1*> \brief \b DPBT05 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 DPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, 12* XACT, LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER UPLO 16* INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS 17* .. 18* .. Array Arguments .. 19* DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), 20* $ FERR( * ), RESLTS( * ), X( LDX, * ), 21* $ XACT( LDXACT, * ) 22* .. 23* 24* 25*> \par Purpose: 26* ============= 27*> 28*> \verbatim 29*> 30*> DPBT05 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*> symmetric band matrix. 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 / ( NZ*EPS + (*) ), where 41*> (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 42*> and NZ = max. number of nonzeros in any row of A, plus 1 43*> \endverbatim 44* 45* Arguments: 46* ========== 47* 48*> \param[in] UPLO 49*> \verbatim 50*> UPLO is CHARACTER*1 51*> Specifies whether the upper or lower triangular part of the 52*> symmetric matrix A is stored. 53*> = 'U': Upper triangular 54*> = 'L': Lower triangular 55*> \endverbatim 56*> 57*> \param[in] N 58*> \verbatim 59*> N is INTEGER 60*> The number of rows of the matrices X, B, and XACT, and the 61*> order of the matrix A. N >= 0. 62*> \endverbatim 63*> 64*> \param[in] KD 65*> \verbatim 66*> KD is INTEGER 67*> The number of super-diagonals of the matrix A if UPLO = 'U', 68*> or the number of sub-diagonals if UPLO = 'L'. KD >= 0. 69*> \endverbatim 70*> 71*> \param[in] NRHS 72*> \verbatim 73*> NRHS is INTEGER 74*> The number of columns of the matrices X, B, and XACT. 75*> NRHS >= 0. 76*> \endverbatim 77*> 78*> \param[in] AB 79*> \verbatim 80*> AB is DOUBLE PRECISION array, dimension (LDAB,N) 81*> The upper or lower triangle of the symmetric band matrix A, 82*> stored in the first KD+1 rows of the array. The j-th column 83*> of A is stored in the j-th column of the array AB as follows: 84*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 85*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 86*> \endverbatim 87*> 88*> \param[in] LDAB 89*> \verbatim 90*> LDAB is INTEGER 91*> The leading dimension of the array AB. LDAB >= KD+1. 92*> \endverbatim 93*> 94*> \param[in] B 95*> \verbatim 96*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 97*> The right hand side vectors for the system of linear 98*> equations. 99*> \endverbatim 100*> 101*> \param[in] LDB 102*> \verbatim 103*> LDB is INTEGER 104*> The leading dimension of the array B. LDB >= max(1,N). 105*> \endverbatim 106*> 107*> \param[in] X 108*> \verbatim 109*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 110*> The computed solution vectors. Each vector is stored as a 111*> column of the matrix X. 112*> \endverbatim 113*> 114*> \param[in] LDX 115*> \verbatim 116*> LDX is INTEGER 117*> The leading dimension of the array X. LDX >= max(1,N). 118*> \endverbatim 119*> 120*> \param[in] XACT 121*> \verbatim 122*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 123*> The exact solution vectors. Each vector is stored as a 124*> column of the matrix XACT. 125*> \endverbatim 126*> 127*> \param[in] LDXACT 128*> \verbatim 129*> LDXACT is INTEGER 130*> The leading dimension of the array XACT. LDXACT >= max(1,N). 131*> \endverbatim 132*> 133*> \param[in] FERR 134*> \verbatim 135*> FERR is DOUBLE PRECISION array, dimension (NRHS) 136*> The estimated forward error bounds for each solution vector 137*> X. If XTRUE is the true solution, FERR bounds the magnitude 138*> of the largest entry in (X - XTRUE) divided by the magnitude 139*> of the largest entry in X. 140*> \endverbatim 141*> 142*> \param[in] BERR 143*> \verbatim 144*> BERR is DOUBLE PRECISION array, dimension (NRHS) 145*> The componentwise relative backward error of each solution 146*> vector (i.e., the smallest relative change in any entry of A 147*> or B that makes X an exact solution). 