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LAPACK
3.4.1
LAPACK: Linear Algebra PACKage
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00001 *> \brief \b DPPT05 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 DPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT, 00012 * LDXACT, FERR, BERR, RESLTS ) 00013 * 00014 * .. Scalar Arguments .. 00015 * CHARACTER UPLO 00016 * INTEGER LDB, LDX, LDXACT, N, NRHS 00017 * .. 00018 * .. Array Arguments .. 00019 * DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 00020 * $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 00021 * .. 00022 * 00023 * 00024 *> \par Purpose: 00025 * ============= 00026 *> 00027 *> \verbatim 00028 *> 00029 *> DPPT05 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 matrix in packed storage format. 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] AP 00070 *> \verbatim 00071 *> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) 00072 *> The upper or lower triangle of the symmetric matrix A, packed 00073 *> columnwise in a linear array. The j-th column of A is stored 00074 *> in the array AP as follows: 00075 *> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 00076 *> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 00077 *> \endverbatim 00078 *> 00079 *> \param[in] B 00080 *> \verbatim 00081 *> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 00082 *> The right hand side vectors for the system of linear 00083 *> equations. 00084 *> \endverbatim 00085 *> 00086 *> \param[in] LDB 00087 *> \verbatim 00088 *> LDB is INTEGER 00089 *> The leading dimension of the array B. LDB >= max(1,N). 00090 *> \endverbatim 00091 *> 00092 *> \param[in] X 00093 *> \verbatim 00094 *> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 00095 *> The computed solution vectors. Each vector is stored as a 00096 *> column of the matrix X. 00097 *> \endverbatim 00098 *> 00099 *> \param[in] LDX 00100 *> \verbatim 00101 *> LDX is INTEGER 00102 *> The leading dimension of the array X. LDX >= max(1,N). 00103 *> \endverbatim 00104 *> 00105 *> \param[in] XACT 00106 *> \verbatim 00107 *> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 00108 *> The exact solution vectors. Each vector is stored as a 00109 *> column of the matrix XACT. 00110 *> \endverbatim 00111 *> 00112 *> \param[in] LDXACT 00113 *> \verbatim 00114 *> LDXACT is INTEGER 00115 *> The leading dimension of the array XACT. LDXACT >= max(1,N). 00116 *> \endverbatim 00117 *> 00118 *> \param[in] FERR 00119 *> \verbatim 00120 *> FERR is DOUBLE PRECISION array, dimension (NRHS) 00121 *> The estimated forward error bounds for each solution vector 00122 *> X. If XTRUE is the true solution, FERR bounds the magnitude 00123 *> of the largest entry in (X - XTRUE) divided by the magnitude 00124 *> of the largest entry in X. 00125 *> \endverbatim 00126 *> 00127 *> \param[in] BERR 00128 *> \verbatim 00129 *> BERR is DOUBLE PRECISION array, dimension (NRHS) 00130 *> The componentwise relative backward error of each solution 00131 *> vector (i.e., the smallest relative change in any entry of A 00132 *> or B that makes X an exact solution). 00133 *> \endverbatim 00134 *> 00135 *> \param[out] RESLTS 00136 *> \verbatim 00137 *> RESLTS is DOUBLE PRECISION array, dimension (2) 00138 *> The maximum over the NRHS solution vectors of the ratios: 00139 *> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 00140 *> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 00141 *> \endverbatim 00142 * 00143 * Authors: 00144 * ======== 00145 * 00146 *> \author Univ. of Tennessee 00147 *> \author Univ. of California Berkeley 00148 *> \author Univ. of Colorado Denver 00149 *> \author NAG Ltd. 00150 * 00151 *> \date November 2011 00152 * 00153 *> \ingroup double_lin 00154 * 00155 * ===================================================================== 00156 SUBROUTINE DPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT, 00157 $ LDXACT, FERR, BERR, RESLTS ) 00158 * 00159 * -- LAPACK test routine (version 3.4.0) -- 00160 * -- LAPACK is a software package provided by Univ. of Tennessee, -- 00161 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 00162 * November 2011 00163 * 00164 * .. Scalar Arguments .. 