1*> \brief \b DTPT05 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 DTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 12* XACT, LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER DIAG, TRANS, UPLO 16* INTEGER LDB, LDX, LDXACT, N, NRHS 17* .. 18* .. Array Arguments .. 19* DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 20* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 21* .. 22* 23* 24*> \par Purpose: 25* ============= 26*> 27*> \verbatim 28*> 29*> DTPT05 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*> triangular matrix in packed storage format. 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 matrix A is upper or lower triangular. 50*> = 'U': Upper triangular 51*> = 'L': Lower triangular 52*> \endverbatim 53*> 54*> \param[in] TRANS 55*> \verbatim 56*> TRANS is CHARACTER*1 57*> Specifies the form of the system of equations. 58*> = 'N': A * X = B (No transpose) 59*> = 'T': A'* X = B (Transpose) 60*> = 'C': A'* X = B (Conjugate transpose = Transpose) 61*> \endverbatim 62*> 63*> \param[in] DIAG 64*> \verbatim 65*> DIAG is CHARACTER*1 66*> Specifies whether or not the matrix A is unit triangular. 67*> = 'N': Non-unit triangular 68*> = 'U': Unit triangular 69*> \endverbatim 70*> 71*> \param[in] N 72*> \verbatim 73*> N is INTEGER 74*> The number of rows of the matrices X, B, and XACT, and the 75*> order of the matrix A. N >= 0. 76*> \endverbatim 77*> 78*> \param[in] NRHS 79*> \verbatim 80*> NRHS is INTEGER 81*> The number of columns of the matrices X, B, and XACT. 82*> NRHS >= 0. 83*> \endverbatim 84*> 85*> \param[in] AP 86*> \verbatim 87*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) 88*> The upper or lower triangular matrix A, packed columnwise in 89*> a linear array. The j-th column of A is stored in the array 90*> AP as follows: 91*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 92*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 93*> If DIAG = 'U', the diagonal elements of A are not referenced 94*> and are assumed to be 1. 95*> \endverbatim 96*> 97*> \param[in] B 98*> \verbatim 99*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 100*> The right hand side vectors for the system of linear 101*> equations. 102*> \endverbatim 103*> 104*> \param[in] LDB 105*> \verbatim 106*> LDB is INTEGER 107*> The leading dimension of the array B. LDB >= max(1,N). 108*> \endverbatim 109*> 110*> \param[in] X 111*> \verbatim 112*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 113*> The computed solution vectors. Each vector is stored as a 114*> column of the matrix X. 115*> \endverbatim 116*> 117*> \param[in] LDX 118*> \verbatim 119*> LDX is INTEGER 120*> The leading dimension of the array X. LDX >= max(1,N). 121*> \endverbatim 122*> 123*> \param[in] XACT 124*> \verbatim 125*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 126*> The exact solution vectors. Each vector is stored as a 127*> column of the matrix XACT. 128*> \endverbatim 129*> 130*> \param[in] LDXACT 131*> \verbatim 132*> LDXACT is INTEGER 133*> The leading dimension of the array XACT. LDXACT >= max(1,N). 134*> \endverbatim 135*> 136*> \param[in] FERR 137*> \verbatim 138*> FERR is DOUBLE PRECISION array, dimension (NRHS) 139*> The estimated forward error bounds for each solution vector 140*> X. If XTRUE is the true solution, FERR bounds the magnitude 141*> of the largest entry in (X - XTRUE) divided by the magnitude 142*> of the largest entry in X. 143*> \endverbatim 144*> 145*> \param[in] BERR 146*> \verbatim 147*> BERR is DOUBLE PRECISION array, dimension (NRHS) 148*> The componentwise relative backward error of each solution 149*> vector (i.e., the smallest relative change in any entry of A 150*> or B that makes X an exact solution). 