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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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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