LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sgtt05.f
Go to the documentation of this file.
00001 *> \brief \b SGTT05
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 SGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
00012 *                          XACT, LDXACT, FERR, BERR, RESLTS )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          TRANS
00016 *       INTEGER            LDB, LDX, LDXACT, N, NRHS
00017 *       ..
00018 *       .. Array Arguments ..
00019 *       REAL               B( LDB, * ), BERR( * ), D( * ), DL( * ),
00020 *      $                   DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ),
00021 *      $                   XACT( LDXACT, * )
00022 *       ..
00023 *  
00024 *
00025 *> \par Purpose:
00026 *  =============
00027 *>
00028 *> \verbatim
00029 *>
00030 *> SGTT05 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 *> general tridiagonal matrix of order n and op(A) = A or A**T,
00033 *> depending on TRANS.
00034 *>
00035 *> RESLTS(1) = test of the error bound
00036 *>           = norm(X - XACT) / ( norm(X) * FERR )
00037 *>
00038 *> A large value is returned if this ratio is not less than one.
00039 *>
00040 *> RESLTS(2) = residual from the iterative refinement routine
00041 *>           = the maximum of BERR / ( NZ*EPS + (*) ), where
00042 *>             (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
00043 *>             and NZ = max. number of nonzeros in any row of A, plus 1
00044 *> \endverbatim
00045 *
00046 *  Arguments:
00047 *  ==========
00048 *
00049 *> \param[in] TRANS
00050 *> \verbatim
00051 *>          TRANS is CHARACTER*1
00052 *>          Specifies the form of the system of equations.
00053 *>          = 'N':  A * X = B     (No transpose)
00054 *>          = 'T':  A**T * X = B  (Transpose)
00055 *>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
00056 *> \endverbatim
00057 *>
00058 *> \param[in] N
00059 *> \verbatim
00060 *>          N is INTEGER
00061 *>          The number of rows of the matrices X and XACT.  N >= 0.
00062 *> \endverbatim
00063 *>
00064 *> \param[in] NRHS
00065 *> \verbatim
00066 *>          NRHS is INTEGER
00067 *>          The number of columns of the matrices X and XACT.  NRHS >= 0.
00068 *> \endverbatim
00069 *>
00070 *> \param[in] DL
00071 *> \verbatim
00072 *>          DL is REAL array, dimension (N-1)
00073 *>          The (n-1) sub-diagonal elements of A.
00074 *> \endverbatim
00075 *>
00076 *> \param[in] D
00077 *> \verbatim
00078 *>          D is REAL array, dimension (N)
00079 *>          The diagonal elements of A.
00080 *> \endverbatim
00081 *>
00082 *> \param[in] DU
00083 *> \verbatim
00084 *>          DU is REAL array, dimension (N-1)
00085 *>          The (n-1) super-diagonal elements of A.
00086 *> \endverbatim
00087 *>
00088 *> \param[in] B
00089 *> \verbatim
00090 *>          B is REAL 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 REAL 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 REAL 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 / ( NZ*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 single_lin
00163 *
00164 *  =====================================================================
00165       SUBROUTINE SGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX,
00166      $                   XACT, 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          TRANS
00175       INTEGER            LDB, LDX, LDXACT, N, NRHS
00176 *     ..
00177 *     .. Array Arguments ..
00178       REAL               B( LDB, * ), BERR( * ), D( * ), DL( * ),
00179      $                   DU( * ), FERR( * ), RESLTS( * ), 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            NOTRAN
00191       INTEGER            I, IMAX, J, K, NZ
00192       REAL               AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM
00193 *     ..
00194 *     .. External Functions ..
00195       LOGICAL            LSAME
00196       INTEGER            ISAMAX
00197       REAL               SLAMCH
00198       EXTERNAL           LSAME, ISAMAX, SLAMCH
00199 *     ..
00200 *     .. Intrinsic Functions ..
00201       INTRINSIC          ABS, MAX, MIN
00202 *     ..
00203 *     .. Executable Statements ..