00165 CHARACTER UPLO 00166 INTEGER LDB, LDX, LDXACT, N, NRHS 00167 * .. 00168 * .. Array Arguments .. 00169 DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 00170 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 00171 * .. 00172 * 00173 * ===================================================================== 00174 * 00175 * .. Parameters .. 00176 DOUBLE PRECISION ZERO, ONE 00177 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 00178 * .. 00179 * .. Local Scalars .. 00180 LOGICAL UPPER 00181 INTEGER I, IMAX, J, JC, K 00182 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 00183 * .. 00184 * .. External Functions .. 00185 LOGICAL LSAME 00186 INTEGER IDAMAX 00187 DOUBLE PRECISION DLAMCH 00188 EXTERNAL LSAME, IDAMAX, DLAMCH 00189 * .. 00190 * .. Intrinsic Functions .. 00191 INTRINSIC ABS, MAX, MIN 00192 * .. 00193 * .. Executable Statements .. 00194 * 00195 * Quick exit if N = 0 or NRHS = 0. 00196 * 00197 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 00198 RESLTS( 1 ) = ZERO 00199 RESLTS( 2 ) = ZERO 00200 RETURN 00201 END IF 00202 * 00203 EPS = DLAMCH( 'Epsilon' ) 00204 UNFL = DLAMCH( 'Safe minimum' ) 00205 OVFL = ONE / UNFL 00206 UPPER = LSAME( UPLO, 'U' ) 00207 * 00208 * Test 1: Compute the maximum of 00209 * norm(X - XACT) / ( norm(X) * FERR ) 00210 * over all the vectors X and XACT using the infinity-norm. 00211 * 00212 ERRBND = ZERO 00213 DO 30 J = 1, NRHS 00214 IMAX = IDAMAX( N, X( 1, J ), 1 ) 00215 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 00216 DIFF = ZERO 00217 DO 10 I = 1, N 00218 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 00219 10 CONTINUE 00220 * 00221 IF( XNORM.GT.ONE ) THEN 00222 GO TO 20 00223 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 00224 GO TO 20 00225 ELSE 00226 ERRBND = ONE / EPS 00227 GO TO 30 00228 END IF 00229 * 00230 20 CONTINUE 00231 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 00232 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 00233 ELSE 00234 ERRBND = ONE / EPS 00235 END IF 00236 30 CONTINUE 00237 RESLTS( 1 ) = ERRBND 00238 * 00239 * Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 00240 * (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 00241 * 00242 DO 90 K = 1, NRHS 00243 DO 80 I = 1, N 00244 TMP = ABS( B( I, K ) ) 00245 IF( UPPER ) THEN 00246 JC = ( ( I-1 )*I ) / 2 00247 DO 40 J = 1, I 00248 TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) ) 00249 40 CONTINUE 00250 JC = JC + I 00251 DO 50 J = I + 1, N 00252 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 00253 JC = JC + J 00254 50 CONTINUE 00255 ELSE 00256 JC = I 00257 DO 60 J = 1, I - 1 00258 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 00259 JC = JC + N - J 00260 60 CONTINUE 00261 DO 70 J = I, N 00262 TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) ) 00263 70 CONTINUE 00264 END IF 00265 IF( I.EQ.1 ) THEN 00266 AXBI = TMP 00267 ELSE 00268 AXBI = MIN( AXBI, TMP ) 00269 END IF 00270 80 CONTINUE 00271 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 00272 $ MAX( AXBI, ( N+1 )*UNFL ) ) 00273 IF( K.EQ.1 ) THEN 00274 RESLTS( 2 ) = TMP 00275 ELSE 00276 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 00277 END IF 00278 90 CONTINUE 00279 * 00280 RETURN 00281 * 00282 * End of DPPT05 00283 * 00284 END