1*> \brief \b DTBT05 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 DTBT05( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, 12* LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER DIAG, TRANS, 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*> DTBT05 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*> triangular 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 matrix A is upper or lower triangular. 52*> = 'U': Upper triangular 53*> = 'L': Lower triangular 54*> \endverbatim 55*> 56*> \param[in] TRANS 57*> \verbatim 58*> TRANS is CHARACTER*1 59*> Specifies the form of the system of equations. 60*> = 'N': A * X = B (No transpose) 61*> = 'T': A'* X = B (Transpose) 62*> = 'C': A'* X = B (Conjugate transpose = Transpose) 63*> \endverbatim 64*> 65*> \param[in] DIAG 66*> \verbatim 67*> DIAG is CHARACTER*1 68*> Specifies whether or not the matrix A is unit triangular. 69*> = 'N': Non-unit triangular 70*> = 'U': Unit triangular 71*> \endverbatim 72*> 73*> \param[in] N 74*> \verbatim 75*> N is INTEGER 76*> The number of rows of the matrices X, B, and XACT, and the 77*> order of the matrix A. N >= 0. 78*> \endverbatim 79*> 80*> \param[in] KD 81*> \verbatim 82*> KD is INTEGER 83*> The number of super-diagonals of the matrix A if UPLO = 'U', 84*> or the number of sub-diagonals if UPLO = 'L'. KD >= 0. 85*> \endverbatim 86*> 87*> \param[in] NRHS 88*> \verbatim 89*> NRHS is INTEGER 90*> The number of columns of the matrices X, B, and XACT. 91*> NRHS >= 0. 92*> \endverbatim 93*> 94*> \param[in] AB 95*> \verbatim 96*> AB is DOUBLE PRECISION array, dimension (LDAB,N) 97*> The upper or lower triangular band matrix A, stored in the 98*> first kd+1 rows of the array. The j-th column of A is stored 99*> in the j-th column of the array AB as follows: 100*> if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; 101*> if UPLO = 'L', AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd). 102*> If DIAG = 'U', the diagonal elements of A are not referenced 103*> and are assumed to be 1. 104*> \endverbatim 105*> 106*> \param[in] LDAB 107*> \verbatim 108*> LDAB is INTEGER 109*> The leading dimension of the array AB. LDAB >= KD+1. 110*> \endverbatim 111*> 112*> \param[in] B 113*> \verbatim 114*> B is DOUBLE PRECISION array, dimension (LDB,NRHS) 115*> The right hand side vectors for the system of linear 116*> equations. 117*> \endverbatim 118*> 119*> \param[in] LDB 120*> \verbatim 121*> LDB is INTEGER 122*> The leading dimension of the array B. LDB >= max(1,N). 123*> \endverbatim 124*> 125*> \param[in] X 126*> \verbatim 127*> X is DOUBLE PRECISION array, dimension (LDX,NRHS) 128*> The computed solution vectors. Each vector is stored as a 129*> column of the matrix X. 130*> \endverbatim 131*> 132*> \param[in] LDX 133*> \verbatim 134*> LDX is INTEGER 135*> The leading dimension of the array X. LDX >= max(1,N). 136*> \endverbatim 137*> 138*> \param[in] XACT 139*> \verbatim 140*> XACT is DOUBLE PRECISION array, dimension (LDX,NRHS) 141*> The exact solution vectors. Each vector is stored as a 142*> column of the matrix XACT. 143*> \endverbatim 144*> 145*> \param[in] LDXACT 146*> \verbatim 147*> LDXACT is INTEGER 148*> The leading dimension of the array XACT. LDXACT >= max(1,N). 149*> \endverbatim 150*> 151*> \param[in] FERR 152*> \verbatim 153*> FERR is DOUBLE PRECISION array, dimension (NRHS) 154*> The estimated forward error bounds for each solution vector 155*> X. If XTRUE is the true solution, FERR bounds the magnitude 156*> of the largest entry in (X - XTRUE) divided by the magnitude 157*> of the largest entry in X. 158*> \endverbatim 159*> 160*> \param[in] BERR 161*> \verbatim 162*> BERR is DOUBLE PRECISION array, dimension (NRHS) 163*> The componentwise relative backward error of each solution 164*> vector (i.e., the smallest relative change in any entry of A 165*> or B that makes X an exact solution). 