spbt05.f

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*> \brief \b SPBT05
*
*  =========== DOCUMENTATION ===========
*
* Online html documentation available at 
*            http://www.netlib.org/lapack/explore-html/ 
*
*  Definition:
*  ===========
*
*       SUBROUTINE SPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX,
*                          XACT, LDXACT, FERR, BERR, RESLTS )
* 
*       .. Scalar Arguments ..
*       CHARACTER          UPLO
*       INTEGER            KD, LDAB, LDB, LDX, LDXACT, N, NRHS
*       ..
*       .. Array Arguments ..
*       REAL               AB( LDAB, * ), B( LDB, * ), BERR( * ),
*      $                   FERR( * ), RESLTS( * ), X( LDX, * ),
*      $                   XACT( LDXACT, * )
*       ..
*  
*
*> \par Purpose:
*  =============
*>
*> \verbatim
*>
*> SPBT05 tests the error bounds from iterative refinement for the
*> computed solution to a system of equations A*X = B, where A is a
*> symmetric band 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 / ( NZ*EPS + (*) ), where
*>             (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
*>             and NZ = max. number of nonzeros in any row of A, plus 1
*> \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] KD
*> \verbatim
*>          KD is INTEGER
*>          The number of super-diagonals of the matrix A if UPLO = 'U',
*>          or the number of sub-diagonals if UPLO = 'L'.  KD >= 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] AB
*> \verbatim
*>          AB is REAL array, dimension (LDAB,N)
*>          The upper or lower triangle of the symmetric band matrix A,
*>          stored in the first KD+1 rows of the array.  The j-th column
*>          of A is stored in the j-th column of the array AB as follows:
*>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
*>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
*> \endverbatim
*>
*> \param[in] LDAB
*> \verbatim
*>          LDAB is INTEGER
*>          The leading dimension of the array AB.  LDAB >= KD+1.
*> \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 / ( NZ*EPS + (*) )
*> \endverbatim
*
*  Authors:
*  ========
*
*> \author Univ. of Tennessee 
*> \author Univ. of California Berkeley 
*> \author Univ. of Colorado Denver 
*> \author NAG Ltd. 
*
*> \date November 2011
*
*> \ingroup single_lin
*
*  =====================================================================
      SUBROUTINE spbt05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX,
     $                   xact, ldxact, ferr, berr, reslts )
*
*  -- LAPACK test routine (version 3.4.0) --
*  -- LAPACK is a software package provided by Univ. of Tennessee,    --
*  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
*     November 2011
*
*     .. Scalar Arguments ..
      CHARACTER          uplo
      INTEGER            kd, ldab, ldb, ldx, ldxact, n, nrhs
*     ..
*     .. Array Arguments ..
      REAL               ab( ldab, * ), 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, nz
      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' )
      nz = 2*max( kd, n-1 ) + 1
*
*     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
*
         IF( xnorm.GT.one ) THEN
            go to 20
         ELSE IF( diff.LE.ovfl*xnorm ) THEN
            go to 20
         ELSE
            errbnd = one / eps
            go to 30
         END IF
*
   20    continue
         IF( diff / xnorm.LE.ferr( 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(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 = max( i-kd, 1 ), i
                  tmp = tmp + abs( ab( kd+1-i+j, i ) )*abs( x( j, k ) )
   40          continue
               DO 50 j = i + 1, min( i+kd, n )
                  tmp = tmp + abs( ab( kd+1+i-j, j ) )*abs( x( j, k ) )
   50          continue
            ELSE
               DO 60 j = max( i-kd, 1 ), i - 1
                  tmp = tmp + abs( ab( 1+i-j, j ) )*abs( x( j, k ) )
   60          continue
               DO 70 j = i, min( i+kd, n )
                  tmp = tmp + abs( ab( 1+j-i, i ) )*abs( x( j, k ) )
   70          continue
            END IF
            IF( i.EQ.1 ) THEN
               axbi = tmp
            ELSE
               axbi = min( axbi, tmp )
            END IF
   80    continue
         tmp = berr( k ) / ( nz*eps+nz*unfl / max( axbi, nz*unfl ) )
         IF( k.EQ.1 ) THEN
            reslts( 2 ) = tmp
         ELSE
            reslts( 2 ) = max( reslts( 2 ), tmp )
         END IF
   90 continue
*
      return
*
*     End of SPBT05
*
      END