1*> \brief \b SPOT05 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 SPOT05( 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* REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 20* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 21* .. 22* 23* 24*> \par Purpose: 25* ============= 26*> 27*> \verbatim 28*> 29*> SPOT05 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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 REAL 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*> \ingroup single_lin 160* 161* ===================================================================== 162 SUBROUTINE SPOT05( UPLO, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 163 $ LDXACT, FERR, BERR, RESLTS ) 164* 165* -- LAPACK test routine -- 166* -- LAPACK is a software package provided by Univ. of Tennessee, -- 167* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 168* 169* .. Scalar Arguments .. 170 CHARACTER UPLO 171 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 172* .. 173* .. Array Arguments .. 174 REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 175 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 176* .. 177* 178* ===================================================================== 179* 180* .. Parameters .. 181 REAL ZERO, ONE 182 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 183* .. 184* .. Local Scalars .. 185 LOGICAL UPPER 186 INTEGER I, IMAX, J, K 187 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 188* .. 189* .. External Functions .. 190 LOGICAL LSAME 191 INTEGER ISAMAX 192 REAL SLAMCH 193 EXTERNAL LSAME, ISAMAX, SLAMCH 194* .. 195* .. Intrinsic Functions .. 196 INTRINSIC ABS, MAX, MIN 197* .. 198* .. Executable Statements .. 199* 200* Quick exit if N = 0 or NRHS = 0. 201* 202 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 203 RESLTS( 1 ) = ZERO 204 RESLTS( 2 ) = ZERO 205 RETURN 206 END IF 207* 208 EPS = SLAMCH( 'Epsilon' ) 209 UNFL = SLAMCH( 'Safe minimum' ) 210 OVFL = ONE / UNFL 211 UPPER = LSAME( UPLO, 'U' ) 212* 213* Test 1: Compute the maximum of 214* norm(X - XACT) / ( norm(X) * FERR ) 215* over all the vectors X and XACT using the infinity-norm. 216* 217 ERRBND = ZERO 218 DO 30 J = 1, NRHS 219 IMAX = ISAMAX( N, X( 1, J ), 1 ) 220 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 221 DIFF = ZERO 222 DO 10 I = 1, N 223 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 224 10 CONTINUE 225* 226 IF( XNORM.GT.ONE ) THEN 227 GO TO 20 228 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 229 GO TO 20 230 ELSE 231 ERRBND = ONE / EPS 232 GO TO 30 233 END IF 234* 235 20 CONTINUE 236 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 237 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 238 ELSE 239 ERRBND = ONE / EPS 240 END IF 241 30 CONTINUE 242 RESLTS( 1 ) = ERRBND 243* 244* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 245* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 246* 247 DO 90 K = 1, NRHS 248 DO 80 I = 1, N 249 TMP = ABS( B( I, K ) ) 250 IF( UPPER ) THEN 251 DO 40 J = 1, I 252 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 253 40 CONTINUE 254 DO 50 J = I + 1, N 255 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 256 50 CONTINUE 257 ELSE 258 DO 60 J = 1, I - 1 259 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 260 60 CONTINUE 261 DO 70 J = I, N 262 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 263 70 CONTINUE 264 END IF 265 IF( I.EQ.1 ) THEN 266 AXBI = TMP 267 ELSE 268 AXBI = MIN( AXBI, TMP ) 269 END IF 270 80 CONTINUE 271 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 272 $ MAX( AXBI, ( N+1 )*UNFL ) ) 273 IF( K.EQ.1 ) THEN 274 RESLTS( 2 ) = TMP 275 ELSE 276 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 277 END IF 278 90 CONTINUE 279* 280 RETURN 281* 282* End of SPOT05 283* 284 END 285