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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b SPOT05 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 SPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 00012 * LDXACT, FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER UPLO 00016 * INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 00017 * .. 00018 * .. Array Arguments .. 00019 * REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 00020 * $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 00021 * .. 00022 * 00023 * 00024 *> \par Purpose: 00025 * ============= 00026 *> 00027 *> \verbatim 00028 *> 00029 *> SPOT05 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 *> symmetric n by n matrix. 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 / ( (n+1)*EPS + (*) ), where 00040 *> (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00041 *> \endverbatim 00042 * 00043 * Arguments: 00044 * ========== 00045 * 00046 *> \param[in] UPLO 00047 *> \verbatim 00048 *> UPLO is CHARACTER*1 00049 *> Specifies whether the upper or lower triangular part of the 00050 *> symmetric matrix A is stored. 00051 *> = 'U': Upper triangular 00052 *> = 'L': Lower triangular 00053 *> \endverbatim 00054 *> 00055 *> \param[in] N 00056 *> \verbatim 00057 *> N is INTEGER 00058 *> The number of rows of the matrices X, B, and XACT, and the 00059 *> order of the matrix A. N >= 0. 00060 *> \endverbatim 00061 *> 00062 *> \param[in] NRHS 00063 *> \verbatim 00064 *> NRHS is INTEGER 00065 *> The number of columns of the matrices X, B, and XACT. 00066 *> NRHS >= 0. 00067 *> \endverbatim 00068 *> 00069 *> \param[in] A 00070 *> \verbatim 00071 *> A is REAL array, dimension (LDA,N) 00072 *> The symmetric matrix A. If UPLO = 'U', the leading n by n 00073 *> upper triangular part of A contains the upper triangular part 00074 *> of the matrix A, and the strictly lower triangular part of A 00075 *> is not referenced. If UPLO = 'L', the leading n by n lower 00076 *> triangular part of A contains the lower triangular part of 00077 *> the matrix A, and the strictly upper triangular part of A is 00078 *> not referenced. 00079 *> \endverbatim 00080 *> 00081 *> \param[in] LDA 00082 *> \verbatim 00083 *> LDA is INTEGER 00084 *> The leading dimension of the array A. LDA >= max(1,N). 00085 *> \endverbatim 00086 *> 00087 *> \param[in] B 00088 *> \verbatim 00089 *> B is REAL array, dimension (LDB,NRHS) 00090 *> The right hand side vectors for the system of linear 00091 *> equations. 00092 *> \endverbatim 00093 *> 00094 *> \param[in] LDB 00095 *> \verbatim 00096 *> LDB is INTEGER 00097 *> The leading dimension of the array B. LDB >= max(1,N). 00098 *> \endverbatim 00099 *> 00100 *> \param[in] X 00101 *> \verbatim 00102 *> X is REAL array, dimension (LDX,NRHS) 00103 *> The computed solution vectors. Each vector is stored as a 00104 *> column of the matrix X. 00105 *> \endverbatim 00106 *> 00107 *> \param[in] LDX 00108 *> \verbatim 00109 *> LDX is INTEGER 00110 *> The leading dimension of the array X. LDX >= max(1,N). 00111 *> \endverbatim 00112 *> 00113 *> \param[in] XACT 00114 *> \verbatim 00115 *> XACT is REAL array, dimension (LDX,NRHS) 00116 *> The exact solution vectors. Each vector is stored as a 00117 *> column of the matrix XACT. 00118 *> \endverbatim 00119 *> 00120 *> \param[in] LDXACT 00121 *> \verbatim 00122 *> LDXACT is INTEGER 00123 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00124 *> \endverbatim 00125 *> 00126 *> \param[in] FERR 00127 *> \verbatim 00128 *> FERR is REAL array, dimension (NRHS) 00129 *> The estimated forward error bounds for each solution vector 00130 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00131 *> of the largest entry in (X - XTRUE) divided by the magnitude 00132 *> of the largest entry in X. 00133 *> \endverbatim 00134 *> 00135 *> \param[in] BERR 00136 *> \verbatim 00137 *> BERR is REAL array, dimension (NRHS) 00138 *> The componentwise relative backward error of each solution 00139 *> vector (i.e., the smallest relative change in any entry of A 00140 *> or B that makes X an exact solution). 