LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
spot05.f
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00001 *> \brief \b SPOT05
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 SPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
00012 *                          LDXACT, FERR, BERR, RESLTS )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          UPLO
00016 *       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
00017 *       ..
00018 *       .. Array Arguments ..
00019 *       REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
00020 *      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
00021 *       ..
00022 *  
00023 *
00024 *> \par Purpose:
00025 *  =============
00026 *>
00027 *> \verbatim
00028 *>
00029 *> SPOT05 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 n by n matrix.
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 / ( (n+1)*EPS + (*) ), where
00040 *>             (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00041 *> \endverbatim
00042 *
00043 *  Arguments:
00044 *  ==========
00045 *
00046 *> \param[in] UPLO
00047 *> \verbatim
00048 *>          UPLO is CHARACTER*1
00049 *>          Specifies whether the upper or lower triangular part of the
00050 *>          symmetric matrix A is stored.
00051 *>          = 'U':  Upper triangular
00052 *>          = 'L':  Lower triangular
00053 *> \endverbatim
00054 *>
00055 *> \param[in] N
00056 *> \verbatim
00057 *>          N is INTEGER
00058 *>          The number of rows of the matrices X, B, and XACT, and the
00059 *>          order of the matrix A.  N >= 0.
00060 *> \endverbatim
00061 *>
00062 *> \param[in] NRHS
00063 *> \verbatim
00064 *>          NRHS is INTEGER
00065 *>          The number of columns of the matrices X, B, and XACT.
00066 *>          NRHS >= 0.
00067 *> \endverbatim
00068 *>
00069 *> \param[in] A
00070 *> \verbatim
00071 *>          A is REAL array, dimension (LDA,N)
00072 *>          The symmetric matrix A.  If UPLO = 'U', the leading n by n
00073 *>          upper triangular part of A contains the upper triangular part
00074 *>          of the matrix A, and the strictly lower triangular part of A
00075 *>          is not referenced.  If UPLO = 'L', the leading n by n lower
00076 *>          triangular part of A contains the lower triangular part of
00077 *>          the matrix A, and the strictly upper triangular part of A is
00078 *>          not referenced.
00079 *> \endverbatim
00080 *>
00081 *> \param[in] LDA
00082 *> \verbatim
00083 *>          LDA is INTEGER
00084 *>          The leading dimension of the array A.  LDA >= max(1,N).
00085 *> \endverbatim
00086 *>
00087 *> \param[in] B
00088 *> \verbatim
00089 *>          B is REAL array, dimension (LDB,NRHS)
00090 *>          The right hand side vectors for the system of linear
00091 *>          equations.
00092 *> \endverbatim
00093 *>
00094 *> \param[in] LDB
00095 *> \verbatim
00096 *>          LDB is INTEGER
00097 *>          The leading dimension of the array B.  LDB >= max(1,N).
00098 *> \endverbatim
00099 *>
00100 *> \param[in] X
00101 *> \verbatim
00102 *>          X is REAL array, dimension (LDX,NRHS)
00103 *>          The computed solution vectors.  Each vector is stored as a
00104 *>          column of the matrix X.
00105 *> \endverbatim
00106 *>
00107 *> \param[in] LDX
00108 *> \verbatim
00109 *>          LDX is INTEGER
00110 *>          The leading dimension of the array X.  LDX >= max(1,N).
00111 *> \endverbatim
00112 *>
00113 *> \param[in] XACT
00114 *> \verbatim
00115 *>          XACT is REAL array, dimension (LDX,NRHS)
00116 *>          The exact solution vectors.  Each vector is stored as a
00117 *>          column of the matrix XACT.
00118 *> \endverbatim
00119 *>
00120 *> \param[in] LDXACT
00121 *> \verbatim
00122 *>          LDXACT is INTEGER
00123 *>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00124 *> \endverbatim
00125 *>
00126 *> \param[in] FERR
00127 *> \verbatim
00128 *>          FERR is REAL array, dimension (NRHS)
00129 *>          The estimated forward error bounds for each solution vector
00130 *>          X.  If XTRUE is the true solution, FERR bounds the magnitude
00131 *>          of the largest entry in (X - XTRUE) divided by the magnitude
00132 *>          of the largest entry in X.
00133 *> \endverbatim
00134 *>
00135 *> \param[in] BERR
00136 *> \verbatim
00137 *>          BERR is REAL array, dimension (NRHS)
00138 *>          The componentwise relative backward error of each solution
00139 *>          vector (i.e., the smallest relative change in any entry of A
00140 *>          or B that makes X an exact solution).
