1*> \brief \b DPOT05 2* 3* =========== DOCUMENTATION =========== 4* 5* Online html documentation available at 6* http://www.netlib.org/lapack/explore-html/ 7* 8* Definition: 9* =========== 10* 11* SUBROUTINE DPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 12* LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER UPLO 16* INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 17* .. 18* .. Array Arguments .. 19* DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 20* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 21* .. 22* 23* 24*> \par Purpose: 25* ============= 26*> 27*> \verbatim 28*> 29*> DPOT05 tests the error bounds from iterative refinement for the 30*> computed solution to a system of equations A*X = B, where A is a 31*> symmetric n by n matrix. 32*> 33*> RESLTS(1) = test of the error bound 34*> = norm(X - XACT) / ( norm(X) * FERR ) 35*> 36*> A large value is returned if this ratio is not less than one. 37*> 38*> RESLTS(2) = residual from the iterative refinement routine 39*> = the maximum of BERR / ( (n+1)*EPS + (*) ), where 40*> (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 41*> \endverbatim 42* 43* Arguments: 44* ========== 45* 46*> \param[in] UPLO 47*> \verbatim 48*> UPLO is CHARACTER*1 49*> Specifies whether the upper or lower triangular part of the 50*> symmetric matrix A is stored. 51*> = 'U': Upper triangular 52*> = 'L': Lower triangular 53*> \endverbatim 54*> 55*> \param[in] N 56*> \verbatim 57*> N is INTEGER 58*> The number of rows of the matrices X, B, and XACT, and the 59*> order of the matrix A. N >= 0. 60*> \endverbatim 61*> 62*> \param[in] NRHS 63*> \verbatim 64*> NRHS is INTEGER 65*> The number of columns of the matrices X, B, and XACT. 66*> NRHS >= 0. 67*> \endverbatim 68*> 69*> \param[in] A 70*> \verbatim 71*> A is DOUBLE PRECISION array, dimension (LDA,N) 72*> The symmetric matrix A. If UPLO = 'U', the leading n by n 73*> upper triangular part of A contains the upper triangular part 74*> of the matrix A, and the strictly lower triangular part of A 75*> is not referenced. If UPLO = 'L', the leading n by n lower 76*> triangular part of A contains the lower triangular part of 77*> the matrix A, and the strictly upper triangular part of A is 78*> not referenced. 79*> \endverbatim 80*> 81*> \param[in] LDA 82*> \verbatim 83*> LDA is INTEGER 84*> The leading dimension of the array A. LDA >= max(1,N). 85*> \endverbatim 86*> 87*> \param[in] B 88*> \verbatim 89*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 90*> The right hand side vectors for the system of linear 91*> equations. 92*> \endverbatim 93*> 94*> \param[in] LDB 95*> \verbatim 96*> LDB is INTEGER 97*> The leading dimension of the array B. LDB >= max(1,N). 98*> \endverbatim 99*> 100*> \param[in] X 101*> \verbatim 102*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 103*> The computed solution vectors. Each vector is stored as a 104*> column of the matrix X. 105*> \endverbatim 106*> 107*> \param[in] LDX 108*> \verbatim 109*> LDX is INTEGER 110*> The leading dimension of the array X. LDX >= max(1,N). 111*> \endverbatim 112*> 113*> \param[in] XACT 114*> \verbatim 115*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 116*> The exact solution vectors. Each vector is stored as a 117*> column of the matrix XACT. 118*> \endverbatim 119*> 120*> \param[in] LDXACT 121*> \verbatim 122*> LDXACT is INTEGER 123*> The leading dimension of the array XACT. LDXACT >= max(1,N). 124*> \endverbatim 125*> 126*> \param[in] FERR 127*> \verbatim 128*> FERR is DOUBLE PRECISION array, dimension (NRHS) 129*> The estimated forward error bounds for each solution vector 130*> X. If XTRUE is the true solution, FERR bounds the magnitude 131*> of the largest entry in (X - XTRUE) divided by the magnitude 132*> of the largest entry in X. 133*> \endverbatim 134*> 135*> \param[in] BERR 136*> \verbatim 137*> BERR is DOUBLE PRECISION array, dimension (NRHS) 138*> The componentwise relative backward error of each solution 139*> vector (i.e., the smallest relative change in any entry of A 140*> or B that makes X an exact solution). 