LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
sget07.f
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00001 *> \brief \b SGET07
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 SGET07( TRANS, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
00012 *                          LDXACT, FERR, CHKFERR, BERR, RESLTS )
00013 * 
00014 *       .. Scalar Arguments ..
00015 *       CHARACTER          TRANS
00016 *       LOGICAL            CHKFERR
00017 *       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
00018 *       ..
00019 *       .. Array Arguments ..
00020 *       REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
00021 *      $                   RESLTS( * ), X( LDX, * ), XACT( LDXACT, * )
00022 *       ..
00023 *  
00024 *
00025 *> \par Purpose:
00026 *  =============
00027 *>
00028 *> \verbatim
00029 *>
00030 *> SGET07 tests the error bounds from iterative refinement for the
00031 *> computed solution to a system of equations op(A)*X = B, where A is a
00032 *> general n by n matrix and op(A) = A or A**T, depending on TRANS.
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(op(A))*abs(X) +abs(b))_i )
00042 *> \endverbatim
00043 *
00044 *  Arguments:
00045 *  ==========
00046 *
00047 *> \param[in] TRANS
00048 *> \verbatim
00049 *>          TRANS is CHARACTER*1
00050 *>          Specifies the form of the system of equations.
00051 *>          = 'N':  A * X = B     (No transpose)
00052 *>          = 'T':  A**T * X = B  (Transpose)
00053 *>          = 'C':  A**H * X = B  (Conjugate transpose = Transpose)
00054 *> \endverbatim
00055 *>
00056 *> \param[in] N
00057 *> \verbatim
00058 *>          N is INTEGER
00059 *>          The number of rows of the matrices X and XACT.  N >= 0.
00060 *> \endverbatim
00061 *>
00062 *> \param[in] NRHS
00063 *> \verbatim
00064 *>          NRHS is INTEGER
00065 *>          The number of columns of the matrices X and XACT.  NRHS >= 0.
00066 *> \endverbatim
00067 *>
00068 *> \param[in] A
00069 *> \verbatim
00070 *>          A is REAL array, dimension (LDA,N)
00071 *>          The original n by n matrix A.
00072 *> \endverbatim
00073 *>
00074 *> \param[in] LDA
00075 *> \verbatim
00076 *>          LDA is INTEGER
00077 *>          The leading dimension of the array A.  LDA >= max(1,N).
00078 *> \endverbatim
00079 *>
00080 *> \param[in] B
00081 *> \verbatim
00082 *>          B is REAL 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 REAL 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 REAL 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] CHKFERR
00129 *> \verbatim
00130 *>          CHKFERR is LOGICAL
00131 *>          Set to .TRUE. to check FERR, .FALSE. not to check FERR.
00132 *>          When the test system is ill-conditioned, the "true"
00133 *>          solution in XACT may be incorrect.
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 single_lin
00163 *
00164 *  =====================================================================
00165       SUBROUTINE SGET07( TRANS, N, NRHS, A, LDA, B, LDB, X, LDX, XACT,
00166      $                   LDXACT, FERR, CHKFERR, 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       LOGICAL            CHKFERR
00176       INTEGER            LDA, LDB, LDX, LDXACT, N, NRHS
00177 *     ..
00178 *     .. Array Arguments ..
00179       REAL               A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ),
00180      $                   RESLTS( * ), X( LDX, * ), 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
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 *
00218 *     Test 1:  Compute the maximum of
00219 *        norm(X - XACT) / ( norm(X) * FERR )
00220 *     over all the vectors X and XACT using the infinity-norm.
00221 *
00222       ERRBND = ZERO
00223       IF( CHKFERR ) THEN
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       END IF
00249       RESLTS( 1 ) = ERRBND
00250 *
00251 *     Test 2:  Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where
00252 *     (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i )
00253 *
00254       DO 70 K = 1, NRHS
00255          DO 60 I = 1, N
00256             TMP = ABS( B( I, K ) )
00257             IF( NOTRAN ) THEN
00258                DO 40 J = 1, N
00259                   TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) )
00260    40          CONTINUE
00261             ELSE
00262                DO 50 J = 1, N
00263                   TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) )
00264    50          CONTINUE
00265             END IF
00266             IF( I.EQ.1 ) THEN
00267                AXBI = TMP
00268             ELSE
00269                AXBI = MIN( AXBI, TMP )
00270             END IF
00271    60    CONTINUE
00272          TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL /
00273      $         MAX( AXBI, ( N+1 )*UNFL ) )
00274          IF( K.EQ.1 ) THEN
00275             RESLTS( 2 ) = TMP
00276          ELSE
00277             RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP )
00278          END IF
00279    70 CONTINUE
00280 *
00281       RETURN
00282 *
00283 *     End of SGET07
00284 *
00285       END
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