00141 *> \endverbatim
00142 *>
00143 *> \param[out] RESLTS
00144 *> \verbatim
00145 *>          RESLTS is REAL array, dimension (2)
00146 *>          The maximum over the NRHS solution vectors of the ratios:
00147 *>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00148 *>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
00149 *> \endverbatim
00150 *
00151 *  Authors:
00152 *  ========
00153 *
00154 *> \author Univ. of Tennessee 
00155 *> \author Univ. of California Berkeley 
00156 *> \author Univ. of Colorado Denver 
00157 *> \author NAG Ltd. 
00158 *
00159 *> \date November 2011
00160 *
00161 *> \ingroup single_lin
00162 *
00163 *  =====================================================================
00164       SUBROUTINE SPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
00165      $                   LDXACT, FERR, BERR, RESLTS )
00166 *
00167 *  -- LAPACK test routine (version 3.4.0) --
00168 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00169 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00170 *     November 2011
00171 *
00172 *     .. Scalar Arguments ..
00173       CHARACTER          UPLO
00174       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
00175 *     ..
00176 *     .. Array Arguments ..
00177       REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
00178      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
00179 *     ..
00180 *
00181 *  =====================================================================
00182 *
00183 *     .. Parameters ..
00184       REAL               ZERO, ONE
00185       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00186 *     ..
00187 *     .. Local Scalars ..
00188       LOGICAL            UPPER
00189       INTEGER            I, IMAX, J, K
00190       REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00191 *     ..
00192 *     .. External Functions ..
00193       LOGICAL            LSAME
00194       INTEGER            ISAMAX
00195       REAL               SLAMCH
00196       EXTERNAL           LSAME, ISAMAX, SLAMCH
00197 *     ..
00198 *     .. Intrinsic Functions ..
00199       INTRINSIC          ABS, MAX, MIN
00200 *     ..
00201 *     .. Executable Statements ..
00202 *
00203 *     Quick exit if N = 0 or NRHS = 0.
00204 *
00205       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00206          RESLTS( 1 ) = ZERO
00207          RESLTS( 2 ) = ZERO
00208          RETURN
00209       END IF
00210 *
00211       EPS = SLAMCH( 'Epsilon' )
00212       UNFL = SLAMCH( 'Safe minimum' )
00213       OVFL = ONE / UNFL
00214       UPPER = LSAME( UPLO, 'U' )
00215 *
00216 *     Test 1:  Compute the maximum of
00217 *        norm(X - XACT) / ( norm(X) * FERR )
00218 *     over all the vectors X and XACT using the infinity-norm.
00219 *
00220       ERRBND = ZERO
00221       DO 30 J = 1, NRHS
00222          IMAX = ISAMAX( N, X( 1, J ), 1 )
00223          XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
00224          DIFF = ZERO
00225          DO 10 I = 1, N
00226             DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
00227    10    CONTINUE
00228 *
00229          IF( XNORM.GT.ONE ) THEN
00230             GO TO 20
00231          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00232             GO TO 20
00233          ELSE
00234             ERRBND = ONE / EPS
00235             GO TO 30
00236          END IF
00237 *
00238    20    CONTINUE
00239          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00240             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00241          ELSE
00242             ERRBND = ONE / EPS
00243          END IF
00244    30 CONTINUE
00245       RESLTS( 1 ) = ERRBND
00246 *
00247 *     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
00248 *     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00249 *
00250       DO 90 K = 1, NRHS
00251          DO 80 I = 1, N
00252             TMP = ABS( B( I, K ) )
00253             IF( UPPER ) THEN
00254                DO 40 J = 1, I
00255                   TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
00256    40          CONTINUE
00257                DO 50 J = I + 1, N
00258                   TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
00259    50          CONTINUE
00260             ELSE
00261                DO 60 J = 1, I - 1
00262                   TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
00263    60          CONTINUE
00264                DO 70 J = I, N
00265                   TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
00266    70          CONTINUE
00267             END IF
00268             IF( I.EQ.1 ) THEN
00269                AXBI = TMP
00270             ELSE
00271                AXBI = MIN( AXBI, TMP )
00272             END IF
00273    80    CONTINUE
00274          TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
00275      $         MAX( AXBI, ( N+1 )*UNFL ) )
00276          IF( K.EQ.1 ) THEN
00277             RESLTS( 2 ) = TMP
00278          ELSE
00279             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00280          END IF
00281    90 CONTINUE
00282 *
00283       RETURN
00284 *
00285 *     End of SPOT05
00286 *
00287       END
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