LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cpot05.f
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00001 *> \brief \b CPOT05
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 CPOT05( 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               BERR( * ), FERR( * ), RESLTS( * )
00020 *       COMPLEX            A( LDA, * ), B( LDB, * ), X( LDX, * ),
00021 *      $                   XACT( LDXACT, * )
00022 *       ..
00023 *  
00024 *
00025 *> \par Purpose:
00026 *  =============
00027 *>
00028 *> \verbatim
00029 *>
00030 *> CPOT05 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 *> Hermitian n by n 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 / ( (n+1)*EPS + (*) ), where
00041 *>             (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00042 *> \endverbatim
00043 *
00044 *  Arguments:
00045 *  ==========
00046 *
00047 *> \param[in] UPLO
00048 *> \verbatim
00049 *>          UPLO is CHARACTER*1
00050 *>          Specifies whether the upper or lower triangular part of the
00051 *>          Hermitian matrix A is stored.
00052 *>          = 'U':  Upper triangular
00053 *>          = 'L':  Lower triangular
00054 *> \endverbatim
00055 *>
00056 *> \param[in] N
00057 *> \verbatim
00058 *>          N is INTEGER
00059 *>          The number of rows of the matrices X, B, and XACT, and the
00060 *>          order of the matrix A.  N >= 0.
00061 *> \endverbatim
00062 *>
00063 *> \param[in] NRHS
00064 *> \verbatim
00065 *>          NRHS is INTEGER
00066 *>          The number of columns of the matrices X, B, and XACT.
00067 *>          NRHS >= 0.
00068 *> \endverbatim
00069 *>
00070 *> \param[in] A
00071 *> \verbatim
00072 *>          A is COMPLEX array, dimension (LDA,N)
00073 *>          The Hermitian matrix A.  If UPLO = 'U', the leading n by n
00074 *>          upper triangular part of A contains the upper triangular part
00075 *>          of the matrix A, and the strictly lower triangular part of A
00076 *>          is not referenced.  If UPLO = 'L', the leading n by n lower
00077 *>          triangular part of A contains the lower triangular part of
00078 *>          the matrix A, and the strictly upper triangular part of A is
00079 *>          not referenced.
00080 *> \endverbatim
00081 *>
00082 *> \param[in] LDA
00083 *> \verbatim
00084 *>          LDA is INTEGER
00085 *>          The leading dimension of the array A.  LDA >= max(1,N).
00086 *> \endverbatim
00087 *>
00088 *> \param[in] B
00089 *> \verbatim
00090 *>          B is COMPLEX array, dimension (LDB,NRHS)
00091 *>          The right hand side vectors for the system of linear
00092 *>          equations.
00093 *> \endverbatim
00094 *>
00095 *> \param[in] LDB
00096 *> \verbatim
00097 *>          LDB is INTEGER
00098 *>          The leading dimension of the array B.  LDB >= max(1,N).
00099 *> \endverbatim
00100 *>
00101 *> \param[in] X
00102 *> \verbatim
00103 *>          X is COMPLEX array, dimension (LDX,NRHS)
00104 *>          The computed solution vectors.  Each vector is stored as a
00105 *>          column of the matrix X.
00106 *> \endverbatim
00107 *>
00108 *> \param[in] LDX
00109 *> \verbatim
00110 *>          LDX is INTEGER
00111 *>          The leading dimension of the array X.  LDX >= max(1,N).
00112 *> \endverbatim
00113 *>
00114 *> \param[in] XACT
00115 *> \verbatim
00116 *>          XACT is COMPLEX array, dimension (LDX,NRHS)
00117 *>          The exact solution vectors.  Each vector is stored as a
00118 *>          column of the matrix XACT.
00119 *> \endverbatim
00120 *>
00121 *> \param[in] LDXACT
00122 *> \verbatim
00123 *>          LDXACT is INTEGER
00124 *>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00125 *> \endverbatim
00126 *>
00127 *> \param[in] FERR
00128 *> \verbatim
00129 *>          FERR is REAL array, dimension (NRHS)
00130 *>          The estimated forward error bounds for each solution vector
00131 *>          X.  If XTRUE is the true solution, FERR bounds the magnitude
00132 *>          of the largest entry in (X - XTRUE) divided by the magnitude
00133 *>          of the largest entry in X.
00134 *> \endverbatim
00135 *>
00136 *> \param[in] BERR
00137 *> \verbatim
00138 *>          BERR is REAL array, dimension (NRHS)
00139 *>          The componentwise relative backward error of each solution
00140 *>          vector (i.e., the smallest relative change in any entry of A
00141 *>          or B that makes X an exact solution).
