LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zptt05.f
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00001 *> \brief \b ZPTT05
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 ZPTT05( 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 *       DOUBLE PRECISION   BERR( * ), D( * ), FERR( * ), RESLTS( * )
00019 *       COMPLEX*16         B( LDB, * ), E( * ), X( LDX, * ),
00020 *      $                   XACT( LDXACT, * )
00021 *       ..
00022 *  
00023 *
00024 *> \par Purpose:
00025 *  =============
00026 *>
00027 *> \verbatim
00028 *>
00029 *> ZPTT05 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 *> Hermitian 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 DOUBLE PRECISION 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 COMPLEX*16 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 complex16_lin
00148 *
00149 *  =====================================================================
00150       SUBROUTINE ZPTT05( 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       DOUBLE PRECISION   BERR( * ), D( * ), FERR( * ), RESLTS( * )
00163       COMPLEX*16         B( LDB, * ), E( * ), X( LDX, * ),
00164      $                   XACT( LDXACT, * )
00165 *     ..
00166 *
00167 *  =====================================================================
00168 *
00169 *     .. Parameters ..
00170       DOUBLE PRECISION   ZERO, ONE
00171       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00172 *     ..
00173 *     .. Local Scalars ..
00174       INTEGER            I, IMAX, J, K, NZ
00175       DOUBLE PRECISION   AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00176       COMPLEX*16         ZDUM
00177 *     ..
00178 *     .. External Functions ..
00179       INTEGER            IZAMAX
00180       DOUBLE PRECISION   DLAMCH
00181       EXTERNAL           IZAMAX, DLAMCH
00182 *     ..
00183 *     .. Intrinsic Functions ..
00184       INTRINSIC          ABS, DBLE, DIMAG, MAX, MIN
00185 *     ..
00186 *     .. Statement Functions ..
00187       DOUBLE PRECISION   CABS1
00188 *     ..
00189 *     .. Statement Function definitions ..
00190       CABS1( ZDUM ) = ABS( DBLE( ZDUM ) ) + ABS( DIMAG( ZDUM ) )
00191 *     ..
00192 *     .. Executable Statements ..
00193 *
00194 *     Quick exit if N = 0 or NRHS = 0.
00195 *
00196       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00197          RESLTS( 1 ) = ZERO
00198          RESLTS( 2 ) = ZERO
00199          RETURN
00200       END IF
00201 *
00202       EPS = DLAMCH( 'Epsilon' )
00203       UNFL = DLAMCH( 'Safe minimum' )
00204       OVFL = ONE / UNFL
00205       NZ = 4
00206 *
00207 *     Test 1:  Compute the maximum of
00208 *        norm(X - XACT) / ( norm(X) * FERR )
00209 *     over all the vectors X and XACT using the infinity-norm.
00210 *
00211       ERRBND = ZERO
00212       DO 30 J = 1, NRHS
00213          IMAX = IZAMAX( N, X( 1, J ), 1 )
00214          XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL )
00215          DIFF = ZERO
00216          DO 10 I = 1, N
00217             DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) )
00218    10    CONTINUE
00219 *
00220          IF( XNORM.GT.ONE ) THEN
00221             GO TO 20
00222          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00223             GO TO 20
00224          ELSE
00225             ERRBND = ONE / EPS
00226             GO TO 30
00227          END IF
00228 *
00229    20    CONTINUE
00230          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00231             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00232          ELSE
00233             ERRBND = ONE / EPS
00234          END IF
00235    30 CONTINUE
00236       RESLTS( 1 ) = ERRBND
00237 *
00238 *     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
00239 *     (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i )
00240 *
00241       DO 50 K = 1, NRHS
00242          IF( N.EQ.1 ) THEN
00243             AXBI = CABS1( B( 1, K ) ) + CABS1( D( 1 )*X( 1, K ) )
00244          ELSE
00245             AXBI = CABS1( B( 1, K ) ) + CABS1( D( 1 )*X( 1, K ) ) +
00246      $             CABS1( E( 1 ) )*CABS1( X( 2, K ) )
00247             DO 40 I = 2, N - 1
00248                TMP = CABS1( B( I, K ) ) + CABS1( E( I-1 ) )*
00249      $               CABS1( X( I-1, K ) ) + CABS1( D( I )*X( I, K ) ) +
00250      $               CABS1( E( I ) )*CABS1( X( I+1, K ) )
00251                AXBI = MIN( AXBI, TMP )
00252    40       CONTINUE
00253             TMP = CABS1( B( N, K ) ) + CABS1( E( N-1 ) )*
00254      $            CABS1( X( N-1, K ) ) + CABS1( D( N )*X( N, K ) )
00255             AXBI = MIN( AXBI, TMP )
00256          END IF
00257          TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
00258          IF( K.EQ.1 ) THEN
00259             RESLTS( 2 ) = TMP
00260          ELSE
00261             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00262          END IF
00263    50 CONTINUE
00264 *
00265       RETURN
00266 *
00267 *     End of ZPTT05
00268 *
00269       END
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