1*> \brief \b STRT05 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 STRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, 12* LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER DIAG, TRANS, UPLO 16* INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 17* .. 18* .. Array Arguments .. 19* REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 20* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 21* .. 22* 23* 24*> \par Purpose: 25* ============= 26*> 27*> \verbatim 28*> 29*> STRT05 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*> triangular n by n matrix. 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 matrix A is upper or lower triangular. 50*> = 'U': Upper triangular 51*> = 'L': Lower triangular 52*> \endverbatim 53*> 54*> \param[in] TRANS 55*> \verbatim 56*> TRANS is CHARACTER*1 57*> Specifies the form of the system of equations. 58*> = 'N': A * X = B (No transpose) 59*> = 'T': A'* X = B (Transpose) 60*> = 'C': A'* X = B (Conjugate transpose = Transpose) 61*> \endverbatim 62*> 63*> \param[in] DIAG 64*> \verbatim 65*> DIAG is CHARACTER*1 66*> Specifies whether or not the matrix A is unit triangular. 67*> = 'N': Non-unit triangular 68*> = 'U': Unit triangular 69*> \endverbatim 70*> 71*> \param[in] N 72*> \verbatim 73*> N is INTEGER 74*> The number of rows of the matrices X, B, and XACT, and the 75*> order of the matrix A. N >= 0. 76*> \endverbatim 77*> 78*> \param[in] NRHS 79*> \verbatim 80*> NRHS is INTEGER 81*> The number of columns of the matrices X, B, and XACT. 82*> NRHS >= 0. 83*> \endverbatim 84*> 85*> \param[in] A 86*> \verbatim 87*> A is REAL array, dimension (LDA,N) 88*> The triangular matrix A. If UPLO = 'U', the leading n by n 89*> upper triangular part of the array A contains the upper 90*> triangular matrix, and the strictly lower triangular part of 91*> A is not referenced. If UPLO = 'L', the leading n by n lower 92*> triangular part of the array A contains the lower triangular 93*> matrix, and the strictly upper triangular part of A is not 94*> referenced. If DIAG = 'U', the diagonal elements of A are 95*> also not referenced and are assumed to be 1. 96*> \endverbatim 97*> 98*> \param[in] LDA 99*> \verbatim 100*> LDA is INTEGER 101*> The leading dimension of the array A. LDA >= max(1,N). 102*> \endverbatim 103*> 104*> \param[in] B 105*> \verbatim 106*> B is REAL array, dimension (LDB,NRHS) 107*> The right hand side vectors for the system of linear 108*> equations. 109*> \endverbatim 110*> 111*> \param[in] LDB 112*> \verbatim 113*> LDB is INTEGER 114*> The leading dimension of the array B. LDB >= max(1,N). 115*> \endverbatim 116*> 117*> \param[in] X 118*> \verbatim 119*> X is REAL array, dimension (LDX,NRHS) 120*> The computed solution vectors. Each vector is stored as a 121*> column of the matrix X. 122*> \endverbatim 123*> 124*> \param[in] LDX 125*> \verbatim 126*> LDX is INTEGER 127*> The leading dimension of the array X. LDX >= max(1,N). 128*> \endverbatim 129*> 130*> \param[in] XACT 131*> \verbatim 132*> XACT is REAL array, dimension (LDX,NRHS) 133*> The exact solution vectors. Each vector is stored as a 134*> column of the matrix XACT. 135*> \endverbatim 136*> 137*> \param[in] LDXACT 138*> \verbatim 139*> LDXACT is INTEGER 140*> The leading dimension of the array XACT. LDXACT >= max(1,N). 141*> \endverbatim 142*> 143*> \param[in] FERR 144*> \verbatim 145*> FERR is REAL array, dimension (NRHS) 146*> The estimated forward error bounds for each solution vector 147*> X. If XTRUE is the true solution, FERR bounds the magnitude 148*> of the largest entry in (X - XTRUE) divided by the magnitude 149*> of the largest entry in X. 150*> \endverbatim 151*> 152*> \param[in] BERR 153*> \verbatim 154*> BERR is REAL array, dimension (NRHS) 155*> The componentwise relative backward error of each solution 156*> vector (i.e., the smallest relative change in any entry of A 157*> or B that makes X an exact solution). 