00141 *> \endverbatim 00142 *> 00143 *> \param[out] RESLTS 00144 *> \verbatim 00145 *> RESLTS is REAL array, dimension (2) 00146 *> The maximum over the NRHS solution vectors of the ratios: 00147 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00148 *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00149 *> \endverbatim 00150 * 00151 * Authors: 00152 * ======== 00153 * 00154 *> \author Univ. of Tennessee 00155 *> \author Univ. of California Berkeley 00156 *> \author Univ. of Colorado Denver 00157 *> \author NAG Ltd. 00158 * 00159 *> \date November 2011 00160 * 00161 *> \ingroup single_lin 00162 * 00163 * ===================================================================== 00164 SUBROUTINE SPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 00165 $ LDXACT, FERR, BERR, RESLTS ) 00166 * 00167 * -- LAPACK test routine (version 3.4.0) -- 00168 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00169 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00170 * November 2011 00171 * 00172 * .. Scalar Arguments .. 00173 CHARACTER UPLO 00174 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 00175 * .. 00176 * .. Array Arguments .. 00177 REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 00178 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 00179 * .. 00180 * 00181 * ===================================================================== 00182 * 00183 * .. Parameters .. 00184 REAL ZERO, ONE 00185 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00186 * .. 00187 * .. Local Scalars .. 00188 LOGICAL UPPER 00189 INTEGER I, IMAX, J, K 00190 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00191 * .. 00192 * .. External Functions .. 00193 LOGICAL LSAME 00194 INTEGER ISAMAX 00195 REAL SLAMCH 00196 EXTERNAL LSAME, ISAMAX, SLAMCH 00197 * .. 00198 * .. Intrinsic Functions .. 00199 INTRINSIC ABS, MAX, MIN 00200 * .. 00201 * .. Executable Statements .. 00202 * 00203 * Quick exit if N = 0 or NRHS = 0. 00204 * 00205 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00206 RESLTS( 1 ) = ZERO 00207 RESLTS( 2 ) = ZERO 00208 RETURN 00209 END IF 00210 * 00211 EPS = SLAMCH( 'Epsilon' ) 00212 UNFL = SLAMCH( 'Safe minimum' ) 00213 OVFL = ONE / UNFL 00214 UPPER = LSAME( UPLO, 'U' ) 00215 * 00216 * Test 1: Compute the maximum of 00217 * norm(X - XACT) / ( norm(X) * FERR ) 00218 * over all the vectors X and XACT using the infinity-norm. 00219 * 00220 ERRBND = ZERO 00221 DO 30 J = 1, NRHS 00222 IMAX = ISAMAX( N, X( 1, J ), 1 ) 00223 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00224 DIFF = ZERO 00225 DO 10 I = 1, N 00226 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00227 10 CONTINUE 00228 * 00229 IF( XNORM.GT.ONE ) THEN 00230 GO TO 20 00231 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00232 GO TO 20 00233 ELSE 00234 ERRBND = ONE / EPS 00235 GO TO 30 00236 END IF 00237 * 00238 20 CONTINUE 00239 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00240 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00241 ELSE 00242 ERRBND = ONE / EPS 00243 END IF 00244 30 CONTINUE 00245 RESLTS( 1 ) = ERRBND 00246 * 00247 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00248 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00249 * 00250 DO 90 K = 1, NRHS 00251 DO 80 I = 1, N 00252 TMP = ABS( B( I, K ) ) 00253 IF( UPPER ) THEN 00254 DO 40 J = 1, I 00255 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 00256 40 CONTINUE 00257 DO 50 J = I + 1, N 00258 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 00259 50 CONTINUE 00260 ELSE 00261 DO 60 J = 1, I - 1 00262 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 00263 60 CONTINUE 00264 DO 70 J = I, N 00265 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 00266 70 CONTINUE 00267 END IF 00268 IF( I.EQ.1 ) THEN 00269 AXBI = TMP 00270 ELSE 00271 AXBI = MIN( AXBI, TMP ) 00272 END IF 00273 80 CONTINUE 00274 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00275 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00276 IF( K.EQ.1 ) THEN 00277 RESLTS( 2 ) = TMP 00278 ELSE 00279 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00280 END IF 00281 90 CONTINUE 00282 * 00283 RETURN 00284 * 00285 * End of SPOT05 00286 * 00287 END