESLTS )
*
*       .. Scalar Arguments ..
*       CHARACTER          TRANS
*       INTEGER            LDB, LDX, LDXACT, N, NRHS
*       ..
*       .. Array Arguments ..
*       DOUBLE PRECISION   BERR( * ), FERR( * ), RESLTS( * )
*       COMPLEX*16         B( LDB, * ), D( * ), DL( * ), DU( * ),
*      $                   X( LDX, * ), XACT( LDXACT, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> ZGTT05 tests the error bounds from iterative refinement for the
*> computed solution to a system of equations A*X = B, where A is a
*> general tridiagonal matrix of order n and op(A) = A or A**T,
*> depending on TRANS.
*>
*> RESLTS(1) = test of the error bound
*>           = norm(X - XACT) / ( norm(X) * FERR )
*>
*> A large value is returned if this ratio is not less than one.
*>
*> RESLTS(2) = residual from the iterative refinement routine
*>           = the maximum of BERR / ( NZ*EPS + (*) ), where
*>             (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
*>             and NZ = max. number of nonzeros in any row of A, plus 1
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] TRANS
*> \verbatim
*>          TRANS is CHARACTER*1
*>          Specifies the form of the system of equations.
*>          = 'N':  A * X = B     (No transpose)
*>          = 'T':  A**T * X = B  (Transpose)
*>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows of the matrices X and XACT.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of columns of the matrices X and XACT.  NRHS >= 0.
*> \endverbatim
*>
*> \param[in] DL
*> \verbatim
*>          DL is COMPLEX*16 array, dimension (N-1)
*>          The (n-1) sub-diagonal elements of A.
*> \endverbatim
*>
*> \param[in] D
*> \verbatim
*>          D is COMPLEX*16 array, dimension (N)
*>          The diagonal elements of A.
*> \endverbatim
*>
*> \param[in] DU
*> \verbatim
*>          DU is COMPLEX*16 array, dimension (N-1)
*>          The (n-1) super-diagonal elements of A.
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is COMPLEX*16 array, dimension (LDB,NRHS)
*>          The right hand side vectors for the system of linear
*>          equations.
*> \endverbatim
*>
*> \param[in] LDB
*> \verbatim
*>          LDB is INTEGER
*>          The leading dimension of the array B.  LDB >= max(1,N).
*> \endverbatim
*>
*> \param[in] X
*> \verbatim
*>          X is COMPLEX*16 array, dimension (LDX,NRHS)
*>          The computed solution vectors.  Each vector is stored as a
*>          column of the matrix X.
*> \endverbatim
*>
*> \param[in] LDX
*> \verbatim
*>          LDX is INTEGER
*>          The leading dimension of the array X.  LDX >= max(1,N).
*> \endverbatim
*>
*> \param[in] XACT
*> \verbatim
*>          XACT is COMPLEX*16 array, dimension (LDX,NRHS)
*>          The exact solution vectors.  Each vector is stored as a
*>          column of the matrix XACT.
*> \endverbatim
*>
*> \param[in] LDXACT
*> \verbatim
*>          LDXACT is INTEGER
*>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
*> \endverbatim
*>
*> \param[in] FERR
*> \verbatim
*>          FERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The estimated forward error bounds for each solution vector
*>          X.  If XTRUE is the true solution, FERR bounds the magnitude
*>          of the largest entry in (X - XTRUE) divided by the magnitude
*>          of the largest entry in X.
*> \endverbatim
*>
*> \param[in] BERR
*> \verbatim
*>          BERR is DOUBLE PRECISION array, dimension (NRHS)
*>          The componentwise relative backward error of each solution
*>          vector (i.e., the smallest relative change in any entry of A
*>          or B that makes X an exact solution).
*> \endverbatim
*>
*> \param[out] RESLTS
*> \verbatim
*>          RESLTS is DOUBLE PRECISION array, dimension (2)
*>          The maximum over the NRHS solution vectors of the ratios:
*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
*>          RESLTS(2) = BERR / ( NZ*EPS + (*) )
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup complex16_lin
*
*  =====================================================================
      SUBROUTINE zgtt05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
     $                   XACT, LDXACT, FERR, BERR, RESLTS )
*
*  -- LAPACK test routine --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*
*     .. Scalar Arguments ..
      CHARACTER          TRANS
      INTEGER            LDB, LDX, LDXACT, N, NRHS
*     ..
*     .. Array Arguments ..
