LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
spbt05.f
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00001 *> \brief \b SPBT05
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 SPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX,
00012 *                          XACT, LDXACT, FERR, BERR, RESLTS )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          UPLO
00016 *       INTEGER            KD, LDAB, LDB, LDX, LDXACT, N, NRHS
00017 *       ..
00018 *       .. Array Arguments ..
00019 *       REAL               AB( LDAB, * ), B( LDB, * ), BERR( * ),
00020 *      $                   FERR( * ), RESLTS( * ), X( LDX, * ),
00021 *      $                   XACT( LDXACT, * )
00022 *       ..
00023 *  
00024 *
00025 *> \par Purpose:
00026 *  =============
00027 *>
00028 *> \verbatim
00029 *>
00030 *> SPBT05 tests the error bounds from iterative refinement for the
00031 *> computed solution to a system of equations A*X = B, where A is a
00032 *> symmetric band matrix.
00033 *>
00034 *> RESLTS(1) = test of the error bound
00035 *>           = norm(X - XACT) / ( norm(X) * FERR )
00036 *>
00037 *> A large value is returned if this ratio is not less than one.
00038 *>
00039 *> RESLTS(2) = residual from the iterative refinement routine
00040 *>           = the maximum of BERR / ( NZ*EPS + (*) ), where
00041 *>             (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00042 *>             and NZ = max. number of nonzeros in any row of A, plus 1
00043 *> \endverbatim
00044 *
00045 *  Arguments:
00046 *  ==========
00047 *
00048 *> \param[in] UPLO
00049 *> \verbatim
00050 *>          UPLO is CHARACTER*1
00051 *>          Specifies whether the upper or lower triangular part of the
00052 *>          symmetric matrix A is stored.
00053 *>          = 'U':  Upper triangular
00054 *>          = 'L':  Lower triangular
00055 *> \endverbatim
00056 *>
00057 *> \param[in] N
00058 *> \verbatim
00059 *>          N is INTEGER
00060 *>          The number of rows of the matrices X, B, and XACT, and the
00061 *>          order of the matrix A.  N >= 0.
00062 *> \endverbatim
00063 *>
00064 *> \param[in] KD
00065 *> \verbatim
00066 *>          KD is INTEGER
00067 *>          The number of super-diagonals of the matrix A if UPLO = 'U',
00068 *>          or the number of sub-diagonals if UPLO = 'L'.  KD >= 0.
00069 *> \endverbatim
00070 *>
00071 *> \param[in] NRHS
00072 *> \verbatim
00073 *>          NRHS is INTEGER
00074 *>          The number of columns of the matrices X, B, and XACT.
00075 *>          NRHS >= 0.
00076 *> \endverbatim
00077 *>
00078 *> \param[in] AB
00079 *> \verbatim
00080 *>          AB is REAL array, dimension (LDAB,N)
00081 *>          The upper or lower triangle of the symmetric band matrix A,
00082 *>          stored in the first KD+1 rows of the array.  The j-th column
00083 *>          of A is stored in the j-th column of the array AB as follows:
00084 *>          if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
00085 *>          if UPLO = 'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).
00086 *> \endverbatim
00087 *>
00088 *> \param[in] LDAB
00089 *> \verbatim
00090 *>          LDAB is INTEGER
00091 *>          The leading dimension of the array AB.  LDAB >= KD+1.
00092 *> \endverbatim
00093 *>
00094 *> \param[in] B
00095 *> \verbatim
00096 *>          B is REAL array, dimension (LDB,NRHS)
00097 *>          The right hand side vectors for the system of linear
00098 *>          equations.
00099 *> \endverbatim
00100 *>
00101 *> \param[in] LDB
00102 *> \verbatim
00103 *>          LDB is INTEGER
00104 *>          The leading dimension of the array B.  LDB >= max(1,N).
00105 *> \endverbatim
00106 *>
00107 *> \param[in] X
00108 *> \verbatim
00109 *>          X is REAL array, dimension (LDX,NRHS)
00110 *>          The computed solution vectors.  Each vector is stored as a
00111 *>          column of the matrix X.
00112 *> \endverbatim
00113 *>
00114 *> \param[in] LDX
00115 *> \verbatim
00116 *>          LDX is INTEGER
00117 *>          The leading dimension of the array X.  LDX >= max(1,N).
00118 *> \endverbatim
00119 *>
00120 *> \param[in] XACT
00121 *> \verbatim
00122 *>          XACT is REAL array, dimension (LDX,NRHS)
00123 *>          The exact solution vectors.  Each vector is stored as a
00124 *>          column of the matrix XACT.
00125 *> \endverbatim
00126 *>
00127 *> \param[in] LDXACT
00128 *> \verbatim
00129 *>          LDXACT is INTEGER
00130 *>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00131 *> \endverbatim
00132 *>
00133 *> \param[in] FERR
00134 *> \verbatim
00135 *>          FERR is REAL array, dimension (NRHS)
00136 *>          The estimated forward error bounds for each solution vector
00137 *>          X.  If XTRUE is the true solution, FERR bounds the magnitude
00138 *>          of the largest entry in (X - XTRUE) divided by the magnitude
00139 *>          of the largest entry in X.
00140 *> \endverbatim
00141 *>
00142 *> \param[in] BERR
00143 *> \verbatim
00144 *>          BERR is REAL array, dimension (NRHS)
00145 *>          The componentwise relative backward error of each solution
00146 *>          vector (i.e., the smallest relative change in any entry of A
00147 *>          or B that makes X an exact solution).
