1*> \brief \b DGTT05 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 DGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX, 12* XACT, LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER TRANS 16* INTEGER LDB, LDX, LDXACT, N, NRHS 17* .. 18* .. Array Arguments .. 19* DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DL( * ), 20* $ DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ), 21* $ XACT( LDXACT, * ) 22* .. 23* 24* 25*> \par Purpose: 26* ============= 27*> 28*> \verbatim 29*> 30*> DGTT05 tests the error bounds from iterative refinement for the 31*> computed solution to a system of equations A*X = B, where A is a 32*> general tridiagonal matrix of order n and op(A) = A or A**T, 33*> depending on TRANS. 34*> 35*> RESLTS(1) = test of the error bound 36*> = norm(X - XACT) / ( norm(X) * FERR ) 37*> 38*> A large value is returned if this ratio is not less than one. 39*> 40*> RESLTS(2) = residual from the iterative refinement routine 41*> = the maximum of BERR / ( NZ*EPS + (*) ), where 42*> (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 43*> and NZ = max. number of nonzeros in any row of A, plus 1 44*> \endverbatim 45* 46* Arguments: 47* ========== 48* 49*> \param[in] TRANS 50*> \verbatim 51*> TRANS is CHARACTER*1 52*> Specifies the form of the system of equations. 53*> = 'N': A * X = B (No transpose) 54*> = 'T': A**T * X = B (Transpose) 55*> = 'C': A**H * X = B (Conjugate transpose = Transpose) 56*> \endverbatim 57*> 58*> \param[in] N 59*> \verbatim 60*> N is INTEGER 61*> The number of rows of the matrices X and XACT. N >= 0. 62*> \endverbatim 63*> 64*> \param[in] NRHS 65*> \verbatim 66*> NRHS is INTEGER 67*> The number of columns of the matrices X and XACT. NRHS >= 0. 68*> \endverbatim 69*> 70*> \param[in] DL 71*> \verbatim 72*> DL is DOUBLE PRECISION array, dimension (N-1) 73*> The (n-1) sub-diagonal elements of A. 74*> \endverbatim 75*> 76*> \param[in] D 77*> \verbatim 78*> D is DOUBLE PRECISION array, dimension (N) 79*> The diagonal elements of A. 80*> \endverbatim 81*> 82*> \param[in] DU 83*> \verbatim 84*> DU is DOUBLE PRECISION array, dimension (N-1) 85*> The (n-1) super-diagonal elements of A. 86*> \endverbatim 87*> 88*> \param[in] B 89*> \verbatim 90*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 91*> The right hand side vectors for the system of linear 92*> equations. 93*> \endverbatim 94*> 95*> \param[in] LDB 96*> \verbatim 97*> LDB is INTEGER 98*> The leading dimension of the array B. LDB >= max(1,N). 99*> \endverbatim 100*> 101*> \param[in] X 102*> \verbatim 103*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 104*> The computed solution vectors. Each vector is stored as a 105*> column of the matrix X. 106*> \endverbatim 107*> 108*> \param[in] LDX 109*> \verbatim 110*> LDX is INTEGER 111*> The leading dimension of the array X. LDX >= max(1,N). 112*> \endverbatim 113*> 114*> \param[in] XACT 115*> \verbatim 116*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 117*> The exact solution vectors. Each vector is stored as a 118*> column of the matrix XACT. 119*> \endverbatim 120*> 121*> \param[in] LDXACT 122*> \verbatim 123*> LDXACT is INTEGER 124*> The leading dimension of the array XACT. LDXACT >= max(1,N). 125*> \endverbatim 126*> 127*> \param[in] FERR 128*> \verbatim 129*> FERR is DOUBLE PRECISION array, dimension (NRHS) 130*> The estimated forward error bounds for each solution vector 131*> X. If XTRUE is the true solution, FERR bounds the magnitude 132*> of the largest entry in (X - XTRUE) divided by the magnitude 133*> of the largest entry in X. 134*> \endverbatim 135*> 136*> \param[in] BERR 137*> \verbatim 138*> BERR is DOUBLE PRECISION 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 DOUBLE PRECISION 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 / ( NZ*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 double_lin 161* 162* ===================================================================== 163 SUBROUTINE DGTT05( TRANS, N, NRHS, DL, D, DU, B, LDB, X, LDX, 164 $ XACT, LDXACT, FERR, 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 INTEGER LDB, LDX, LDXACT, N, NRHS 173* .. 174* .. Array Arguments .. 175 DOUBLE PRECISION B( LDB, * ), BERR( * ), D( * ), DL( * ), 176 $ DU( * ), FERR( * ), RESLTS( * ), X( LDX, * ), 177 $ XACT( LDXACT, * ) 178* .. 179* 180* ===================================================================== 181* 182* .. Parameters .. 183 DOUBLE PRECISION ZERO, ONE 184 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 185* .. 186* .. Local Scalars .. 187 LOGICAL NOTRAN 188 INTEGER I, IMAX, J, K, NZ 189 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 190* .. 191* .. External Functions .. 192 LOGICAL LSAME 193 INTEGER IDAMAX 194 DOUBLE PRECISION DLAMCH 195 EXTERNAL LSAME, IDAMAX, DLAMCH 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 = DLAMCH( 'Epsilon' ) 211 UNFL = DLAMCH( 'Safe minimum' ) 212 OVFL = ONE / UNFL 213 NOTRAN = LSAME( TRANS, 'N' ) 214 NZ = 4 215* 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 / ( NZ*EPS + (*) ), where 248* (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 249* 250 DO 60 K = 1, NRHS 251 IF( NOTRAN ) THEN 252 IF( N.EQ.1 ) THEN 253 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) 254 ELSE 255 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) + 256 $ ABS( DU( 1 )*X( 2, K ) ) 257 DO 40 I = 2, N - 1 258 TMP = ABS( B( I, K ) ) + ABS( DL( I-1 )*X( I-1, K ) ) 259 $ + ABS( D( I )*X( I, K ) ) + 260 $ ABS( DU( I )*X( I+1, K ) ) 261 AXBI = MIN( AXBI, TMP ) 262 40 CONTINUE 263 TMP = ABS( B( N, K ) ) + ABS( DL( N-1 )*X( N-1, K ) ) + 264 $ ABS( D( N )*X( N, K ) ) 265 AXBI = MIN( AXBI, TMP ) 266 END IF 267 ELSE 268 IF( N.EQ.1 ) THEN 269 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) 270 ELSE 271 AXBI = ABS( B( 1, K ) ) + ABS( D( 1 )*X( 1, K ) ) + 272 $ ABS( DL( 1 )*X( 2, K ) ) 273 DO 50 I = 2, N - 1 274 TMP = ABS( B( I, K ) ) + ABS( DU( I-1 )*X( I-1, K ) ) 275 $ + ABS( D( I )*X( I, K ) ) + 276 $ ABS( DL( I )*X( I+1, K ) ) 277 AXBI = MIN( AXBI, TMP ) 278 50 CONTINUE 279 TMP = ABS( B( N, K ) ) + ABS( DU( N-1 )*X( N-1, K ) ) + 280 $ ABS( D( N )*X( N, K ) ) 281 AXBI = MIN( AXBI, TMP ) 282 END IF 283 END IF 284 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 285 IF( K.EQ.1 ) THEN 286 RESLTS( 2 ) = TMP 287 ELSE 288 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 289 END IF 290 60 CONTINUE 291* 292 RETURN 293* 294* End of DGTT05 295* 296 END 297