1*> \brief \b DPPT05 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 DPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT, 12* LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER 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*> DPPT05 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 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 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] AP 70*> \verbatim 71*> AP is DOUBLE PRECISION array, dimension (N*(N+1)/2) 72*> The upper or lower triangle of the symmetric matrix A, packed 73*> columnwise in a linear array. The j-th column of A is stored 74*> in the array AP as follows: 75*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 76*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 77*> \endverbatim 78*> 79*> \param[in] B 80*> \verbatim 81*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 82*> The right hand side vectors for the system of linear 83*> equations. 84*> \endverbatim 85*> 86*> \param[in] LDB 87*> \verbatim 88*> LDB is INTEGER 89*> The leading dimension of the array B. LDB >= max(1,N). 90*> \endverbatim 91*> 92*> \param[in] X 93*> \verbatim 94*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 95*> The computed solution vectors. Each vector is stored as a 96*> column of the matrix X. 97*> \endverbatim 98*> 99*> \param[in] LDX 100*> \verbatim 101*> LDX is INTEGER 102*> The leading dimension of the array X. LDX >= max(1,N). 103*> \endverbatim 104*> 105*> \param[in] XACT 106*> \verbatim 107*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 108*> The exact solution vectors. Each vector is stored as a 109*> column of the matrix XACT. 110*> \endverbatim 111*> 112*> \param[in] LDXACT 113*> \verbatim 114*> LDXACT is INTEGER 115*> The leading dimension of the array XACT. LDXACT >= max(1,N). 116*> \endverbatim 117*> 118*> \param[in] FERR 119*> \verbatim 120*> FERR is DOUBLE PRECISION array, dimension (NRHS) 121*> The estimated forward error bounds for each solution vector 122*> X. If XTRUE is the true solution, FERR bounds the magnitude 123*> of the largest entry in (X - XTRUE) divided by the magnitude 124*> of the largest entry in X. 125*> \endverbatim 126*> 127*> \param[in] BERR 128*> \verbatim 129*> BERR is DOUBLE PRECISION array, dimension (NRHS) 130*> The componentwise relative backward error of each solution 131*> vector (i.e., the smallest relative change in any entry of A 132*> or B that makes X an exact solution). 133*> \endverbatim 134*> 135*> \param[out] RESLTS 136*> \verbatim 137*> RESLTS is DOUBLE PRECISION array, dimension (2) 138*> The maximum over the NRHS solution vectors of the ratios: 139*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 140*> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 141*> \endverbatim 142* 143* Authors: 144* ======== 145* 146*> \author Univ. of Tennessee 147*> \author Univ. of California Berkeley 148*> \author Univ. of Colorado Denver 149*> \author NAG Ltd. 150* 151*> \ingroup double_lin 152* 153* ===================================================================== 154 SUBROUTINE DPPT05( UPLO, N, NRHS, AP, B, LDB, X, LDX, XACT, 155 $ LDXACT, FERR, BERR, RESLTS ) 156* 157* -- LAPACK test routine -- 158* -- LAPACK is a software package provided by Univ. of Tennessee, -- 159* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 160* 161* .. Scalar Arguments .. 162 CHARACTER UPLO 163 INTEGER LDB, LDX, LDXACT, N, NRHS 164* .. 165* .. Array Arguments .. 166 DOUBLE PRECISION AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 167 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 168* .. 169* 170* ===================================================================== 171* 172* .. Parameters .. 173 DOUBLE PRECISION ZERO, ONE 174 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 175* .. 176* .. Local Scalars .. 177 LOGICAL UPPER 178 INTEGER I, IMAX, J, JC, K 179 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 180* .. 181* .. External Functions .. 182 LOGICAL LSAME 183 INTEGER IDAMAX 184 DOUBLE PRECISION DLAMCH 185 EXTERNAL LSAME, IDAMAX, DLAMCH 186* .. 187* .. Intrinsic Functions .. 188 INTRINSIC ABS, MAX, MIN 189* .. 190* .. Executable Statements .. 191* 192* Quick exit if N = 0 or NRHS = 0. 193* 194 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 195 RESLTS( 1 ) = ZERO 196 RESLTS( 2 ) = ZERO 197 RETURN 198 END IF 199* 200 EPS = DLAMCH( 'Epsilon' ) 201 UNFL = DLAMCH( 'Safe minimum' ) 202 OVFL = ONE / UNFL 203 UPPER = LSAME( UPLO, 'U' ) 204* 205* Test 1: Compute the maximum of 206* norm(X - XACT) / ( norm(X) * FERR ) 207* over all the vectors X and XACT using the infinity-norm. 208* 209 ERRBND = ZERO 210 DO 30 J = 1, NRHS 211 IMAX = IDAMAX( N, X( 1, J ), 1 ) 212 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 213 DIFF = ZERO 214 DO 10 I = 1, N 215 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 216 10 CONTINUE 217* 218 IF( XNORM.GT.ONE ) THEN 219 GO TO 20 220 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 221 GO TO 20 222 ELSE 223 ERRBND = ONE / EPS 224 GO TO 30 225 END IF 226* 227 20 CONTINUE 228 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 229 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 230 ELSE 231 ERRBND = ONE / EPS 232 END IF 233 30 CONTINUE 234 RESLTS( 1 ) = ERRBND 235* 236* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 237* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 238* 239 DO 90 K = 1, NRHS 240 DO 80 I = 1, N 241 TMP = ABS( B( I, K ) ) 242 IF( UPPER ) THEN 243 JC = ( ( I-1 )*I ) / 2 244 DO 40 J = 1, I 245 TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) ) 246 40 CONTINUE 247 JC = JC + I 248 DO 50 J = I + 1, N 249 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 250 JC = JC + J 251 50 CONTINUE 252 ELSE 253 JC = I 254 DO 60 J = 1, I - 1 255 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 256 JC = JC + N - J 257 60 CONTINUE 258 DO 70 J = I, N 259 TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) ) 260 70 CONTINUE 261 END IF 262 IF( I.EQ.1 ) THEN 263 AXBI = TMP 264 ELSE 265 AXBI = MIN( AXBI, TMP ) 266 END IF 267 80 CONTINUE 268 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 269 $ MAX( AXBI, ( N+1 )*UNFL ) ) 270 IF( K.EQ.1 ) THEN 271 RESLTS( 2 ) = TMP 272 ELSE 273 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 274 END IF 275 90 CONTINUE 276* 277 RETURN 278* 279* End of DPPT05 280* 281 END 282