1*> \brief \b SGET07 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 SGET07( TRANS, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 12* LDXACT, FERR, CHKFERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER TRANS 16* LOGICAL CHKFERR 17* INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 18* .. 19* .. Array Arguments .. 20* REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 21* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 22* .. 23* 24* 25*> \par Purpose: 26* ============= 27*> 28*> \verbatim 29*> 30*> SGET07 tests the error bounds from iterative refinement for the 31*> computed solution to a system of equations op(A)*X = B, where A is a 32*> general n by n matrix and op(A) = A or A**T, depending on TRANS. 33*> 34*> RESLTS(1) = test of the error bound 35*> = norm(X - XACT) / ( norm(X) * FERR ) 36*> 37*> A large value is returned if this ratio is not less than one. 38*> 39*> RESLTS(2) = residual from the iterative refinement routine 40*> = the maximum of BERR / ( (n+1)*EPS + (*) ), where 41*> (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 42*> \endverbatim 43* 44* Arguments: 45* ========== 46* 47*> \param[in] TRANS 48*> \verbatim 49*> TRANS is CHARACTER*1 50*> Specifies the form of the system of equations. 51*> = 'N': A * X = B (No transpose) 52*> = 'T': A**T * X = B (Transpose) 53*> = 'C': A**H * X = B (Conjugate transpose = Transpose) 54*> \endverbatim 55*> 56*> \param[in] N 57*> \verbatim 58*> N is INTEGER 59*> The number of rows of the matrices X and XACT. N >= 0. 60*> \endverbatim 61*> 62*> \param[in] NRHS 63*> \verbatim 64*> NRHS is INTEGER 65*> The number of columns of the matrices X and XACT. NRHS >= 0. 66*> \endverbatim 67*> 68*> \param[in] A 69*> \verbatim 70*> A is REAL array, dimension (LDA,N) 71*> The original n by n matrix A. 72*> \endverbatim 73*> 74*> \param[in] LDA 75*> \verbatim 76*> LDA is INTEGER 77*> The leading dimension of the array A. LDA >= max(1,N). 78*> \endverbatim 79*> 80*> \param[in] B 81*> \verbatim 82*> B is REAL array, dimension (LDB,NRHS) 83*> The right hand side vectors for the system of linear 84*> equations. 85*> \endverbatim 86*> 87*> \param[in] LDB 88*> \verbatim 89*> LDB is INTEGER 90*> The leading dimension of the array B. LDB >= max(1,N). 91*> \endverbatim 92*> 93*> \param[in] X 94*> \verbatim 95*> X is REAL array, dimension (LDX,NRHS) 96*> The computed solution vectors. Each vector is stored as a 97*> column of the matrix X. 98*> \endverbatim 99*> 100*> \param[in] LDX 101*> \verbatim 102*> LDX is INTEGER 103*> The leading dimension of the array X. LDX >= max(1,N). 104*> \endverbatim 105*> 106*> \param[in] XACT 107*> \verbatim 108*> XACT is REAL array, dimension (LDX,NRHS) 109*> The exact solution vectors. Each vector is stored as a 110*> column of the matrix XACT. 111*> \endverbatim 112*> 113*> \param[in] LDXACT 114*> \verbatim 115*> LDXACT is INTEGER 116*> The leading dimension of the array XACT. LDXACT >= max(1,N). 117*> \endverbatim 118*> 119*> \param[in] FERR 120*> \verbatim 121*> FERR is REAL array, dimension (NRHS) 122*> The estimated forward error bounds for each solution vector 123*> X. If XTRUE is the true solution, FERR bounds the magnitude 124*> of the largest entry in (X - XTRUE) divided by the magnitude 125*> of the largest entry in X. 126*> \endverbatim 127*> 128*> \param[in] CHKFERR 129*> \verbatim 130*> CHKFERR is LOGICAL 131*> Set to .TRUE. to check FERR, .FALSE. not to check FERR. 132*> When the test system is ill-conditioned, the "true" 133*> solution in XACT may be incorrect. 