LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
cppt05.f
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00001 *> \brief \b CPPT05
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 CPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT,
00012 *                          LDXACT, FERR, BERR, RESLTS )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          UPLO
00016 *       INTEGER            LDB, LDX, LDXACT, N, NRHS
00017 *       ..
00018 *       .. Array Arguments ..
00019 *       REAL               BERR( * ), FERR( * ), RESLTS( * )
00020 *       COMPLEX            AP( * ), B( LDB, * ), X( LDX, * ),
00021 *      $                   XACT( LDXACT, * )
00022 *       ..
00023 *  
00024 *
00025 *> \par Purpose:
00026 *  =============
00027 *>
00028 *> \verbatim
00029 *>
00030 *> CPPT05 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 matrix in packed storage format.
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] AP
00071 *> \verbatim
00072 *>          AP is COMPLEX array, dimension (N*(N+1)/2)
00073 *>          The upper or lower triangle of the Hermitian matrix A, packed
00074 *>          columnwise in a linear array.  The j-th column of A is stored
00075 *>          in the array AP as follows:
00076 *>          if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j;
00077 *>          if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n.
00078 *> \endverbatim
00079 *>
00080 *> \param[in] B
00081 *> \verbatim
00082 *>          B is COMPLEX array, dimension (LDB,NRHS)
00083 *>          The right hand side vectors for the system of linear
00084 *>          equations.
00085 *> \endverbatim
00086 *>
00087 *> \param[in] LDB
00088 *> \verbatim
00089 *>          LDB is INTEGER
00090 *>          The leading dimension of the array B.  LDB >= max(1,N).
00091 *> \endverbatim
00092 *>
00093 *> \param[in] X
00094 *> \verbatim
00095 *>          X is COMPLEX array, dimension (LDX,NRHS)
00096 *>          The computed solution vectors.  Each vector is stored as a
00097 *>          column of the matrix X.
00098 *> \endverbatim
00099 *>
00100 *> \param[in] LDX
00101 *> \verbatim
00102 *>          LDX is INTEGER
00103 *>          The leading dimension of the array X.  LDX >= max(1,N).
00104 *> \endverbatim
00105 *>
00106 *> \param[in] XACT
00107 *> \verbatim
00108 *>          XACT is COMPLEX array, dimension (LDX,NRHS)
00109 *>          The exact solution vectors.  Each vector is stored as a
00110 *>          column of the matrix XACT.
00111 *> \endverbatim
00112 *>
00113 *> \param[in] LDXACT
00114 *> \verbatim
00115 *>          LDXACT is INTEGER
00116 *>          The leading dimension of the array XACT.  LDXACT >= max(1,N).
00117 *> \endverbatim
00118 *>
00119 *> \param[in] FERR
00120 *> \verbatim
00121 *>          FERR is REAL array, dimension (NRHS)
00122 *>          The estimated forward error bounds for each solution vector
00123 *>          X.  If XTRUE is the true solution, FERR bounds the magnitude
00124 *>          of the largest entry in (X - XTRUE) divided by the magnitude
00125 *>          of the largest entry in X.
00126 *> \endverbatim
00127 *>
00128 *> \param[in] BERR
00129 *> \verbatim
00130 *>          BERR is REAL array, dimension (NRHS)
00131 *>          The componentwise relative backward error of each solution
00132 *>          vector (i.e., the smallest relative change in any entry of A
00133 *>          or B that makes X an exact solution).
00134 *> \endverbatim
00135 *>
00136 *> \param[out] RESLTS
00137 *> \verbatim
00138 *>          RESLTS is REAL array, dimension (2)
00139 *>          The maximum over the NRHS solution vectors of the ratios:
00140 *>          RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR )
00141 *>          RESLTS(2) = BERR / ( (n+1)*EPS + (*) )
00142 *> \endverbatim
00143 *
00144 *  Authors:
00145 *  ========
00146 *
00147 *> \author Univ. of Tennessee 
00148 *> \author Univ. of California Berkeley 
00149 *> \author Univ. of Colorado Denver 
00150 *> \author NAG Ltd. 
00151 *
00152 *> \date November 2011
00153 *
00154 *> \ingroup complex_lin
00155 *
00156 *  =====================================================================
00157       SUBROUTINE CPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT,
00158      $                   LDXACT, FERR, BERR, RESLTS )
00159 *
00160 *  -- LAPACK test routine (version 3.4.0) --
00161 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00162 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00163 *     November 2011
00164 *
00165 *     .. Scalar Arguments ..
