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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b CTRT05 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 CTRT05( 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 BERR( * ), FERR( * ), RESLTS( * ) 00020 * COMPLEX A( LDA, * ), B( LDB, * ), X( LDX, * ), 00021 * $ XACT( LDXACT, * ) 00022 * .. 00023 * 00024 * 00025 *> \par Purpose: 00026 * ============= 00027 *> 00028 *> \verbatim 00029 *> 00030 *> CTRT05 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 *> triangular n by n matrix. 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(A)*abs(X) +abs(b))_i ) 00042 *> \endverbatim 00043 * 00044 * Arguments: 00045 * ========== 00046 * 00047 *> \param[in] UPLO 00048 *> \verbatim 00049 *> UPLO is CHARACTER*1 00050 *> Specifies whether the matrix A is upper or lower triangular. 00051 *> = 'U': Upper triangular 00052 *> = 'L': Lower triangular 00053 *> \endverbatim 00054 *> 00055 *> \param[in] TRANS 00056 *> \verbatim 00057 *> TRANS is CHARACTER*1 00058 *> Specifies the form of the system of equations. 00059 *> = 'N': A * X = B (No transpose) 00060 *> = 'T': A'* X = B (Transpose) 00061 *> = 'C': A'* X = B (Conjugate transpose = Transpose) 00062 *> \endverbatim 00063 *> 00064 *> \param[in] DIAG 00065 *> \verbatim 00066 *> DIAG is CHARACTER*1 00067 *> Specifies whether or not the matrix A is unit triangular. 00068 *> = 'N': Non-unit triangular 00069 *> = 'U': Unit triangular 00070 *> \endverbatim 00071 *> 00072 *> \param[in] N 00073 *> \verbatim 00074 *> N is INTEGER 00075 *> The number of rows of the matrices X, B, and XACT, and the 00076 *> order of the matrix A. N >= 0. 00077 *> \endverbatim 00078 *> 00079 *> \param[in] NRHS 00080 *> \verbatim 00081 *> NRHS is INTEGER 00082 *> The number of columns of the matrices X, B, and XACT. 00083 *> NRHS >= 0. 00084 *> \endverbatim 00085 *> 00086 *> \param[in] A 00087 *> \verbatim 00088 *> A is COMPLEX array, dimension (LDA,N) 00089 *> The triangular matrix A. If UPLO = 'U', the leading n by n 00090 *> upper triangular part of the array A contains the upper 00091 *> triangular matrix, and the strictly lower triangular part of 00092 *> A is not referenced. If UPLO = 'L', the leading n by n lower 00093 *> triangular part of the array A contains the lower triangular 00094 *> matrix, and the strictly upper triangular part of A is not 00095 *> referenced. If DIAG = 'U', the diagonal elements of A are 00096 *> also not referenced and are assumed to be 1. 00097 *> \endverbatim 00098 *> 00099 *> \param[in] LDA 00100 *> \verbatim 00101 *> LDA is INTEGER 00102 *> The leading dimension of the array A. LDA >= max(1,N). 00103 *> \endverbatim 00104 *> 00105 *> \param[in] B 00106 *> \verbatim 00107 *> B is COMPLEX array, dimension (LDB,NRHS) 00108 *> The right hand side vectors for the system of linear 00109 *> equations. 00110 *> \endverbatim 00111 *> 00112 *> \param[in] LDB 00113 *> \verbatim 00114 *> LDB is INTEGER 00115 *> The leading dimension of the array B. LDB >= max(1,N). 00116 *> \endverbatim 00117 *> 00118 *> \param[in] X 00119 *> \verbatim 00120 *> X is COMPLEX array, dimension (LDX,NRHS) 00121 *> The computed solution vectors. Each vector is stored as a 00122 *> column of the matrix X. 00123 *> \endverbatim 00124 *> 00125 *> \param[in] LDX 00126 *> \verbatim 00127 *> LDX is INTEGER 00128 *> The leading dimension of the array X. LDX >= max(1,N). 00129 *> \endverbatim 00130 *> 00131 *> \param[in] XACT 00132 *> \verbatim 00133 *> XACT is COMPLEX array, dimension (LDX,NRHS) 00134 *> The exact solution vectors. Each vector is stored as a 00135 *> column of the matrix XACT. 00136 *> \endverbatim 00137 *> 00138 *> \param[in] LDXACT 00139 *> \verbatim 00140 *> LDXACT is INTEGER 00141 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00142 *> \endverbatim 00143 *> 00144 *> \param[in] FERR 00145 *> \verbatim 00146 *> FERR is REAL array, dimension (NRHS) 00147 *> The estimated forward error bounds for each solution vector 00148 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00149 *> of the largest entry in (X - XTRUE) divided by the magnitude 00150 *> of the largest entry in X. 00151 *> \endverbatim 00152 *> 00153 *> \param[in] BERR 00154 *> \verbatim 00155 *> BERR is REAL array, dimension (NRHS) 00156 *> The componentwise relative backward error of each solution 00157 *> vector (i.e., the smallest relative change in any entry of A 00158 *> or B that makes X an exact solution). 