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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b STRT05 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 STRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, 00012 * LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER DIAG, TRANS, 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 *> STRT05 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 *> triangular 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 matrix A is upper or lower triangular. 00050 *> = 'U': Upper triangular 00051 *> = 'L': Lower triangular 00052 *> \endverbatim 00053 *> 00054 *> \param[in] TRANS 00055 *> \verbatim 00056 *> TRANS is CHARACTER*1 00057 *> Specifies the form of the system of equations. 00058 *> = 'N': A * X = B (No transpose) 00059 *> = 'T': A'* X = B (Transpose) 00060 *> = 'C': A'* X = B (Conjugate transpose = Transpose) 00061 *> \endverbatim 00062 *> 00063 *> \param[in] DIAG 00064 *> \verbatim 00065 *> DIAG is CHARACTER*1 00066 *> Specifies whether or not the matrix A is unit triangular. 00067 *> = 'N': Non-unit triangular 00068 *> = 'U': Unit triangular 00069 *> \endverbatim 00070 *> 00071 *> \param[in] N 00072 *> \verbatim 00073 *> N is INTEGER 00074 *> The number of rows of the matrices X, B, and XACT, and the 00075 *> order of the matrix A. N >= 0. 00076 *> \endverbatim 00077 *> 00078 *> \param[in] NRHS 00079 *> \verbatim 00080 *> NRHS is INTEGER 00081 *> The number of columns of the matrices X, B, and XACT. 00082 *> NRHS >= 0. 00083 *> \endverbatim 00084 *> 00085 *> \param[in] A 00086 *> \verbatim 00087 *> A is REAL array, dimension (LDA,N) 00088 *> The triangular matrix A. If UPLO = 'U', the leading n by n 00089 *> upper triangular part of the array A contains the upper 00090 *> triangular matrix, and the strictly lower triangular part of 00091 *> A is not referenced. If UPLO = 'L', the leading n by n lower 00092 *> triangular part of the array A contains the lower triangular 00093 *> matrix, and the strictly upper triangular part of A is not 00094 *> referenced. If DIAG = 'U', the diagonal elements of A are 00095 *> also not referenced and are assumed to be 1. 00096 *> \endverbatim 00097 *> 00098 *> \param[in] LDA 00099 *> \verbatim 00100 *> LDA is INTEGER 00101 *> The leading dimension of the array A. LDA >= max(1,N). 00102 *> \endverbatim 00103 *> 00104 *> \param[in] B 00105 *> \verbatim 00106 *> B is REAL array, dimension (LDB,NRHS) 00107 *> The right hand side vectors for the system of linear 00108 *> equations. 00109 *> \endverbatim 00110 *> 00111 *> \param[in] LDB 00112 *> \verbatim 00113 *> LDB is INTEGER 00114 *> The leading dimension of the array B. LDB >= max(1,N). 00115 *> \endverbatim 00116 *> 00117 *> \param[in] X 00118 *> \verbatim 00119 *> X is REAL array, dimension (LDX,NRHS) 00120 *> The computed solution vectors. Each vector is stored as a 00121 *> column of the matrix X. 00122 *> \endverbatim 00123 *> 00124 *> \param[in] LDX 00125 *> \verbatim 00126 *> LDX is INTEGER 00127 *> The leading dimension of the array X. LDX >= max(1,N). 00128 *> \endverbatim 00129 *> 00130 *> \param[in] XACT 00131 *> \verbatim 00132 *> XACT is REAL array, dimension (LDX,NRHS) 00133 *> The exact solution vectors. Each vector is stored as a 00134 *> column of the matrix XACT. 00135 *> \endverbatim 00136 *> 00137 *> \param[in] LDXACT 00138 *> \verbatim 00139 *> LDXACT is INTEGER 00140 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00141 *> \endverbatim 00142 *> 00143 *> \param[in] FERR 00144 *> \verbatim 00145 *> FERR is REAL array, dimension (NRHS) 00146 *> The estimated forward error bounds for each solution vector 00147 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00148 *> of the largest entry in (X - XTRUE) divided by the magnitude 00149 *> of the largest entry in X. 00150 *> \endverbatim 00151 *> 00152 *> \param[in] BERR 00153 *> \verbatim 00154 *> BERR is REAL array, dimension (NRHS) 00155 *> The componentwise relative backward error of each solution 00156 *> vector (i.e., the smallest relative change in any entry of A 00157 *> or B that makes X an exact solution). 