00166       CHARACTER          UPLO
00167       INTEGER            LDB, LDX, LDXACT, N, NRHS
00168 *     ..
00169 *     .. Array Arguments ..
00170       REAL               BERR( * ), FERR( * ), RESLTS( * )
00171       COMPLEX            AP( * ), B( LDB, * ), X( LDX, * ),
00172      $                   XACT( LDXACT, * )
00173 *     ..
00174 *
00175 *  =====================================================================
00176 *
00177 *     .. Parameters ..
00178       REAL               ZERO, ONE
00179       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00180 *     ..
00181 *     .. Local Scalars ..
00182       LOGICAL            UPPER
00183       INTEGER            I, IMAX, J, JC, K
00184       REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00185       COMPLEX            ZDUM
00186 *     ..
00187 *     .. External Functions ..
00188       LOGICAL            LSAME
00189       INTEGER            ICAMAX
00190       REAL               SLAMCH
00191       EXTERNAL           LSAME, ICAMAX, SLAMCH
00192 *     ..
00193 *     .. Intrinsic Functions ..
00194       INTRINSIC          ABS, AIMAG, MAX, MIN, REAL
00195 *     ..
00196 *     .. Statement Functions ..
00197       REAL               CABS1
00198 *     ..
00199 *     .. Statement Function definitions ..
00200       CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) )
00201 *     ..
00202 *     .. Executable Statements ..
00203 *
00204 *     Quick exit if N = 0 or NRHS = 0.
00205 *
00206       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00207          RESLTS( 1 ) = ZERO
00208          RESLTS( 2 ) = ZERO
00209          RETURN
00210       END IF
00211 *
00212       EPS = SLAMCH( 'Epsilon' )
00213       UNFL = SLAMCH( 'Safe minimum' )
00214       OVFL = ONE / UNFL
00215       UPPER = LSAME( UPLO, 'U' )
00216 *
00217 *     Test 1:  Compute the maximum of
00218 *        norm(X - XACT) / ( norm(X) * FERR )
00219 *     over all the vectors X and XACT using the infinity-norm.
00220 *
00221       ERRBND = ZERO
00222       DO 30 J = 1, NRHS
00223          IMAX = ICAMAX( N, X( 1, J ), 1 )
00224          XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
00225          DIFF = ZERO
00226          DO 10 I = 1, N
00227             DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
00228    10    CONTINUE
00229 *
00230          IF( XNORM.GT.ONE ) THEN
00231             GO TO 20
00232          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00233             GO TO 20
00234          ELSE
00235             ERRBND = ONE / EPS
00236             GO TO 30
00237          END IF
00238 *
00239    20    CONTINUE
00240          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00241             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00242          ELSE
00243             ERRBND = ONE / EPS
00244          END IF
00245    30 CONTINUE
00246       RESLTS( 1 ) = ERRBND
00247 *
00248 *     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
00249 *     (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00250 *
00251       DO 90 K = 1, NRHS
00252          DO 80 I = 1, N
00253             TMP = CABS1( B( I, K ) )
00254             IF( UPPER ) THEN
00255                JC = ( ( I-1 )*I ) / 2
00256                DO 40 J = 1, I - 1
00257                   TMP = TMP + CABS1( AP( JC+J ) )*CABS1( X( J, K ) )
00258    40          CONTINUE
00259                TMP = TMP + ABS( REAL( AP( JC+I ) ) )*CABS1( X( I, K ) )
00260                JC = JC + I + I
00261                DO 50 J = I + 1, N
00262                   TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
00263                   JC = JC + J
00264    50          CONTINUE
00265             ELSE
00266                JC = I
00267                DO 60 J = 1, I - 1
00268                   TMP = TMP + CABS1( AP( JC ) )*CABS1( X( J, K ) )
00269                   JC = JC + N - J
00270    60          CONTINUE
00271                TMP = TMP + ABS( REAL( AP( JC ) ) )*CABS1( X( I, K ) )
00272                DO 70 J = I + 1, N
00273                   TMP = TMP + CABS1( AP( JC+J-I ) )*CABS1( X( J, K ) )
00274    70          CONTINUE
00275             END IF
00276             IF( I.EQ.1 ) THEN
00277                AXBI = TMP
00278             ELSE
00279                AXBI = MIN( AXBI, TMP )
00280             END IF
00281    80    CONTINUE
00282          TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
00283      $         MAX( AXBI, ( N+1 )*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 CPPT05
00294 *
00295       END
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