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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b CPOT05 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 CPOT05( 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 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 *> CPOT05 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 *> Hermitian 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 upper or lower triangular part of the 00051 *> Hermitian matrix A is stored. 00052 *> = 'U': Upper triangular 00053 *> = 'L': Lower triangular 00054 *> \endverbatim 00055 *> 00056 *> \param[in] N 00057 *> \verbatim 00058 *> N is INTEGER 00059 *> The number of rows of the matrices X, B, and XACT, and the 00060 *> order of the matrix A. N >= 0. 00061 *> \endverbatim 00062 *> 00063 *> \param[in] NRHS 00064 *> \verbatim 00065 *> NRHS is INTEGER 00066 *> The number of columns of the matrices X, B, and XACT. 00067 *> NRHS >= 0. 00068 *> \endverbatim 00069 *> 00070 *> \param[in] A 00071 *> \verbatim 00072 *> A is COMPLEX array, dimension (LDA,N) 00073 *> The Hermitian matrix A. If UPLO = 'U', the leading n by n 00074 *> upper triangular part of A contains the upper triangular part 00075 *> of the matrix A, and the strictly lower triangular part of A 00076 *> is not referenced. If UPLO = 'L', the leading n by n lower 00077 *> triangular part of A contains the lower triangular part of 00078 *> the matrix A, and the strictly upper triangular part of A is 00079 *> not referenced. 00080 *> \endverbatim 00081 *> 00082 *> \param[in] LDA 00083 *> \verbatim 00084 *> LDA is INTEGER 00085 *> The leading dimension of the array A. LDA >= max(1,N). 00086 *> \endverbatim 00087 *> 00088 *> \param[in] B 00089 *> \verbatim 00090 *> B is COMPLEX 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 COMPLEX 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 COMPLEX 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 REAL 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 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 complex_lin 00163 * 00164 * ===================================================================== 00165 SUBROUTINE CPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 00166 $ 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 UPLO 00175 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 00176 * .. 00177 * .. Array Arguments .. 00178 REAL BERR( * ), FERR( * ), RESLTS( * ) 00179 COMPLEX A( LDA, * ), B( LDB, * ), X( LDX, * ), 00180 $ 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 UPPER 00191 INTEGER I, IMAX, J, K 00192 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00193 COMPLEX ZDUM 00194 * .. 00195 * .. External Functions .. 00196 LOGICAL LSAME 00197 INTEGER ICAMAX 00198 REAL SLAMCH 00199 EXTERNAL LSAME, ICAMAX, SLAMCH 00200 * .. 00201 * .. Intrinsic Functions .. 00202 INTRINSIC ABS, AIMAG, MAX, MIN, REAL 00203 * .. 00204 * .. Statement Functions .. 00205 REAL CABS1 00206 * .. 00207 * .. Statement Function definitions .. 00208 CABS1( ZDUM ) = ABS( REAL( ZDUM ) ) + ABS( AIMAG( ZDUM ) ) 00209 * .. 00210 * .. Executable Statements .. 00211 * 00212 * Quick exit if N = 0 or NRHS = 0. 00213 * 00214 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00215 RESLTS( 1 ) = ZERO 00216 RESLTS( 2 ) = ZERO 00217 RETURN 00218 END IF 00219 * 00220 EPS = SLAMCH( 'Epsilon' ) 00221 UNFL = SLAMCH( 'Safe minimum' ) 00222 OVFL = ONE / UNFL 00223 UPPER = LSAME( UPLO, 'U' ) 00224 * 00225 * Test 1: Compute the maximum of 00226 * norm(X - XACT) / ( norm(X) * FERR ) 00227 * over all the vectors X and XACT using the infinity-norm. 00228 * 00229 ERRBND = ZERO 00230 DO 30 J = 1, NRHS 00231 IMAX = ICAMAX( N, X( 1, J ), 1 ) 00232 XNORM = MAX( CABS1( X( IMAX, J ) ), UNFL ) 00233 DIFF = ZERO 00234 DO 10 I = 1, N 00235 DIFF = MAX( DIFF, CABS1( X( I, J )-XACT( I, J ) ) ) 00236 10 CONTINUE 00237 * 00238 IF( XNORM.GT.ONE ) THEN 00239 GO TO 20 00240 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00241 GO TO 20 00242 ELSE 00243 ERRBND = ONE / EPS 00244 GO TO 30 00245 END IF 00246 * 00247 20 CONTINUE 00248 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00249 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00250 ELSE 00251 ERRBND = ONE / EPS 00252 END IF 00253 30 CONTINUE 00254 RESLTS( 1 ) = ERRBND 00255 * 00256 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00257 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00258 * 00259 DO 90 K = 1, NRHS 00260 DO 80 I = 1, N 00261 TMP = CABS1( B( I, K ) ) 00262 IF( UPPER ) THEN 00263 DO 40 J = 1, I - 1 00264 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) ) 00265 40 CONTINUE 00266 TMP = TMP + ABS( REAL( A( I, I ) ) )*CABS1( X( I, K ) ) 00267 DO 50 J = I + 1, N 00268 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) ) 00269 50 CONTINUE 00270 ELSE 00271 DO 60 J = 1, I - 1 00272 TMP = TMP + CABS1( A( I, J ) )*CABS1( X( J, K ) ) 00273 60 CONTINUE 00274 TMP = TMP + ABS( REAL( A( I, I ) ) )*CABS1( X( I, K ) ) 00275 DO 70 J = I + 1, N 00276 TMP = TMP + CABS1( A( J, I ) )*CABS1( X( J, K ) ) 00277 70 CONTINUE 00278 END IF 00279 IF( I.EQ.1 ) THEN 00280 AXBI = TMP 00281 ELSE 00282 AXBI = MIN( AXBI, TMP ) 00283 END IF 00284 80 CONTINUE 00285 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00286 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00287 IF( K.EQ.1 ) THEN 00288 RESLTS( 2 ) = TMP 00289 ELSE 00290 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00291 END IF 00292 90 CONTINUE 00293 * 00294 RETURN 00295 * 00296 * End of CPOT05 00297 * 00298 END