00204 *
00205 *     Quick exit if N = 0 or NRHS = 0.
00206 *
00207       IF( N.LE.0 .OR. NRHS.LE.0 ) THEN
00208          RESLTS( 1 ) = ZERO
00209          RESLTS( 2 ) = ZERO
00210          RETURN
00211       END IF
00212 *
00213       EPS = SLAMCH( 'Epsilon' )
00214       UNFL = SLAMCH( 'Safe minimum' )
00215       OVFL = ONE / UNFL
00216       NOTRAN = LSAME( TRANS, 'N' )
00217       NZ = 4
00218 *
00219 *     Test 1:  Compute the maximum of
00220 *        norm(X - XACT) / ( norm(X) * FERR )
00221 *     over all the vectors X and XACT using the infinity-norm.
00222 *
00223       ERRBND = ZERO
00224       DO 30 J = 1, NRHS
00225          IMAX = ISAMAX( N, X( 1, J ), 1 )
00226          XNORM = MAX( ABS( X( IMAX, J ) ), UNFL )
00227          DIFF = ZERO
00228          DO 10 I = 1, N
00229             DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) )
00230    10    CONTINUE
00231 *
00232          IF( XNORM.GT.ONE ) THEN
00233             GO TO 20
00234          ELSE IF( DIFF.LE.OVFL*XNORM ) THEN
00235             GO TO 20
00236          ELSE
00237             ERRBND = ONE / EPS
00238             GO TO 30
00239          END IF
00240 *
00241    20    CONTINUE
00242          IF( DIFF / XNORM.LE.FERR( J ) ) THEN
00243             ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) )
00244          ELSE
00245             ERRBND = ONE / EPS
00246          END IF
00247    30 CONTINUE
00248       RESLTS( 1 ) = ERRBND
00249 *
00250 *     Test 2:  Compute the maximum of BERR / ( NZ*EPS + (*) ), where
00251 *     (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
00252 *
00253       DO 60 K = 1, NRHS
00254          IF( NOTRAN ) THEN
00255             IF( N.EQ.1 ) THEN
00256                AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
00257             ELSE
00258                AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
00259      $                ABS( DU( 1 )*X( 2, K ) )
00260                DO 40 I = 2, N - 1
00261                   TMP = ABS( B( I, K ) ) + ABS( DL( I-1 )*X( I-1, K ) )
00262      $                   + ABS( D( I )*X( I, K ) ) +
00263      $                  ABS( DU( I )*X( I+1, K ) )
00264                   AXBI = MIN( AXBI, TMP )
00265    40          CONTINUE
00266                TMP = ABS( B( N, K ) ) + ABS( DL( N-1 )*X( N-1, K ) ) +
00267      $               ABS( D( N )*X( N, K ) )
00268                AXBI = MIN( AXBI, TMP )
00269             END IF
00270          ELSE
00271             IF( N.EQ.1 ) THEN
00272                AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) )
00273             ELSE
00274                AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) +
00275      $                ABS( DL( 1 )*X( 2, K ) )
00276                DO 50 I = 2, N - 1
00277                   TMP = ABS( B( I, K ) ) + ABS( DU( I-1 )*X( I-1, K ) )
00278      $                   + ABS( D( I )*X( I, K ) ) +
00279      $                  ABS( DL( I )*X( I+1, K ) )
00280                   AXBI = MIN( AXBI, TMP )
00281    50          CONTINUE
00282                TMP = ABS( B( N, K ) ) + ABS( DU( N-1 )*X( N-1, K ) ) +
00283      $               ABS( D( N )*X( N, K ) )
00284                AXBI = MIN( AXBI, TMP )
00285             END IF
00286          END IF
00287          TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) )
00288          IF( K.EQ.1 ) THEN
00289             RESLTS( 2 ) = TMP
00290          ELSE
00291             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00292          END IF
00293    60 CONTINUE
00294 *
00295       RETURN
00296 *
00297 *     End of SGTT05
00298 *
00299       END
 All Files Functions