spot05.f

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*> \brief \b SPOT05
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at
*            http://www.netlib.org/lapack/explore-html/
*
*  Definition:
*  ===========
*
*       SUBROUTINE SPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
*                          LDXACT, FERR, BERR, RESLTS )
*
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
*       ..
*       .. Array Arguments ..
*       REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
*      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
*       ..
*
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SPOT05 tests the error bounds from iterative refinement for the
*> computed solution to a system of equations A*X = B, where A is a
*> symmetric n by n matrix.
*>
*> 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 / ( (n+1)*EPS + (*) ), where
*>             (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
*> \endverbatim
*
*  Arguments:
*  ==========
*
*> \param[in] UPLO
*> \verbatim
*>          UPLO is CHARACTER*1
*>          Specifies whether the upper or lower triangular part of the
*>          symmetric matrix A is stored.
*>          = 'U':  Upper triangular
*>          = 'L':  Lower triangular
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*>          N is INTEGER
*>          The number of rows of the matrices X, B, and XACT, and the
*>          order of the matrix A.  N >= 0.
*> \endverbatim
*>
*> \param[in] NRHS
*> \verbatim
*>          NRHS is INTEGER
*>          The number of columns of the matrices X, B, and XACT.
*>          NRHS >= 0.
*> \endverbatim
*>
*> \param[in] A
*> \verbatim
*>          A is REAL array, dimension (LDA,N)
*>          The symmetric matrix A.  If UPLO = 'U', the leading n by n
*>          upper triangular part of A contains the upper triangular part
*>          of the matrix A, and the strictly lower triangular part of A
*>          is not referenced.  If UPLO = 'L', the leading n by n lower
*>          triangular part of A contains the lower triangular part of
*>          the matrix A, and the strictly upper triangular part of A is
*>          not referenced.
*> \endverbatim
*>
*> \param[in] LDA
*> \verbatim
*>          LDA is INTEGER
*>          The leading dimension of the array A.  LDA >= max(1,N).
*> \endverbatim
*>
*> \param[in] B
*> \verbatim
*>          B is REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL array, dimension (2)
*>          The maximum over the NRHS solution vectors of the ratios:
*>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
*>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \ingroup single_lin
*
*  =====================================================================
      SUBROUTINE spot05( UPLO, N, NRHS, A, LDA, 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          UPLO
      INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
*     ..
*     .. Array Arguments ..
      REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
     $                   reslts( * ), x( ldx, * ), xact( ldxact, * )
*     ..
*
*  =====================================================================
*
*     .. Parameters ..
      REAL               ZERO, ONE
      parameter( zero = 0.0e+0, one = 1.0e+0 )
*     ..
*     .. Local Scalars ..
      LOGICAL            UPPER
      INTEGER            I, IMAX, J, K
      REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
*     ..
*     .. External Functions ..
      LOGICAL            LSAME
      INTEGER            ISAMAX
      REAL               SLAMCH
      EXTERNAL           lsame, isamax, slamch
*     ..
*     .. Intrinsic Functions ..
      INTRINSIC          abs, max, min
*     ..
*     .. 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 = slamch( 'epsilon' )
      UNFL = SLAMCH( 'safe minimum' )
      OVFL = ONE / UNFL
      UPPER = LSAME( UPLO, 'u' )
*
*     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 = ISAMAX( N, X( 1, J ), 1 )
         XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
         DIFF = ZERO
         DO 10 I = 1, N
            DIFF = MAX( DIFF, ABS( 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 / ( (n+1)*EPS + (*) ), where
*     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
*
      DO 90 K = 1, NRHS
         DO 80 I = 1, N
            TMP = ABS( B( I, K ) )
            IF( UPPER ) THEN
               DO 40 J = 1, I
                  TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
   40          CONTINUE
               DO 50 J = I + 1, N
                  TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
   50          CONTINUE
            ELSE
               DO 60 J = 1, I - 1
                  TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
   60          CONTINUE
               DO 70 J = I, N
                  TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
   70          CONTINUE
            END IF
.EQ.            IF( I1 ) THEN
               AXBI = TMP
            ELSE
               AXBI = MIN( AXBI, TMP )
            END IF
   80    CONTINUE
         TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
     $         MAX( AXBI, ( N+1 )*UNFL ) )
.EQ.         IF( K1 ) THEN
            RESLTS( 2 ) = TMP
         ELSE
            RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
         END IF
   90 CONTINUE
*
      RETURN
*
*     End of SPOT05
*
      END