151*> \endverbatim 152*> 153*> \param[out] RESLTS 154*> \verbatim 155*> RESLTS is DOUBLE PRECISION array, dimension (2) 156*> The maximum over the NRHS solution vectors of the ratios: 157*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 158*> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 159*> \endverbatim 160* 161* Authors: 162* ======== 163* 164*> \author Univ. of Tennessee 165*> \author Univ. of California Berkeley 166*> \author Univ. of Colorado Denver 167*> \author NAG Ltd. 168* 169*> \ingroup double_lin 170* 171* ===================================================================== 172 SUBROUTINE DTPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 173 $ XACT, LDXACT, FERR, BERR, RESLTS ) 174* 175* -- LAPACK test routine -- 176* -- LAPACK is a software package provided by Univ. of Tennessee, -- 177* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 178* 179* .. Scalar Arguments .. 180 CHARACTER DIAG, TRANS, UPLO 181 INTEGER LDB, LDX, LDXACT, N, NRHS 182* .. 183* .. Array Arguments .. 184 DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 185 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 186* .. 187* 188* ===================================================================== 189* 190* .. Parameters .. 191 DOUBLE PRECISION ZERO, ONE 192 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 193* .. 194* .. Local Scalars .. 195 LOGICAL NOTRAN, UNIT, UPPER 196 INTEGER I, IFU, IMAX, J, JC, K 197 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 198* .. 199* .. External Functions .. 200 LOGICAL LSAME 201 INTEGER IDAMAX 202 DOUBLE PRECISION DLAMCH 203 EXTERNAL LSAME, IDAMAX, DLAMCH 204* .. 205* .. Intrinsic Functions .. 206 INTRINSIC ABS, MAX, MIN 207* .. 208* .. Executable Statements .. 209* 210* Quick exit if N = 0 or NRHS = 0. 211* 212 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 213 RESLTS( 1 ) = ZERO 214 RESLTS( 2 ) = ZERO 215 RETURN 216 END IF 217* 218 EPS = DLAMCH( 'Epsilon' ) 219 UNFL = DLAMCH( 'Safe minimum' ) 220 OVFL = ONE / UNFL 221 UPPER = LSAME( UPLO, 'U' ) 222 NOTRAN = LSAME( TRANS, 'N' ) 223 UNIT = LSAME( DIAG, 'U' ) 224* 225* Test 1: Compute the maximum of 226* norm(X - XACT) / ( norm(X) * FERR ) 227* over all the vectors X and XACT using the infinity-norm. 228* 229 ERRBND = ZERO 230 DO 30 J = 1, NRHS 231 IMAX = IDAMAX( N, X( 1, J ), 1 ) 232 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 233 DIFF = ZERO 234 DO 10 I = 1, N 235 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 236 10 CONTINUE 237* 238 IF( XNORM.GT.ONE ) THEN 239 GO TO 20 240 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 241 GO TO 20 242 ELSE 243 ERRBND = ONE / EPS 244 GO TO 30 245 END IF 246* 247 20 CONTINUE 248 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 249 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 250 ELSE 251 ERRBND = ONE / EPS 252 END IF 253 30 CONTINUE 254 RESLTS( 1 ) = ERRBND 255* 256* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 257* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 258* 259 IFU = 0 260 IF( UNIT ) 261 $ IFU = 1 262 DO 90 K = 1, NRHS 263 DO 80 I = 1, N 264 TMP = ABS( B( I, K ) ) 265 IF( UPPER ) THEN 266 JC = ( ( I-1 )*I ) / 2 267 IF( .NOT.NOTRAN ) THEN 268 DO 40 J = 1, I - IFU 269 TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) ) 270 40 CONTINUE 271 IF( UNIT ) 272 $ TMP = TMP + ABS( X( I, K ) ) 273 ELSE 274 JC = JC + I 275 IF( UNIT ) THEN 276 TMP = TMP + ABS( X( I, K ) ) 277 JC = JC + I 278 END IF 279 DO 50 J = I + IFU, N 280 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 281 JC = JC + J 282 50 CONTINUE 283 END IF 284 ELSE 285 IF( NOTRAN ) THEN 286 JC = I 287 DO 60 J = 1, I - IFU 288 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 289 JC = JC + N - J 290 60 CONTINUE 291 IF( UNIT ) 292 $ TMP = TMP + ABS( X( I, K ) ) 293 ELSE 294 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2 295 IF( UNIT ) 296 $ TMP = TMP + ABS( X( I, K ) ) 297 DO 70 J = I + IFU, N 298 TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) ) 299 70 CONTINUE 300 END IF 301 END IF 302 IF( I.EQ.1 ) THEN 303 AXBI = TMP 304 ELSE 305 AXBI = MIN( AXBI, TMP ) 306 END IF 307 80 CONTINUE 308 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 309 $ MAX( AXBI, ( N+1 )*UNFL ) ) 310 IF( K.EQ.1 ) THEN 311 RESLTS( 2 ) = TMP 312 ELSE 313 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 314 END IF 315 90 CONTINUE 316* 317 RETURN 318* 319* End of DTPT05 320* 321 END 322