1*> \brief \b DGBT05 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 DGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X, 12* LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER TRANS 16* INTEGER KL, KU, 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*> DGBT05 tests the error bounds from iterative refinement for the 31*> computed solution to a system of equations op(A)*X = B, where A is a 32*> general band matrix of order n with kl subdiagonals and ku 33*> superdiagonals and op(A) = A or A**T, 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, B, and XACT, and the 62*> order of the matrix A. N >= 0. 63*> \endverbatim 64*> 65*> \param[in] KL 66*> \verbatim 67*> KL is INTEGER 68*> The number of subdiagonals within the band of A. KL >= 0. 69*> \endverbatim 70*> 71*> \param[in] KU 72*> \verbatim 73*> KU is INTEGER 74*> The number of superdiagonals within the band of A. KU >= 0. 75*> \endverbatim 76*> 77*> \param[in] NRHS 78*> \verbatim 79*> NRHS is INTEGER 80*> The number of columns of the matrices X, B, and XACT. 81*> NRHS >= 0. 82*> \endverbatim 83*> 84*> \param[in] AB 85*> \verbatim 86*> AB is DOUBLE PRECISION array, dimension (LDAB,N) 87*> The original band matrix A, stored in rows 1 to KL+KU+1. 88*> The j-th column of A is stored in the j-th column of the 89*> array AB as follows: 90*> AB(ku+1+i-j,j) = A(i,j) for max(1,j-ku)<=i<=min(n,j+kl). 91*> \endverbatim 92*> 93*> \param[in] LDAB 94*> \verbatim 95*> LDAB is INTEGER 96*> The leading dimension of the array AB. LDAB >= KL+KU+1. 97*> \endverbatim 98*> 99*> \param[in] B 100*> \verbatim 101*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 102*> The right hand side vectors for the system of linear 103*> equations. 104*> \endverbatim 105*> 106*> \param[in] LDB 107*> \verbatim 108*> LDB is INTEGER 109*> The leading dimension of the array B. LDB >= max(1,N). 110*> \endverbatim 111*> 112*> \param[in] X 113*> \verbatim 114*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 115*> The computed solution vectors. Each vector is stored as a 116*> column of the matrix X. 117*> \endverbatim 118*> 119*> \param[in] LDX 120*> \verbatim 121*> LDX is INTEGER 122*> The leading dimension of the array X. LDX >= max(1,N). 123*> \endverbatim 124*> 125*> \param[in] XACT 126*> \verbatim 127*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 128*> The exact solution vectors. Each vector is stored as a 129*> column of the matrix XACT. 130*> \endverbatim 131*> 132*> \param[in] LDXACT 133*> \verbatim 134*> LDXACT is INTEGER 135*> The leading dimension of the array XACT. LDXACT >= max(1,N). 136*> \endverbatim 137*> 138*> \param[in] FERR 139*> \verbatim 140*> FERR is DOUBLE PRECISION array, dimension (NRHS) 141*> The estimated forward error bounds for each solution vector 142*> X. If XTRUE is the true solution, FERR bounds the magnitude 143*> of the largest entry in (X - XTRUE) divided by the magnitude 144*> of the largest entry in X. 145*> \endverbatim 146*> 147*> \param[in] BERR 148*> \verbatim 149*> BERR is DOUBLE PRECISION array, dimension (NRHS) 150*> The componentwise relative backward error of each solution 151*> vector (i.e., the smallest relative change in any entry of A 152*> or B that makes X an exact solution). 