1*> \brief \b STPT05 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 STPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 12* XACT, LDXACT, FERR, BERR, RESLTS ) 13* 14* .. Scalar Arguments .. 15* CHARACTER DIAG, TRANS, UPLO 16* INTEGER LDB, LDX, LDXACT, N, NRHS 17* .. 18* .. Array Arguments .. 19* REAL AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 20* $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 21* .. 22* 23* 24*> \par Purpose: 25* ============= 26*> 27*> \verbatim 28*> 29*> STPT05 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 matrix in packed storage format. 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] AP 86*> \verbatim 87*> AP is REAL array, dimension (N*(N+1)/2) 88*> The upper or lower triangular matrix A, packed columnwise in 89*> a linear array. The j-th column of A is stored in the array 90*> AP as follows: 91*> if UPLO = 'U', AP(i + (j-1)*j/2) = A(i,j) for 1<=i<=j; 92*> if UPLO = 'L', AP(i + (j-1)*(2n-j)/2) = A(i,j) for j<=i<=n. 93*> If DIAG = 'U', the diagonal elements of A are not referenced 94*> and are assumed to be 1. 95*> \endverbatim 96*> 97*> \param[in] B 98*> \verbatim 99*> B is REAL array, dimension (LDB,NRHS) 100*> The right hand side vectors for the system of linear 101*> equations. 102*> \endverbatim 103*> 104*> \param[in] LDB 105*> \verbatim 106*> LDB is INTEGER 107*> The leading dimension of the array B. LDB >= max(1,N). 108*> \endverbatim 109*> 110*> \param[in] X 111*> \verbatim 112*> X is REAL array, dimension (LDX,NRHS) 113*> The computed solution vectors. Each vector is stored as a 114*> column of the matrix X. 115*> \endverbatim 116*> 117*> \param[in] LDX 118*> \verbatim 119*> LDX is INTEGER 120*> The leading dimension of the array X. LDX >= max(1,N). 121*> \endverbatim 122*> 123*> \param[in] XACT 124*> \verbatim 125*> XACT is REAL array, dimension (LDX,NRHS) 126*> The exact solution vectors. Each vector is stored as a 127*> column of the matrix XACT. 128*> \endverbatim 129*> 130*> \param[in] LDXACT 131*> \verbatim 132*> LDXACT is INTEGER 133*> The leading dimension of the array XACT. LDXACT >= max(1,N). 134*> \endverbatim 135*> 136*> \param[in] FERR 137*> \verbatim 138*> FERR is REAL array, dimension (NRHS) 139*> The estimated forward error bounds for each solution vector 140*> X. If XTRUE is the true solution, FERR bounds the magnitude 141*> of the largest entry in (X - XTRUE) divided by the magnitude 142*> of the largest entry in X. 143*> \endverbatim 144*> 145*> \param[in] BERR 146*> \verbatim 147*> BERR is REAL array, dimension (NRHS) 148*> The componentwise relative backward error of each solution 149*> vector (i.e., the smallest relative change in any entry of A 150*> or B that makes X an exact solution). 151*> \endverbatim 152*> 153*> \param[out] RESLTS 154*> \verbatim 155*> RESLTS is REAL array, dimension (2) 156*> The maximum over the NRHS solution vectors of the ratios: 157*> RESLTS(1) = norm(X - XACT) / ( norm(X) * FERR ) 158*> RESLTS(2) = BERR / ( (n+1)*EPS + (*) ) 159*> \endverbatim 160* 161* Authors: 162* ======== 163* 164*> \author Univ. of Tennessee 165*> \author Univ. of California Berkeley 166*> \author Univ. of Colorado Denver 167*> \author NAG Ltd. 168* 169*> \date November 2011 170* 171*> \ingroup single_lin 172* 173* ===================================================================== 174 SUBROUTINE STPT05( UPLO, TRANS, DIAG, N, NRHS, AP, B, LDB, X, LDX, 175 $ XACT, LDXACT, FERR, BERR, RESLTS ) 176* 177* -- LAPACK test routine (version 3.4.0) -- 178* -- LAPACK is a software package provided by Univ. of Tennessee, -- 179* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- 180* November 2011 181* 182* .. Scalar Arguments .. 