00159 *> \endverbatim 00160 *> 00161 *> \param[out] RESLTS 00162 *> \verbatim 00163 *> RESLTS is REAL array, dimension (2) 00164 *> The maximum over the NRHS solution vectors of the ratios: 00165 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00166 *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00167 *> \endverbatim 00168 * 00169 * Authors: 00170 * ======== 00171 * 00172 *> \author Univ. of Tennessee 00173 *> \author Univ. of California Berkeley 00174 *> \author Univ. of Colorado Denver 00175 *> \author NAG Ltd. 00176 * 00177 *> \date November 2011 00178 * 00179 *> \ingroup complex_lin 00180 * 00181 * ===================================================================== 00182 SUBROUTINE CTRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, 00183 $ LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 00184 * 00185 * -- LAPACK test routine (version 3.4.0) -- 00186 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00187 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00188 * November 2011 00189 * 00190 * .. Scalar Arguments .. 00191 CHARACTER DIAG, TRANS, UPLO 00192 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 00193 * .. 00194 * .. Array Arguments .. 00195 REAL BERR( * ), FERR( * ), RESLTS( * ) 00196 COMPLEX A( LDA, * ), B( LDB, * ), X( LDX, * ), 00197 $ XACT( LDXACT, * ) 00198 * .. 00199 * 00200 * ===================================================================== 00201 * 00202 * .. Parameters .. 00203 REAL ZERO, ONE 00204 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 00205 * .. 00206 * .. Local Scalars .. 00207 LOGICAL NOTRAN, UNIT, UPPER 00208 INTEGER I, IFU, IMAX, J, K 00209 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00210 COMPLEX ZDUM 00211 * .. 00212 * .. External Functions .. 00213 LOGICAL LSAME 00214 INTEGER ICAMAX 00215 REAL SLAMCH 00216 EXTERNAL LSAME, ICAMAX, SLAMCH 00217 * .. 00218 * .. Intrinsic Functions .. 00219 INTRINSIC ABS, AIMAG, MAX, MIN, REAL 00220 * .. 00221 * .. Statement Functions .. 00222 REAL CABS1 00223 * .. 00224 * .. Statement Function definitions .. 00225 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) ) 00226 * .. 00227 * .. Executable Statements .. 00228 * 00229 * Quick exit if N = 0 or NRHS = 0. 00230 * 00231 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00232 RESLTS( 1 ) = ZERO 00233 RESLTS( 2 ) = ZERO 00234 RETURN 00235 END IF 00236 * 00237 EPS = SLAMCH( 'Epsilon' ) 00238 UNFL = SLAMCH( 'Safe minimum' ) 00239 OVFL = ONE / UNFL 00240 UPPER = LSAME( UPLO, 'U' ) 00241 NOTRAN = LSAME( TRANS, 'N' ) 00242 UNIT = LSAME( DIAG, 'U' ) 00243 * 00244 * Test 1: Compute the maximum of 00245 * norm(X - XACT) / ( norm(X) * FERR ) 00246 * over all the vectors X and XACT using the infinity-norm. 00247 * 00248 ERRBND = ZERO 00249 DO 30 J = 1, NRHS 00250 IMAX = ICAMAX( N, X( 1, J ), 1 ) 00251 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 00252 DIFF = ZERO 00253 DO 10 I = 1, N 00254 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 00255 10 CONTINUE 00256 * 00257 IF( XNORM.GT.ONE ) THEN 00258 GO TO 20 00259 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00260 GO TO 20 00261 ELSE 00262 ERRBND = ONE / EPS 00263 GO TO 30 00264 END IF 00265 * 00266 20 CONTINUE 00267 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00268 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00269 ELSE 00270 ERRBND = ONE / EPS 00271 END IF 00272 30 CONTINUE 00273 RESLTS( 1 ) = ERRBND 00274 * 00275 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00276 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00277 * 00278 IFU = 0 00279 IF( UNIT ) 00280 $ IFU = 1 00281 DO 90 K = 1, NRHS 00282 DO 80 I = 1, N 00283 TMP = CABS1( B( I, K ) ) 00284 IF( UPPER ) THEN 00285 IF( .NOT.NOTRAN ) THEN 00286 DO 40 J = 1, I - IFU 00287 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) ) 00288 40 CONTINUE 00289 IF( UNIT ) 00290 $ TMP = TMP + CABS1( X( I, K ) ) 00291 ELSE 00292 IF( UNIT ) 00293 $ TMP = TMP + CABS1( X( I, K ) ) 00294 DO 50 J = I + IFU, N 00295 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) ) 00296 50 CONTINUE 00297 END IF 00298 ELSE 00299 IF( NOTRAN ) THEN 00300 DO 60 J = 1, I - IFU 00301 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) ) 00302 60 CONTINUE 00303 IF( UNIT ) 00304 $ TMP = TMP + CABS1( X( I, K ) ) 00305 ELSE 00306 IF( UNIT ) 00307 $ TMP = TMP + CABS1( X( I, K ) ) 00308 DO 70 J = I + IFU, N 00309 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) ) 00310 70 CONTINUE 00311 END IF 00312 END IF 00313 IF( I.EQ.1 ) THEN 00314 AXBI = TMP 00315 ELSE 00316 AXBI = MIN( AXBI, TMP ) 00317 END IF 00318 80 CONTINUE 00319 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00320 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00321 IF( K.EQ.1 ) THEN 00322 RESLTS( 2 ) = TMP 00323 ELSE 00324 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00325 END IF 00326 90 CONTINUE 00327 * 00328 RETURN 00329 * 00330 * End of CTRT05 00331 * 00332 END