166*> \endverbatim 167*> 168*> \param[out] RESLTS 169*> \verbatim 170*> RESLTS is DOUBLE PRECISION array, dimension (2) 171*> The maximum over the NRHS solution vectors of the ratios: 172*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 173*> RESLTS(2) = BERR / ( NZ*EPS + (*) ) 174*> \endverbatim 175* 176* Authors: 177* ======== 178* 179*> \author Univ. of Tennessee 180*> \author Univ. of California Berkeley 181*> \author Univ. of Colorado Denver 182*> \author NAG Ltd. 183* 184*> \date November 2011 185* 186*> \ingroup double_lin 187* 188* ===================================================================== 189 SUBROUTINE DTBT05( UPLO, TRANS, DIAG, N, KD, NRHS, AB, LDAB, B, 190 $ LDB, X, LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 191* 192* -- LAPACK test routine (version 3.4.0) -- 193* -- LAPACK is a software package provided by Univ. of Tennessee, -- 194* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 195* November 2011 196* 197* .. Scalar Arguments .. 198 CHARACTER DIAG, TRANS, UPLO 199 INTEGER KD, LDAB, LDB, LDX, LDXACT, N, NRHS 200* .. 201* .. Array Arguments .. 202 DOUBLE PRECISION AB( LDAB, * ), B( LDB, * ), BERR( * ), 203 $ FERR( * ), RESLTS( * ), X( LDX, * ), 204 $ XACT( LDXACT, * ) 205* .. 206* 207* ===================================================================== 208* 209* .. Parameters .. 210 DOUBLE PRECISION ZERO, ONE 211 PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 ) 212* .. 213* .. Local Scalars .. 214 LOGICAL NOTRAN, UNIT, UPPER 215 INTEGER I, IFU, IMAX, J, K, NZ 216 DOUBLE PRECISION AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 217* .. 218* .. External Functions .. 219 LOGICAL LSAME 220 INTEGER IDAMAX 221 DOUBLE PRECISION DLAMCH 222 EXTERNAL LSAME, IDAMAX, DLAMCH 223* .. 224* .. Intrinsic Functions .. 225 INTRINSIC ABS, MAX, MIN 226* .. 227* .. Executable Statements .. 228* 229* Quick exit if N = 0 or NRHS = 0. 230* 231 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 232 RESLTS( 1 ) = ZERO 233 RESLTS( 2 ) = ZERO 234 RETURN 235 END IF 236* 237 EPS = DLAMCH( 'Epsilon' ) 238 UNFL = DLAMCH( 'Safe minimum' ) 239 OVFL = ONE / UNFL 240 UPPER = LSAME( UPLO, 'U' ) 241 NOTRAN = LSAME( TRANS, 'N' ) 242 UNIT = LSAME( DIAG, 'U' ) 243 NZ = MIN( KD, N-1 ) + 1 244* 245* Test 1: Compute the maximum of 246* norm(X - XACT) / ( norm(X) * FERR ) 247* over all the vectors X and XACT using the infinity-norm. 248* 249 ERRBND = ZERO 250 DO 30 J = 1, NRHS 251 IMAX = IDAMAX( N, X( 1, J ), 1 ) 252 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 253 DIFF = ZERO 254 DO 10 I = 1, N 255 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 256 10 CONTINUE 257* 258 IF( XNORM.GT.ONE ) THEN 259 GO TO 20 260 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 261 GO TO 20 262 ELSE 263 ERRBND = ONE / EPS 264 GO TO 30 265 END IF 266* 267 20 CONTINUE 268 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 269 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 270 ELSE 271 ERRBND = ONE / EPS 272 END IF 273 30 CONTINUE 274 RESLTS( 1 ) = ERRBND 275* 276* Test 2: Compute the maximum of BERR / ( NZ*EPS + (*) ), where 277* (*) = NZ*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 278* 279 IFU = 0 280 IF( UNIT ) 281 $ IFU = 1 282 DO 90 K = 1, NRHS 283 DO 80 I = 1, N 284 TMP = ABS( B( I, K ) ) 285 IF( UPPER ) THEN 286 IF( .NOT.NOTRAN ) THEN 287 DO 40 J = MAX( I-KD, 1 ), I - IFU 288 TMP = TMP + ABS( AB( KD+1-I+J, I ) )* 289 $ ABS( X( J, K ) ) 290 40 CONTINUE 291 IF( UNIT ) 292 $ TMP = TMP + ABS( X( I, K ) ) 293 ELSE 294 IF( UNIT ) 295 $ TMP = TMP + ABS( X( I, K ) ) 296 DO 50 J = I + IFU, MIN( I+KD, N ) 297 TMP = TMP + ABS( AB( KD+1+I-J, J ) )* 298 $ ABS( X( J, K ) ) 299 50 CONTINUE 300 END IF 301 ELSE 302 IF( NOTRAN ) THEN 303 DO 60 J = MAX( I-KD, 1 ), I - IFU 304 TMP = TMP + ABS( AB( 1+I-J, J ) )*ABS( X( J, K ) ) 305 60 CONTINUE 306 IF( UNIT ) 307 $ TMP = TMP + ABS( X( I, K ) ) 308 ELSE 309 IF( UNIT ) 310 $ TMP = TMP + ABS( X( I, K ) ) 311 DO 70 J = I + IFU, MIN( I+KD, N ) 312 TMP = TMP + ABS( AB( 1+J-I, I ) )*ABS( X( J, K ) ) 313 70 CONTINUE 314 END IF 315 END IF 316 IF( I.EQ.1 ) THEN 317 AXBI = TMP 318 ELSE 319 AXBI = MIN( AXBI, TMP ) 320 END IF 321 80 CONTINUE 322 TMP = BERR( K ) / ( NZ*EPS+NZ*UNFL / MAX( AXBI, NZ*UNFL ) ) 323 IF( K.EQ.1 ) THEN 324 RESLTS( 2 ) = TMP 325 ELSE 326 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 327 END IF 328 90 CONTINUE 329* 330 RETURN 331* 332* End of DTBT05 333* 334 END 335