00148 *> \endverbatim
00149 *>
00150 *> \param[out] RESLTS
00151 *> \verbatim
00152 *>          RESLTS is REAL array, dimension (2)
00153 *>          The maximum over the NRHS solution vectors of the ratios:
00154 *>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00155 *>          RESLTS(2) = BERR / ( NZ*EPS + (*) )
00156 *> \endverbatim
00157 *
00158 *  Authors:
00159 *  ========
00160 *
00161 *> \author Univ. of Tennessee 
00162 *> \author Univ. of California Berkeley 
00163 *> \author Univ. of Colorado Denver 
00164 *> \author NAG Ltd. 
00165 *
00166 *> \date November 2011
00167 *
00168 *> \ingroup single_lin
00169 *
00170 *  =====================================================================
00171       SUBROUTINE SPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX,
00172      $                   XACT, LDXACT, FERR, BERR, RESLTS )
00173 *
00174 *  -- LAPACK test routine (version 3.4.0) --
00175 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00176 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00177 *     November 2011
00178 *
00179 *     .. Scalar Arguments ..
00180       CHARACTER          UPLO
00181       INTEGER            KD, LDAB, LDB, LDX, LDXACT, N, NRHS
00182 *     ..
00183 *     .. Array Arguments ..
00184       REAL               AB( LDAB, * ), B( LDB, * ), BERR( * ),
00185      $                   FERR( * ), RESLTS( * ), X( LDX, * ),
00186      $                   XACT( LDXACT, * )
00187 *     ..
00188 *
00189 *  =====================================================================
00190 *
00191 *     .. Parameters ..
00192       REAL               ZERO, ONE
00193       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00194 *     ..
00195 *     .. Local Scalars ..
00196       LOGICAL            UPPER
00197       INTEGER            I, IMAX, J, K, NZ
00198       REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00199 *     ..
00200 *     .. External Functions ..
00201       LOGICAL            LSAME
00202       INTEGER            ISAMAX
00203       REAL               SLAMCH
00204       EXTERNAL           LSAME, ISAMAX, SLAMCH
00205 *     ..
00206 *     .. Intrinsic Functions ..
00207       INTRINSIC          ABS, MAX, MIN
00208 *     ..
00209 *     .. Executable Statements ..
00210 *
00211 *     Quick exit if N = 0 or NRHS = 0.
00212 *
00213       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00214          RESLTS( 1 ) = ZERO
00215          RESLTS( 2 ) = ZERO
00216          RETURN
00217       END IF
00218 *
00219       EPS = SLAMCH( 'Epsilon' )
00220       UNFL = SLAMCH( 'Safe minimum' )
00221       OVFL = ONE / UNFL
00222       UPPER = LSAME( UPLO, 'U' )
00223       NZ = 2*MAX( KD, N-1 ) + 1
00224 *
00225 *     Test 1:  Compute the maximum of
00226 *        norm(X - XACT) / ( norm(X) * FERR )
00227 *     over all the vectors X and XACT using the infinity-norm.
00228 *
00229       ERRBND = ZERO
00230       DO 30 J = 1, NRHS
00231          IMAX = ISAMAX( N, X( 1, J ), 1 )
00232          XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
00233          DIFF = ZERO
00234          DO 10 I = 1, N
00235             DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
00236    10    CONTINUE
00237 *
00238          IF( XNORM.GT.ONE ) THEN
00239             GO TO 20
00240          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00241             GO TO 20
00242          ELSE
00243             ERRBND = ONE / EPS
00244             GO TO 30
00245          END IF
00246 *
00247    20    CONTINUE
00248          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00249             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00250          ELSE
00251             ERRBND = ONE / EPS
00252          END IF
00253    30 CONTINUE
00254       RESLTS( 1 ) = ERRBND
00255 *
00256 *     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
00257 *     (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00258 *
00259       DO 90 K = 1, NRHS
00260          DO 80 I = 1, N
00261             TMP = ABS( B( I, K ) )
00262             IF( UPPER ) THEN
00263                DO 40 J = MAX( I-KD, 1 ), I
00264                   TMP = TMP + ABS( AB( KD+1-I+J, I ) )*ABS( X( J, K ) )
00265    40          CONTINUE
00266                DO 50 J = I + 1, MIN( I+KD, N )
00267                   TMP = TMP + ABS( AB( KD+1+I-J, J ) )*ABS( X( J, K ) )
00268    50          CONTINUE
00269             ELSE
00270                DO 60 J = MAX( I-KD, 1 ), I - 1
00271                   TMP = TMP + ABS( AB( 1+I-J, J ) )*ABS( X( J, K ) )
00272    60          CONTINUE
00273                DO 70 J = I, MIN( I+KD, N )
00274                   TMP = TMP + ABS( AB( 1+J-I, I ) )*ABS( X( J, K ) )
00275    70          CONTINUE
00276             END IF
00277             IF( I.EQ.1 ) THEN
00278                AXBI = TMP
00279             ELSE
00280                AXBI = MIN( AXBI, TMP )
00281             END IF
00282    80    CONTINUE
00283          TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
00284          IF( K.EQ.1 ) THEN
00285             RESLTS( 2 ) = TMP
00286          ELSE
00287             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00288          END IF
00289    90 CONTINUE
00290 *
00291       RETURN
00292 *
00293 *     End of SPBT05
00294 *
00295       END
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