148*> \endverbatim 149*> 150*> \param[out] RESLTS 151*> \verbatim 152*> RESLTS is DOUBLE PRECISION array, dimension (2) 153*> The maximum over the NRHS solution vectors of the ratios: 154*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 155*> RESLTS(2) = BERR / ( NZ*EPS + (*) ) 156*> \endverbatim 157* 158* Authors: 159* ======== 160* 161*> \author Univ. of Tennessee 162*> \author Univ. of California Berkeley 163*> \author Univ. of Colorado Denver 164*> \author NAG Ltd. 165* 166*> \ingroup double_lin 167* 168* ===================================================================== 169 SUBROUTINE DPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, 170 $ XACT, LDXACT, FERR, BERR, RESLTS ) 171* 172* -- LAPACK test routine -- 173* -- LAPACK is a software package provided by Univ. of Tennessee, -- 174* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 175* 176* .. Scalar Arguments .. 177 CHARACTER UPLO 178 INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS 179* .. 180* .. Array Arguments .. 181 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), 182 $ FERR( * ), RESLTS( * ), X( LDX, * ), 183 $ XACT( LDXACT, * ) 184* .. 185* 186* ===================================================================== 187* 188* .. Parameters .. 189 DOUBLE PRECISION ZERO, ONE 190 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 191* .. 192* .. Local Scalars .. 193 LOGICAL UPPER 194 INTEGER I, IMAX, J, K, NZ 195 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 196* .. 197* .. External Functions .. 198 LOGICAL LSAME 199 INTEGER IDAMAX 200 DOUBLE PRECISION DLAMCH 201 EXTERNAL LSAME, IDAMAX, DLAMCH 202* .. 203* .. Intrinsic Functions .. 204 INTRINSIC ABS, MAX, MIN 205* .. 206* .. Executable Statements .. 207* 208* Quick exit if N = 0 or NRHS = 0. 209* 210 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 211 RESLTS( 1 ) = ZERO 212 RESLTS( 2 ) = ZERO 213 RETURN 214 END IF 215* 216 EPS = DLAMCH( 'Epsilon' ) 217 UNFL = DLAMCH( 'Safe minimum' ) 218 OVFL = ONE / UNFL 219 UPPER = LSAME( UPLO, 'U' ) 220 NZ = 2*MAX( KD, N-1 ) + 1 221* 222* Test 1: Compute the maximum of 223* norm(X - XACT) / ( norm(X) * FERR ) 224* over all the vectors X and XACT using the infinity-norm. 225* 226 ERRBND = ZERO 227 DO 30 J = 1, NRHS 228 IMAX = IDAMAX( N, X( 1, J ), 1 ) 229 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 230 DIFF = ZERO 231 DO 10 I = 1, N 232 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 233 10 CONTINUE 234* 235 IF( XNORM.GT.ONE ) THEN 236 GO TO 20 237 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 238 GO TO 20 239 ELSE 240 ERRBND = ONE / EPS 241 GO TO 30 242 END IF 243* 244 20 CONTINUE 245 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 246 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 247 ELSE 248 ERRBND = ONE / EPS 249 END IF 250 30 CONTINUE 251 RESLTS( 1 ) = ERRBND 252* 253* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 254* (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 255* 256 DO 90 K = 1, NRHS 257 DO 80 I = 1, N 258 TMP = ABS( B( I, K ) ) 259 IF( UPPER ) THEN 260 DO 40 J = MAX( I-KD, 1 ), I 261 TMP = TMP + ABS( AB( KD+1-I+J, I ) )*ABS( X( J, K ) ) 262 40 CONTINUE 263 DO 50 J = I + 1, MIN( I+KD, N ) 264 TMP = TMP + ABS( AB( KD+1+I-J, J ) )*ABS( X( J, K ) ) 265 50 CONTINUE 266 ELSE 267 DO 60 J = MAX( I-KD, 1 ), I - 1 268 TMP = TMP + ABS( AB( 1+I-J, J ) )*ABS( X( J, K ) ) 269 60 CONTINUE 270 DO 70 J = I, MIN( I+KD, N ) 271 TMP = TMP + ABS( AB( 1+J-I, I ) )*ABS( X( J, K ) ) 272 70 CONTINUE 273 END IF 274 IF( I.EQ.1 ) THEN 275 AXBI = TMP 276 ELSE 277 AXBI = MIN( AXBI, TMP ) 278 END IF 279 80 CONTINUE 280 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 281 IF( K.EQ.1 ) THEN 282 RESLTS( 2 ) = TMP 283 ELSE 284 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 285 END IF 286 90 CONTINUE 287* 288 RETURN 289* 290* End of DPBT05 291* 292 END 293