LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sptt05.f
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00001 *> \brief \b SPTT05
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *  Definition:
00009 *  ===========
00010 *
00011 *       SUBROUTINE SPTT05( N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT,
00012 *                          FERR, BERR, RESLTS )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       INTEGER            LDB, LDX, LDXACT, N, NRHS
00016 *       ..
00017 *       .. Array Arguments ..
00018 *       REAL               B( LDB, * ), BERR( * ), D( * ), E( * ),
00019 *      $                   FERR( * ), RESLTS( * ), X( LDX, * ),
00020 *      $                   XACT( LDXACT, * )
00021 *       ..
00022 *  
00023 *
00024 *> \par Purpose:
00025 *  =============
00026 *>
00027 *> \verbatim
00028 *>
00029 *> SPTT05 tests the error bounds from iterative refinement for the
00030 *> computed solution to a system of equations A*X = B, where A is a
00031 *> symmetric tridiagonal matrix of order n.
00032 *>
00033 *> RESLTS(1) = test of the error bound
00034 *>           = norm(X - XACT) / ( norm(X) * FERR )
00035 *>
00036 *> A large value is returned if this ratio is not less than one.
00037 *>
00038 *> RESLTS(2) = residual from the iterative refinement routine
00039 *>           = the maximum of BERR / ( NZ*EPS + (*) ), where
00040 *>             (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00041 *>             and NZ = max. number of nonzeros in any row of A, plus 1
00042 *> \endverbatim
00043 *
00044 *  Arguments:
00045 *  ==========
00046 *
00047 *> \param[in] N
00048 *> \verbatim
00049 *>          N is INTEGER
00050 *>          The number of rows of the matrices X, B, and XACT, and the
00051 *>          order of the matrix A.  N >= 0.
00052 *> \endverbatim
00053 *>
00054 *> \param[in] NRHS
00055 *> \verbatim
00056 *>          NRHS is INTEGER
00057 *>          The number of columns of the matrices X, B, and XACT.
00058 *>          NRHS >= 0.
00059 *> \endverbatim
00060 *>
00061 *> \param[in] D
00062 *> \verbatim
00063 *>          D is REAL array, dimension (N)
00064 *>          The n diagonal elements of the tridiagonal matrix A.
00065 *> \endverbatim
00066 *>
00067 *> \param[in] E
00068 *> \verbatim
00069 *>          E is REAL array, dimension (N-1)
00070 *>          The (n-1) subdiagonal elements of the tridiagonal matrix A.
00071 *> \endverbatim
00072 *>
00073 *> \param[in] B
00074 *> \verbatim
00075 *>          B is REAL array, dimension (LDB,NRHS)
00076 *>          The right hand side vectors for the system of linear
00077 *>          equations.
00078 *> \endverbatim
00079 *>
00080 *> \param[in] LDB
00081 *> \verbatim
00082 *>          LDB is INTEGER
00083 *>          The leading dimension of the array B.  LDB >= max(1,N).
00084 *> \endverbatim
00085 *>
00086 *> \param[in] X
00087 *> \verbatim
00088 *>          X is REAL array, dimension (LDX,NRHS)
00089 *>          The computed solution vectors.  Each vector is stored as a
00090 *>          column of the matrix X.
00091 *> \endverbatim
00092 *>
00093 *> \param[in] LDX
00094 *> \verbatim
00095 *>          LDX is INTEGER
00096 *>          The leading dimension of the array X.  LDX >= max(1,N).
00097 *> \endverbatim
00098 *>
00099 *> \param[in] XACT
00100 *> \verbatim
00101 *>          XACT is REAL array, dimension (LDX,NRHS)
00102 *>          The exact solution vectors.  Each vector is stored as a
00103 *>          column of the matrix XACT.
00104 *> \endverbatim
00105 *>
00106 *> \param[in] LDXACT
00107 *> \verbatim
00108 *>          LDXACT is INTEGER
00109 *>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00110 *> \endverbatim
00111 *>
00112 *> \param[in] FERR
00113 *> \verbatim
00114 *>          FERR is REAL array, dimension (NRHS)
00115 *>          The estimated forward error bounds for each solution vector
00116 *>          X.  If XTRUE is the true solution, FERR bounds the magnitude
00117 *>          of the largest entry in (X - XTRUE) divided by the magnitude
00118 *>          of the largest entry in X.
00119 *> \endverbatim
00120 *>
00121 *> \param[in] BERR
00122 *> \verbatim
00123 *>          BERR is REAL array, dimension (NRHS)
00124 *>          The componentwise relative backward error of each solution
00125 *>          vector (i.e., the smallest relative change in any entry of A
00126 *>          or B that makes X an exact solution).