134*> \endverbatim 135*> 136*> \param[in] BERR 137*> \verbatim 138*> BERR is REAL array, dimension (NRHS) 139*> The componentwise relative backward error of each solution 140*> vector (i.e., the smallest relative change in any entry of A 141*> or B that makes X an exact solution). 142*> \endverbatim 143*> 144*> \param[out] RESLTS 145*> \verbatim 146*> RESLTS is REAL array, dimension (2) 147*> The maximum over the NRHS solution vectors of the ratios: 148*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 149*> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 150*> \endverbatim 151* 152* Authors: 153* ======== 154* 155*> \author Univ. of Tennessee 156*> \author Univ. of California Berkeley 157*> \author Univ. of Colorado Denver 158*> \author NAG Ltd. 159* 160*> \ingroup single_lin 161* 162* ===================================================================== 163 SUBROUTINE SGET07( TRANS, N, NRHS, A, LDA, B, LDB, X, LDX, XACT, 164 $ LDXACT, FERR, CHKFERR, BERR, RESLTS ) 165* 166* -- LAPACK test routine -- 167* -- LAPACK is a software package provided by Univ. of Tennessee, -- 168* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 169* 170* .. Scalar Arguments .. 171 CHARACTER TRANS 172 LOGICAL CHKFERR 173 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 174* .. 175* .. Array Arguments .. 176 REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 177 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 178* .. 179* 180* ===================================================================== 181* 182* .. Parameters .. 183 REAL ZERO, ONE 184 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 185* .. 186* .. Local Scalars .. 187 LOGICAL NOTRAN 188 INTEGER I, IMAX, J, K 189 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 190* .. 191* .. External Functions .. 192 LOGICAL LSAME 193 INTEGER ISAMAX 194 REAL SLAMCH 195 EXTERNAL LSAME, ISAMAX, SLAMCH 196* .. 197* .. Intrinsic Functions .. 198 INTRINSIC ABS, MAX, MIN 199* .. 200* .. Executable Statements .. 201* 202* Quick exit if N = 0 or NRHS = 0. 203* 204 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 205 RESLTS( 1 ) = ZERO 206 RESLTS( 2 ) = ZERO 207 RETURN 208 END IF 209* 210 EPS = SLAMCH( 'Epsilon' ) 211 UNFL = SLAMCH( 'Safe minimum' ) 212 OVFL = ONE / UNFL 213 NOTRAN = LSAME( TRANS, 'N' ) 214* 215* Test 1: Compute the maximum of 216* norm(X - XACT) / ( norm(X) * FERR ) 217* over all the vectors X and XACT using the infinity-norm. 218* 219 ERRBND = ZERO 220 IF( CHKFERR ) THEN 221 DO 30 J = 1, NRHS 222 IMAX = ISAMAX( 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 END IF 246 RESLTS( 1 ) = ERRBND 247* 248* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 249* (*) = (n+1)*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 250* 251 DO 70 K = 1, NRHS 252 DO 60 I = 1, N 253 TMP = ABS( B( I, K ) ) 254 IF( NOTRAN ) THEN 255 DO 40 J = 1, N 256 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 257 40 CONTINUE 258 ELSE 259 DO 50 J = 1, N 260 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 261 50 CONTINUE 262 END IF 263 IF( I.EQ.1 ) THEN 264 AXBI = TMP 265 ELSE 266 AXBI = MIN( AXBI, TMP ) 267 END IF 268 60 CONTINUE 269 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 270 $ MAX( AXBI, ( N+1 )*UNFL ) ) 271 IF( K.EQ.1 ) THEN 272 RESLTS( 2 ) = TMP 273 ELSE 274 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 275 END IF 276 70 CONTINUE 277* 278 RETURN 279* 280* End of SGET07 281* 282 END 283