00142 *> \endverbatim
00143 *>
00144 *> \param[out] RESLTS
00145 *> \verbatim
00146 *>          RESLTS is REAL array, dimension (2)
00147 *>          The maximum over the NRHS solution vectors of the ratios:
00148 *>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00149 *>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
00150 *> \endverbatim
00151 *
00152 *  Authors:
00153 *  ========
00154 *
00155 *> \author Univ. of Tennessee 
00156 *> \author Univ. of California Berkeley 
00157 *> \author Univ. of Colorado Denver 
00158 *> \author NAG Ltd. 
00159 *
00160 *> \date November 2011
00161 *
00162 *> \ingroup complex_lin
00163 *
00164 *  =====================================================================
00165       SUBROUTINE CPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
00166      $                   LDXACT, FERR, BERR, RESLTS )
00167 *
00168 *  -- LAPACK test routine (version 3.4.0) --
00169 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00170 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00171 *     November 2011
00172 *
00173 *     .. Scalar Arguments ..
00174       CHARACTER          UPLO
00175       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
00176 *     ..
00177 *     .. Array Arguments ..
00178       REAL               BERR( * ), FERR( * ), RESLTS( * )
00179       COMPLEX            A( LDA, * ), B( LDB, * ), X( LDX, * ),
00180      $                   XACT( LDXACT, * )
00181 *     ..
00182 *
00183 *  =====================================================================
00184 *
00185 *     .. Parameters ..
00186       REAL               ZERO, ONE
00187       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00188 *     ..
00189 *     .. Local Scalars ..
00190       LOGICAL            UPPER
00191       INTEGER            I, IMAX, J, K
00192       REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00193       COMPLEX            ZDUM
00194 *     ..
00195 *     .. External Functions ..
00196       LOGICAL            LSAME
00197       INTEGER            ICAMAX
00198       REAL               SLAMCH
00199       EXTERNAL           LSAME, ICAMAX, SLAMCH
00200 *     ..
00201 *     .. Intrinsic Functions ..
00202       INTRINSIC          ABS, AIMAG, MAX, MIN, REAL
00203 *     ..
00204 *     .. Statement Functions ..
00205       REAL               CABS1
00206 *     ..
00207 *     .. Statement Function definitions ..
00208       CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
00209 *     ..
00210 *     .. Executable Statements ..
00211 *
00212 *     Quick exit if N = 0 or NRHS = 0.
00213 *
00214       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00215          RESLTS( 1 ) = ZERO
00216          RESLTS( 2 ) = ZERO
00217          RETURN
00218       END IF
00219 *
00220       EPS = SLAMCH( 'Epsilon' )
00221       UNFL = SLAMCH( 'Safe minimum' )
00222       OVFL = ONE / UNFL
00223       UPPER = LSAME( UPLO, 'U' )
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 = ICAMAX( N, X( 1, J ), 1 )
00232          XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
00233          DIFF = ZERO
00234          DO 10 I = 1, N
00235             DIFF = MAX( DIFF, CABS1( 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 / ( (n+1)*EPS + (*) ), where
00257 *     (*) = (n+1)*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 = CABS1( B( I, K ) )
00262             IF( UPPER ) THEN
00263                DO 40 J = 1, I - 1
00264                   TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) )
00265    40          CONTINUE
00266                TMP = TMP + ABS( REAL( A( I, I ) ) )*CABS1( X( I, K ) )
00267                DO 50 J = I + 1, N
00268                   TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) )
00269    50          CONTINUE
00270             ELSE
00271                DO 60 J = 1, I - 1
00272                   TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) )
00273    60          CONTINUE
00274                TMP = TMP + ABS( REAL( A( I, I ) ) )*CABS1( X( I, K ) )
00275                DO 70 J = I + 1, N
00276                   TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) )
00277    70          CONTINUE
00278             END IF
00279             IF( I.EQ.1 ) THEN
00280                AXBI = TMP
00281             ELSE
00282                AXBI = MIN( AXBI, TMP )
00283             END IF
00284    80    CONTINUE
00285          TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
00286      $         MAX( AXBI, ( N+1 )*UNFL ) )
00287          IF( K.EQ.1 ) THEN
00288             RESLTS( 2 ) = TMP
00289          ELSE
00290             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00291          END IF
00292    90 CONTINUE
00293 *
00294       RETURN
00295 *
00296 *     End of CPOT05
00297 *
00298       END
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