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*> \date November 2011 167* 168*> \ingroup double_lin 169* 170* ===================================================================== 171 SUBROUTINE DPBT05( UPLO, N, KD, NRHS, AB, LDAB, B, LDB, X, LDX, 172 $ XACT, LDXACT, FERR, BERR, RESLTS ) 173* 174* -- LAPACK test routine (version 3.4.0) -- 175* -- LAPACK is a software package provided by Univ. of Tennessee, -- 176* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 177* November 2011 178* 179* .. Scalar Arguments .. 180 CHARACTER UPLO 181 INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS 182* .. 183* .. Array Arguments .. 184 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), 185 $ FERR( * ), RESLTS( * ), X( LDX, * ), 186 $ XACT( LDXACT, * ) 187* .. 188* 189* ===================================================================== 190* 191* .. Parameters .. 192 DOUBLE PRECISION ZERO, ONE 193 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 194* .. 195* .. Local Scalars .. 196 LOGICAL UPPER 197 INTEGER I, IMAX, J, K, NZ 198 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 199* .. 200* .. External Functions .. 201 LOGICAL LSAME 202 INTEGER IDAMAX 203 DOUBLE PRECISION DLAMCH 204 EXTERNAL LSAME, IDAMAX, DLAMCH 205* .. 206* .. Intrinsic Functions .. 207 INTRINSIC ABS, MAX, MIN 208* .. 209* .. Executable Statements .. 210* 211* Quick exit if N = 0 or NRHS = 0. 212* 213 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 214 RESLTS( 1 ) = ZERO 215 RESLTS( 2 ) = ZERO 216 RETURN 217 END IF 218* 219 EPS = DLAMCH( 'Epsilon' ) 220 UNFL = DLAMCH( 'Safe minimum' ) 221 OVFL = ONE / UNFL 222 UPPER = LSAME( UPLO, 'U' ) 223 NZ = 2*MAX( KD, N-1 ) + 1 224* 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 / ( NZ*EPS + (*) ), where 257* (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 258* 259 DO 90 K = 1, NRHS 260 DO 80 I = 1, N 261 TMP = ABS( B( I, K ) ) 262 IF( UPPER ) THEN 263 DO 40 J = MAX( I-KD, 1 ), I 264 TMP = TMP + ABS( AB( KD+1-I+J, I ) )*ABS( X( J, K ) ) 265 40 CONTINUE 266 DO 50 J = I + 1, MIN( I+KD, N ) 267 TMP = TMP + ABS( AB( KD+1+I-J, J ) )*ABS( X( J, K ) ) 268 50 CONTINUE 269 ELSE 270 DO 60 J = MAX( I-KD, 1 ), I - 1 271 TMP = TMP + ABS( AB( 1+I-J, J ) )*ABS( X( J, K ) ) 272 60 CONTINUE 273 DO 70 J = I, MIN( I+KD, N ) 274 TMP = TMP + ABS( AB( 1+J-I, I ) )*ABS( X( J, K ) ) 275 70 CONTINUE 276 END IF 277 IF( I.EQ.1 ) THEN 278 AXBI = TMP 279 ELSE 280 AXBI = MIN( AXBI, TMP ) 281 END IF 282 80 CONTINUE 283 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 284 IF( K.EQ.1 ) THEN 285 RESLTS( 2 ) = TMP 286 ELSE 287 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 288 END IF 289 90 CONTINUE 290* 291 RETURN 292* 293* End of DPBT05 294* 295 END 296