153*> \endverbatim 154*> 155*> \param[out] RESLTS 156*> \verbatim 157*> RESLTS is DOUBLE PRECISION array, dimension (2) 158*> The maximum over the NRHS solution vectors of the ratios: 159*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 160*> RESLTS(2) = BERR / ( NZ*EPS + (*) ) 161*> \endverbatim 162* 163* Authors: 164* ======== 165* 166*> \author Univ. of Tennessee 167*> \author Univ. of California Berkeley 168*> \author Univ. of Colorado Denver 169*> \author NAG Ltd. 170* 171*> \date November 2011 172* 173*> \ingroup double_lin 174* 175* ===================================================================== 176 SUBROUTINE DGBT05( TRANS, N, KL, KU, NRHS, AB, LDAB, B, LDB, X, 177 $ LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 178* 179* -- LAPACK test routine (version 3.4.0) -- 180* -- LAPACK is a software package provided by Univ. of Tennessee, -- 181* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 182* November 2011 183* 184* .. Scalar Arguments .. 185 CHARACTER TRANS 186 INTEGER KL, KU, LDAB, LDB, LDX, LDXACT, N, NRHS 187* .. 188* .. Array Arguments .. 189 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), 190 $ FERR( * ), RESLTS( * ), X( LDX, * ), 191 $ XACT( LDXACT, * ) 192* .. 193* 194* ===================================================================== 195* 196* .. Parameters .. 197 DOUBLE PRECISION ZERO, ONE 198 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 199* .. 200* .. Local Scalars .. 201 LOGICAL NOTRAN 202 INTEGER I, IMAX, J, K, NZ 203 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 204* .. 205* .. External Functions .. 206 LOGICAL LSAME 207 INTEGER IDAMAX 208 DOUBLE PRECISION DLAMCH 209 EXTERNAL LSAME, IDAMAX, DLAMCH 210* .. 211* .. Intrinsic Functions .. 212 INTRINSIC ABS, MAX, MIN 213* .. 214* .. Executable Statements .. 215* 216* Quick exit if N = 0 or NRHS = 0. 217* 218 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 219 RESLTS( 1 ) = ZERO 220 RESLTS( 2 ) = ZERO 221 RETURN 222 END IF 223* 224 EPS = DLAMCH( 'Epsilon' ) 225 UNFL = DLAMCH( 'Safe minimum' ) 226 OVFL = ONE / UNFL 227 NOTRAN = LSAME( TRANS, 'N' ) 228 NZ = MIN( KL+KU+2, N+1 ) 229* 230* Test 1: Compute the maximum of 231* norm(X - XACT) / ( norm(X) * FERR ) 232* over all the vectors X and XACT using the infinity-norm. 233* 234 ERRBND = ZERO 235 DO 30 J = 1, NRHS 236 IMAX = IDAMAX( N, X( 1, J ), 1 ) 237 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 238 DIFF = ZERO 239 DO 10 I = 1, N 240 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 241 10 CONTINUE 242* 243 IF( XNORM.GT.ONE ) THEN 244 GO TO 20 245 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 246 GO TO 20 247 ELSE 248 ERRBND = ONE / EPS 249 GO TO 30 250 END IF 251* 252 20 CONTINUE 253 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 254 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 255 ELSE 256 ERRBND = ONE / EPS 257 END IF 258 30 CONTINUE 259 RESLTS( 1 ) = ERRBND 260* 261* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 262* (*) = NZ*UNFL / (min_i (abs(op(A))*abs(X) +abs(b))_i ) 263* 264 DO 70 K = 1, NRHS 265 DO 60 I = 1, N 266 TMP = ABS( B( I, K ) ) 267 IF( NOTRAN ) THEN 268 DO 40 J = MAX( I-KL, 1 ), MIN( I+KU, N ) 269 TMP = TMP + ABS( AB( KU+1+I-J, J ) )*ABS( X( J, K ) ) 270 40 CONTINUE 271 ELSE 272 DO 50 J = MAX( I-KU, 1 ), MIN( I+KL, N ) 273 TMP = TMP + ABS( AB( KU+1+J-I, I ) )*ABS( X( J, K ) ) 274 50 CONTINUE 275 END IF 276 IF( I.EQ.1 ) THEN 277 AXBI = TMP 278 ELSE 279 AXBI = MIN( AXBI, TMP ) 280 END IF 281 60 CONTINUE 282 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 283 IF( K.EQ.1 ) THEN 284 RESLTS( 2 ) = TMP 285 ELSE 286 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 287 END IF 288 70 CONTINUE 289* 290 RETURN 291* 292* End of DGBT05 293* 294 END 295