158*> \endverbatim 159*> 160*> \param[out] RESLTS 161*> \verbatim 162*> RESLTS is REAL array, dimension (2) 163*> The maximum over the NRHS solution vectors of the ratios: 164*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 165*> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 166*> \endverbatim 167* 168* Authors: 169* ======== 170* 171*> \author Univ. of Tennessee 172*> \author Univ. of California Berkeley 173*> \author Univ. of Colorado Denver 174*> \author NAG Ltd. 175* 176*> \ingroup single_lin 177* 178* ===================================================================== 179 SUBROUTINE STRT05( UPLO, TRANS, DIAG, N, NRHS, A, LDA, B, LDB, X, 180 $ LDX, XACT, LDXACT, FERR, BERR, RESLTS ) 181* 182* -- LAPACK test routine -- 183* -- LAPACK is a software package provided by Univ. of Tennessee, -- 184* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 185* 186* .. Scalar Arguments .. 187 CHARACTER DIAG, TRANS, UPLO 188 INTEGER LDA, LDB, LDX, LDXACT, N, NRHS 189* .. 190* .. Array Arguments .. 191 REAL A( LDA, * ), B( LDB, * ), BERR( * ), FERR( * ), 192 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 193* .. 194* 195* ===================================================================== 196* 197* .. Parameters .. 198 REAL ZERO, ONE 199 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 200* .. 201* .. Local Scalars .. 202 LOGICAL NOTRAN, UNIT, UPPER 203 INTEGER I, IFU, IMAX, J, K 204 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 205* .. 206* .. External Functions .. 207 LOGICAL LSAME 208 INTEGER ISAMAX 209 REAL SLAMCH 210 EXTERNAL LSAME, ISAMAX, SLAMCH 211* .. 212* .. Intrinsic Functions .. 213 INTRINSIC ABS, MAX, MIN 214* .. 215* .. Executable Statements .. 216* 217* Quick exit if N = 0 or NRHS = 0. 218* 219 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 220 RESLTS( 1 ) = ZERO 221 RESLTS( 2 ) = ZERO 222 RETURN 223 END IF 224* 225 EPS = SLAMCH( 'Epsilon' ) 226 UNFL = SLAMCH( 'Safe minimum' ) 227 OVFL = ONE / UNFL 228 UPPER = LSAME( UPLO, 'U' ) 229 NOTRAN = LSAME( TRANS, 'N' ) 230 UNIT = LSAME( DIAG, 'U' ) 231* 232* Test 1: Compute the maximum of 233* norm(X - XACT) / ( norm(X) * FERR ) 234* over all the vectors X and XACT using the infinity-norm. 235* 236 ERRBND = ZERO 237 DO 30 J = 1, NRHS 238 IMAX = ISAMAX( N, X( 1, J ), 1 ) 239 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 240 DIFF = ZERO 241 DO 10 I = 1, N 242 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 243 10 CONTINUE 244* 245 IF( XNORM.GT.ONE ) THEN 246 GO TO 20 247 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 248 GO TO 20 249 ELSE 250 ERRBND = ONE / EPS 251 GO TO 30 252 END IF 253* 254 20 CONTINUE 255 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 256 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 257 ELSE 258 ERRBND = ONE / EPS 259 END IF 260 30 CONTINUE 261 RESLTS( 1 ) = ERRBND 262* 263* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 264* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 265* 266 IFU = 0 267 IF( UNIT ) 268 $ IFU = 1 269 DO 90 K = 1, NRHS 270 DO 80 I = 1, N 271 TMP = ABS( B( I, K ) ) 272 IF( UPPER ) THEN 273 IF( .NOT.NOTRAN ) THEN 274 DO 40 J = 1, I - IFU 275 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 276 40 CONTINUE 277 IF( UNIT ) 278 $ TMP = TMP + ABS( X( I, K ) ) 279 ELSE 280 IF( UNIT ) 281 $ TMP = TMP + ABS( X( I, K ) ) 282 DO 50 J = I + IFU, N 283 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 284 50 CONTINUE 285 END IF 286 ELSE 287 IF( NOTRAN ) THEN 288 DO 60 J = 1, I - IFU 289 TMP = TMP + ABS( A( I, J ) )*ABS( X( J, K ) ) 290 60 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 70 J = I + IFU, N 297 TMP = TMP + ABS( A( J, I ) )*ABS( X( J, K ) ) 298 70 CONTINUE 299 END IF 300 END IF 301 IF( I.EQ.1 ) THEN 302 AXBI = TMP 303 ELSE 304 AXBI = MIN( AXBI, TMP ) 305 END IF 306 80 CONTINUE 307 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 308 $ MAX( AXBI, ( N+1 )*UNFL ) ) 309 IF( K.EQ.1 ) THEN 310 RESLTS( 2 ) = TMP 311 ELSE 312 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 313 END IF 314 90 CONTINUE 315* 316 RETURN 317* 318* End of STRT05 319* 320 END 321