00127 *> \endverbatim
00128 *>
00129 *> \param[out] RESLTS
00130 *> \verbatim
00131 *>          RESLTS is REAL array, dimension (2)
00132 *>          The maximum over the NRHS solution vectors of the ratios:
00133 *>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00134 *>          RESLTS(2) = BERR / ( NZ*EPS + (*) )
00135 *> \endverbatim
00136 *
00137 *  Authors:
00138 *  ========
00139 *
00140 *> \author Univ. of Tennessee 
00141 *> \author Univ. of California Berkeley 
00142 *> \author Univ. of Colorado Denver 
00143 *> \author NAG Ltd. 
00144 *
00145 *> \date November 2011
00146 *
00147 *> \ingroup single_lin
00148 *
00149 *  =====================================================================
00150       SUBROUTINE SPTT05( N, NRHS, D, E, B, LDB, X, LDX, XACT, LDXACT,
00151      $                   FERR, BERR, RESLTS )
00152 *
00153 *  -- LAPACK test routine (version 3.4.0) --
00154 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00155 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00156 *     November 2011
00157 *
00158 *     .. Scalar Arguments ..
00159       INTEGER            LDB, LDX, LDXACT, N, NRHS
00160 *     ..
00161 *     .. Array Arguments ..
00162       REAL               B( LDB, * ), BERR( * ), D( * ), E( * ),
00163      $                   FERR( * ), RESLTS( * ), X( LDX, * ),
00164      $                   XACT( LDXACT, * )
00165 *     ..
00166 *
00167 *  =====================================================================
00168 *
00169 *     .. Parameters ..
00170       REAL               ZERO, ONE
00171       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00172 *     ..
00173 *     .. Local Scalars ..
00174       INTEGER            I, IMAX, J, K, NZ
00175       REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00176 *     ..
00177 *     .. External Functions ..
00178       INTEGER            ISAMAX
00179       REAL               SLAMCH
00180       EXTERNAL           ISAMAX, SLAMCH
00181 *     ..
00182 *     .. Intrinsic Functions ..
00183       INTRINSIC          ABS, MAX, MIN
00184 *     ..
00185 *     .. Executable Statements ..
00186 *
00187 *     Quick exit if N = 0 or NRHS = 0.
00188 *
00189       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00190          RESLTS( 1 ) = ZERO
00191          RESLTS( 2 ) = ZERO
00192          RETURN
00193       END IF
00194 *
00195       EPS = SLAMCH( 'Epsilon' )
00196       UNFL = SLAMCH( 'Safe minimum' )
00197       OVFL = ONE / UNFL
00198       NZ = 4
00199 *
00200 *     Test 1:  Compute the maximum of
00201 *        norm(X - XACT) / ( norm(X) * FERR )
00202 *     over all the vectors X and XACT using the infinity-norm.
00203 *
00204       ERRBND = ZERO
00205       DO 30 J = 1, NRHS
00206          IMAX = ISAMAX( N, X( 1, J ), 1 )
00207          XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
00208          DIFF = ZERO
00209          DO 10 I = 1, N
00210             DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
00211    10    CONTINUE
00212 *
00213          IF( XNORM.GT.ONE ) THEN
00214             GO TO 20
00215          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00216             GO TO 20
00217          ELSE
00218             ERRBND = ONE / EPS
00219             GO TO 30
00220          END IF
00221 *
00222    20    CONTINUE
00223          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00224             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00225          ELSE
00226             ERRBND = ONE / EPS
00227          END IF
00228    30 CONTINUE
00229       RESLTS( 1 ) = ERRBND
00230 *
00231 *     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
00232 *     (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00233 *
00234       DO 50 K = 1, NRHS
00235          IF( N.EQ.1 ) THEN
00236             AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
00237          ELSE
00238             AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
00239      $             ABS( E( 1 )*X( 2, K ) )
00240             DO 40 I = 2, N - 1
00241                TMP = ABS( B( I, K ) ) + ABS( E( I-1 )*X( I-1, K ) ) +
00242      $               ABS( D( I )*X( I, K ) ) + ABS( E( I )*X( I+1, K ) )
00243                AXBI = MIN( AXBI, TMP )
00244    40       CONTINUE
00245             TMP = ABS( B( N, K ) ) + ABS( E( N-1 )*X( N-1, K ) ) +
00246      $            ABS( D( N )*X( N, K ) )
00247             AXBI = MIN( AXBI, TMP )
00248          END IF
00249          TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
00250          IF( K.EQ.1 ) THEN
00251             RESLTS( 2 ) = TMP
00252          ELSE
00253             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00254          END IF
00255    50 CONTINUE
00256 *
00257       RETURN
00258 *
00259 *     End of SPTT05
00260 *
00261       END
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