      DOUBLE PRECISION   BERR( * ), FERR( * ), RESLTS( * )
      COMPLEX*16         B( LDB, * ), D( * ), DL( * ), DU( * ),
     $                   x( ldx, * ), xact( ldxact, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      DOUBLE PRECISION   ZERO, ONE
      parameter( zero = 0.0d+0, one = 1.0d+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            NOTRAN
      INTEGER            I, IMAX, J, K, NZ
      DOUBLE PRECISION   AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
      COMPLEX*16         ZDUM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            IZAMAX
      DOUBLE PRECISION   DLAMCH
      EXTERNAL           lsame, izamax, dlamch
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, dble, dimag, max, min
*     ..
*     .. Statement Functions ..
      DOUBLE PRECISION   CABS1
*     ..
*     .. Statement Function definitions ..
      cabs1( zdum ) = abs( dble( zdum ) ) + abs( dimag( zdum ) )
*     ..
*     .. Executable Statements ..
*
*     Quick exit if N = 0 or NRHS = 0.
*
      IF( n.LE.0 .OR. nrhs.LE.0 ) THEN
         reslts( 1 ) = zero
         reslts( 2 ) = zero
         RETURN
      END IF
*
      eps = dlamch( 'epsilon' )
      UNFL = DLAMCH( 'safe minimum' )
      OVFL = ONE / UNFL
      NOTRAN = LSAME( TRANS, 'n' )
      NZ = 4
*
*     Test 1:  Compute the maximum of
*        norm(X - XACT) / ( norm(X) * FERR )
*     over all the vectors X and XACT using the infinity-norm.
*
      ERRBND = ZERO
      DO 30 J = 1, NRHS
         IMAX = IZAMAX( N, X( 1, J ), 1 )
         XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
         DIFF = ZERO
         DO 10 I = 1, N
            DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
   10    CONTINUE
*
.GT.         IF( XNORMONE ) THEN
            GO TO 20
.LE.         ELSE IF( DIFFOVFL*XNORM ) THEN
            GO TO 20
         ELSE
            ERRBND = ONE / EPS
            GO TO 30
         END IF
*
   20    CONTINUE
.LE.         IF( DIFF / XNORMFERR( J ) ) THEN
            ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
         ELSE
            ERRBND = ONE / EPS
         END IF
   30 CONTINUE
      RESLTS( 1 ) = ERRBND
*
*     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
*     (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
*
      DO 60 K = 1, NRHS
         IF( NOTRAN ) THEN
.EQ.            IF( N1 ) THEN
               AXBI = CABS1( B( 1, K ) ) +
     $                CABS1( D( 1 ) )*CABS1( X( 1, K ) )
            ELSE
               AXBI = CABS1( B( 1, K ) ) +
     $                CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
     $                CABS1( DU( 1 ) )*CABS1( X( 2, K ) )
               DO 40 I = 2, N - 1
                  TMP = CABS1( B( I, K ) ) +
     $                  CABS1( DL( I-1 ) )*CABS1( X( I-1, K ) ) +
     $                  CABS1( D( I ) )*CABS1( X( I, K ) ) +
     $                  CABS1( DU( I ) )*CABS1( X( I+1, K ) )
                  AXBI = MIN( AXBI, TMP )
   40          CONTINUE
               TMP = CABS1( B( N, K ) ) + CABS1( DL( N-1 ) )*
     $               CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
     $               CABS1( X( N, K ) )
               AXBI = MIN( AXBI, TMP )
            END IF
         ELSE
.EQ.            IF( N1 ) THEN
               AXBI = CABS1( B( 1, K ) ) +
     $                CABS1( D( 1 ) )*CABS1( X( 1, K ) )
            ELSE
               AXBI = CABS1( B( 1, K ) ) +
     $                CABS1( D( 1 ) )*CABS1( X( 1, K ) ) +
     $                CABS1( DL( 1 ) )*CABS1( X( 2, K ) )
               DO 50 I = 2, N - 1
                  TMP = CABS1( B( I, K ) ) +
     $                  CABS1( DU( I-1 ) )*CABS1( X( I-1, K ) ) +
     $                  CABS1( D( I ) )*CABS1( X( I, K ) ) +
     $                  CABS1( DL( I ) )*CABS1( X( I+1, K ) )
                  AXBI = MIN( AXBI, TMP )
   50          CONTINUE
               TMP = CABS1( B( N, K ) ) + CABS1( DU( N-1 ) )*
     $               CABS1( X( N-1, K ) ) + CABS1( D( N ) )*
     $               CABS1( X( N, K ) )
               AXBI = MIN( AXBI, TMP )
            END IF
         END IF
         TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
.EQ.         IF( K1 ) THEN
            RESLTS( 2 ) = TMP
         ELSE
            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
         END IF
   60 CONTINUE
*
      RETURN
*
*     End of ZGTT05
*
      END