00158 *> \endverbatim 00159 *> 00160 *> \param[out] RESLTS 00161 *> \verbatim 00162 *> RESLTS is REAL array, dimension (2) 00163 *> The maximum over the NRHS solution vectors of the ratios: 00164 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00165 *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00166 *> \endverbatim 00167 * 00168 * Authors: 00169 * ======== 00170 * 00171 *> \author Univ. of Tennessee 00172 *> \author Univ. of California Berkeley 00173 *> \author Univ. of Colorado Denver 00174 *> \author NAG Ltd. 00175 * 00176 *> \date November 2011 00177 * 00178 *> \ingroup single_lin 00179 * 00180 * ===================================================================== 00181 SUBROUTINE STRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, 00182 $ LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 00183 * 00184 * -- LAPACK test routine (version 3.4.0) -- 00185 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00186 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00187 * November 2011 00188 * 00189 * .. Scalar Arguments .. 00190 CHARACTER DIAG, TRANS, UPLO 00191 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 00192 * .. 00193 * .. Array Arguments .. 00194 REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 00195 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 00196 * .. 00197 * 00198 * ===================================================================== 00199 * 00200 * .. Parameters .. 00201 REAL ZERO, ONE 00202 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00203 * .. 00204 * .. Local Scalars .. 00205 LOGICAL NOTRAN, UNIT, UPPER 00206 INTEGER I, IFU, IMAX, J, K 00207 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00208 * .. 00209 * .. External Functions .. 00210 LOGICAL LSAME 00211 INTEGER ISAMAX 00212 REAL SLAMCH 00213 EXTERNAL LSAME, ISAMAX, SLAMCH 00214 * .. 00215 * .. Intrinsic Functions .. 00216 INTRINSIC ABS, MAX, MIN 00217 * .. 00218 * .. Executable Statements .. 00219 * 00220 * Quick exit if N = 0 or NRHS = 0. 00221 * 00222 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00223 RESLTS( 1 ) = ZERO 00224 RESLTS( 2 ) = ZERO 00225 RETURN 00226 END IF 00227 * 00228 EPS = SLAMCH( 'Epsilon' ) 00229 UNFL = SLAMCH( 'Safe minimum' ) 00230 OVFL = ONE / UNFL 00231 UPPER = LSAME( UPLO, 'U' ) 00232 NOTRAN = LSAME( TRANS, 'N' ) 00233 UNIT = LSAME( DIAG, 'U' ) 00234 * 00235 * Test 1: Compute the maximum of 00236 * norm(X - XACT) / ( norm(X) * FERR ) 00237 * over all the vectors X and XACT using the infinity-norm. 00238 * 00239 ERRBND = ZERO 00240 DO 30 J = 1, NRHS 00241 IMAX = ISAMAX( N, X( 1, J ), 1 ) 00242 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00243 DIFF = ZERO 00244 DO 10 I = 1, N 00245 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00246 10 CONTINUE 00247 * 00248 IF( XNORM.GT.ONE ) THEN 00249 GO TO 20 00250 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00251 GO TO 20 00252 ELSE 00253 ERRBND = ONE / EPS 00254 GO TO 30 00255 END IF 00256 * 00257 20 CONTINUE 00258 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00259 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00260 ELSE 00261 ERRBND = ONE / EPS 00262 END IF 00263 30 CONTINUE 00264 RESLTS( 1 ) = ERRBND 00265 * 00266 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00267 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00268 * 00269 IFU = 0 00270 IF( UNIT ) 00271 $ IFU = 1 00272 DO 90 K = 1, NRHS 00273 DO 80 I = 1, N 00274 TMP = ABS( B( I, K ) ) 00275 IF( UPPER ) THEN 00276 IF( .NOT.NOTRAN ) THEN 00277 DO 40 J = 1, I - IFU 00278 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 00279 40 CONTINUE 00280 IF( UNIT ) 00281 $ TMP = TMP + ABS( X( I, K ) ) 00282 ELSE 00283 IF( UNIT ) 00284 $ TMP = TMP + ABS( X( I, K ) ) 00285 DO 50 J = I + IFU, N 00286 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 00287 50 CONTINUE 00288 END IF 00289 ELSE 00290 IF( NOTRAN ) THEN 00291 DO 60 J = 1, I - IFU 00292 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 00293 60 CONTINUE 00294 IF( UNIT ) 00295 $ TMP = TMP + ABS( X( I, K ) ) 00296 ELSE 00297 IF( UNIT ) 00298 $ TMP = TMP + ABS( X( I, K ) ) 00299 DO 70 J = I + IFU, N 00300 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 00301 70 CONTINUE 00302 END IF 00303 END IF 00304 IF( I.EQ.1 ) THEN 00305 AXBI = TMP 00306 ELSE 00307 AXBI = MIN( AXBI, TMP ) 00308 END IF 00309 80 CONTINUE 00310 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00311 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00312 IF( K.EQ.1 ) THEN 00313 RESLTS( 2 ) = TMP 00314 ELSE 00315 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00316 END IF 00317 90 CONTINUE 00318 * 00319 RETURN 00320 * 00321 * End of STRT05 00322 * 00323 END