141*> \endverbatim 142*> 143*> \param[out] RESLTS 144*> \verbatim 145*> RESLTS is DOUBLE PRECISION array, dimension (2) 146*> The maximum over the NRHS solution vectors of the ratios: 147*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 148*> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 149*> \endverbatim 150* 151* Authors: 152* ======== 153* 154*> \author Univ. of Tennessee 155*> \author Univ. of California Berkeley 156*> \author Univ. of Colorado Denver 157*> \author NAG Ltd. 158* 159*> \date December 2016 160* 161*> \ingroup double_lin 162* 163* ===================================================================== 164 SUBROUTINE DPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 165 $ LDXACT, FERR, BERR, RESLTS ) 166* 167* -- LAPACK test routine (version 3.7.0) -- 168* -- LAPACK is a software package provided by Univ. of Tennessee, -- 169* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 170* December 2016 171* 172* .. Scalar Arguments .. 173 CHARACTER UPLO 174 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 175* .. 176* .. Array Arguments .. 177 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 178 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 179* .. 180* 181* ===================================================================== 182* 183* .. Parameters .. 184 DOUBLE PRECISION ZERO, ONE 185 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 186* .. 187* .. Local Scalars .. 188 LOGICAL UPPER 189 INTEGER I, IMAX, J, K 190 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 191* .. 192* .. External Functions .. 193 LOGICAL LSAME 194 INTEGER IDAMAX 195 DOUBLE PRECISION DLAMCH 196 EXTERNAL LSAME, IDAMAX, DLAMCH 197* .. 198* .. Intrinsic Functions .. 199 INTRINSIC ABS, MAX, MIN 200* .. 201* .. Executable Statements .. 202* 203* Quick exit if N = 0 or NRHS = 0. 204* 205 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 206 RESLTS( 1 ) = ZERO 207 RESLTS( 2 ) = ZERO 208 RETURN 209 END IF 210* 211 EPS = DLAMCH( 'Epsilon' ) 212 UNFL = DLAMCH( 'Safe minimum' ) 213 OVFL = ONE / UNFL 214 UPPER = LSAME( UPLO, 'U' ) 215* 216* Test 1: Compute the maximum of 217* norm(X - XACT) / ( norm(X) * FERR ) 218* over all the vectors X and XACT using the infinity-norm. 219* 220 ERRBND = ZERO 221 DO 30 J = 1, NRHS 222 IMAX = IDAMAX( N, X( 1, J ), 1 ) 223 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 224 DIFF = ZERO 225 DO 10 I = 1, N 226 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 227 10 CONTINUE 228* 229 IF( XNORM.GT.ONE ) THEN 230 GO TO 20 231 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 232 GO TO 20 233 ELSE 234 ERRBND = ONE / EPS 235 GO TO 30 236 END IF 237* 238 20 CONTINUE 239 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 240 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 241 ELSE 242 ERRBND = ONE / EPS 243 END IF 244 30 CONTINUE 245 RESLTS( 1 ) = ERRBND 246* 247* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 248* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 249* 250 DO 90 K = 1, NRHS 251 DO 80 I = 1, N 252 TMP = ABS( B( I, K ) ) 253 IF( UPPER ) THEN 254 DO 40 J = 1, I 255 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 256 40 CONTINUE 257 DO 50 J = I + 1, N 258 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 259 50 CONTINUE 260 ELSE 261 DO 60 J = 1, I - 1 262 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 263 60 CONTINUE 264 DO 70 J = I, N 265 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 266 70 CONTINUE 267 END IF 268 IF( I.EQ.1 ) THEN 269 AXBI = TMP 270 ELSE 271 AXBI = MIN( AXBI, TMP ) 272 END IF 273 80 CONTINUE 274 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 275 $ MAX( AXBI, ( N+1 )*UNFL ) ) 276 IF( K.EQ.1 ) THEN 277 RESLTS( 2 ) = TMP 278 ELSE 279 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 280 END IF 281 90 CONTINUE 282* 283 RETURN 284* 285* End of DPOT05 286* 287 END 288