183 CHARACTER DIAG, TRANS, UPLO 184 INTEGER LDB, LDX, LDXACT, N, NRHS 185* .. 186* .. Array Arguments .. 187 REAL AP( * ), B( LDB, * ), BERR( * ), FERR( * ), 188 $ RESLTS( * ), X( LDX, * ), XACT( LDXACT, * ) 189* .. 190* 191* ===================================================================== 192* 193* .. Parameters .. 194 REAL ZERO, ONE 195 PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0 ) 196* .. 197* .. Local Scalars .. 198 LOGICAL NOTRAN, UNIT, UPPER 199 INTEGER I, IFU, IMAX, J, JC, K 200 REAL AXBI, DIFF, EPS, ERRBND, OVFL, TMP, UNFL, XNORM 201* .. 202* .. External Functions .. 203 LOGICAL LSAME 204 INTEGER ISAMAX 205 REAL SLAMCH 206 EXTERNAL LSAME, ISAMAX, SLAMCH 207* .. 208* .. Intrinsic Functions .. 209 INTRINSIC ABS, MAX, MIN 210* .. 211* .. Executable Statements .. 212* 213* Quick exit if N = 0 or NRHS = 0. 214* 215 IF( N.LE.0 .OR. NRHS.LE.0 ) THEN 216 RESLTS( 1 ) = ZERO 217 RESLTS( 2 ) = ZERO 218 RETURN 219 END IF 220* 221 EPS = SLAMCH( 'Epsilon' ) 222 UNFL = SLAMCH( 'Safe minimum' ) 223 OVFL = ONE / UNFL 224 UPPER = LSAME( UPLO, 'U' ) 225 NOTRAN = LSAME( TRANS, 'N' ) 226 UNIT = LSAME( DIAG, 'U' ) 227* 228* Test 1: Compute the maximum of 229* norm(X - XACT) / ( norm(X) * FERR ) 230* over all the vectors X and XACT using the infinity-norm. 231* 232 ERRBND = ZERO 233 DO 30 J = 1, NRHS 234 IMAX = ISAMAX( N, X( 1, J ), 1 ) 235 XNORM = MAX( ABS( X( IMAX, J ) ), UNFL ) 236 DIFF = ZERO 237 DO 10 I = 1, N 238 DIFF = MAX( DIFF, ABS( X( I, J )-XACT( I, J ) ) ) 239 10 CONTINUE 240* 241 IF( XNORM.GT.ONE ) THEN 242 GO TO 20 243 ELSE IF( DIFF.LE.OVFL*XNORM ) THEN 244 GO TO 20 245 ELSE 246 ERRBND = ONE / EPS 247 GO TO 30 248 END IF 249* 250 20 CONTINUE 251 IF( DIFF / XNORM.LE.FERR( J ) ) THEN 252 ERRBND = MAX( ERRBND, ( DIFF / XNORM ) / FERR( J ) ) 253 ELSE 254 ERRBND = ONE / EPS 255 END IF 256 30 CONTINUE 257 RESLTS( 1 ) = ERRBND 258* 259* Test 2: Compute the maximum of BERR / ( (n+1)*EPS + (*) ), where 260* (*) = (n+1)*UNFL / (min_i (abs(A)*abs(X) +abs(b))_i ) 261* 262 IFU = 0 263 IF( UNIT ) 264 $ IFU = 1 265 DO 90 K = 1, NRHS 266 DO 80 I = 1, N 267 TMP = ABS( B( I, K ) ) 268 IF( UPPER ) THEN 269 JC = ( ( I-1 )*I ) / 2 270 IF( .NOT.NOTRAN ) THEN 271 DO 40 J = 1, I - IFU 272 TMP = TMP + ABS( AP( JC+J ) )*ABS( X( J, K ) ) 273 40 CONTINUE 274 IF( UNIT ) 275 $ TMP = TMP + ABS( X( I, K ) ) 276 ELSE 277 JC = JC + I 278 IF( UNIT ) THEN 279 TMP = TMP + ABS( X( I, K ) ) 280 JC = JC + I 281 END IF 282 DO 50 J = I + IFU, N 283 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 284 JC = JC + J 285 50 CONTINUE 286 END IF 287 ELSE 288 IF( NOTRAN ) THEN 289 JC = I 290 DO 60 J = 1, I - IFU 291 TMP = TMP + ABS( AP( JC ) )*ABS( X( J, K ) ) 292 JC = JC + N - J 293 60 CONTINUE 294 IF( UNIT ) 295 $ TMP = TMP + ABS( X( I, K ) ) 296 ELSE 297 JC = ( I-1 )*( N-I ) + ( I*( I+1 ) ) / 2 298 IF( UNIT ) 299 $ TMP = TMP + ABS( X( I, K ) ) 300 DO 70 J = I + IFU, N 301 TMP = TMP + ABS( AP( JC+J-I ) )*ABS( X( J, K ) ) 302 70 CONTINUE 303 END IF 304 END IF 305 IF( I.EQ.1 ) THEN 306 AXBI = TMP 307 ELSE 308 AXBI = MIN( AXBI, TMP ) 309 END IF 310 80 CONTINUE 311 TMP = BERR( K ) / ( ( N+1 )*EPS+( N+1 )*UNFL / 312 $ MAX( AXBI, ( N+1 )*UNFL ) ) 313 IF( K.EQ.1 ) THEN 314 RESLTS( 2 ) = TMP 315 ELSE 316 RESLTS( 2 ) = MAX( RESLTS( 2 ), TMP ) 317 END IF 318 90 CONTINUE 319* 320 RETURN 321* 322* End of STPT05 323* 324 END 325