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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DGTT05 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 DGTT05( 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 * DOUBLE PRECISION 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 *> DGTT05 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N) 00079 *> The diagonal elements of A. 00080 *> \endverbatim 00081 *> 00082 *> \param[in] DU 00083 *> \verbatim 00084 *> DU is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 double_lin 00163 * 00164 * ===================================================================== 00165 SUBROUTINE DGTT05( 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 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DL( * ), 00179 $ DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ), 00180 $ XACT( LDXACT, * ) 00181 * .. 00182 * 00183 * ===================================================================== 00184 * 00185 * .. Parameters .. 00186 DOUBLE PRECISION ZERO, ONE 00187 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00188 * .. 00189 * .. Local Scalars .. 00190 LOGICAL NOTRAN 00191 INTEGER I, IMAX, J, K, NZ 00192 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00193 * .. 00194 * .. External Functions .. 00195 LOGICAL LSAME 00196 INTEGER IDAMAX 00197 DOUBLE PRECISION DLAMCH 00198 EXTERNAL LSAME, IDAMAX, DLAMCH 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 = DLAMCH( 'Epsilon' ) 00214 UNFL = DLAMCH( '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 = IDAMAX( 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 DGTT05 00298 * 00299 END