1*> \brief \b DEBCHVXX 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 DEBCHVXX( THRESH, PATH ) 12* 13* .. Scalar Arguments .. 14* DOUBLE PRECISION THRESH 15* CHARACTER*3 PATH 16* .. 17* 18* 19*> \par Purpose: 20* ============= 21*> 22*> \verbatim 23*> 24*> DEBCHVXX will run D**SVXX on a series of Hilbert matrices and then 25*> compare the error bounds returned by D**SVXX to see if the returned 26*> answer indeed falls within those bounds. 27*> 28*> Eight test ratios will be computed. The tests will pass if they are .LT. 29*> THRESH. There are two cases that are determined by 1 / (SQRT( N ) * EPS). 30*> If that value is .LE. to the component wise reciprocal condition number, 31*> it uses the guaranteed case, other wise it uses the unguaranteed case. 32*> 33*> Test ratios: 34*> Let Xc be X_computed and Xt be X_truth. 35*> The norm used is the infinity norm. 36*> 37*> Let A be the guaranteed case and B be the unguaranteed case. 38*> 39*> 1. Normwise guaranteed forward error bound. 40*> A: norm ( abs( Xc - Xt ) / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ) and 41*> ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N),10) * EPS. 42*> If these conditions are met, the test ratio is set to be 43*> ERRBND( *, nwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS. 44*> B: For this case, CGESVXX should just return 1. If it is less than 45*> one, treat it the same as in 1A. Otherwise it fails. (Set test 46*> ratio to ERRBND( *, nwise_i, bnd_i ) * THRESH?) 47*> 48*> 2. Componentwise guaranteed forward error bound. 49*> A: norm ( abs( Xc(j) - Xt(j) ) ) / norm (Xt(j)) .LE. ERRBND( *, cwise_i, bnd_i ) 50*> for all j .AND. ERRBND( *, cwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS. 51*> If these conditions are met, the test ratio is set to be 52*> ERRBND( *, cwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS. 53*> B: Same as normwise test ratio. 54*> 55*> 3. Backwards error. 56*> A: The test ratio is set to BERR/EPS. 57*> B: Same test ratio. 58*> 59*> 4. Reciprocal condition number. 60*> A: A condition number is computed with Xt and compared with the one 61*> returned from CGESVXX. Let RCONDc be the RCOND returned by D**SVXX 62*> and RCONDt be the RCOND from the truth value. Test ratio is set to 63*> MAX(RCONDc/RCONDt, RCONDt/RCONDc). 64*> B: Test ratio is set to 1 / (EPS * RCONDc). 65*> 66*> 5. Reciprocal normwise condition number. 67*> A: The test ratio is set to 68*> MAX(ERRBND( *, nwise_i, cond_i ) / NCOND, NCOND / ERRBND( *, nwise_i, cond_i )). 69*> B: Test ratio is set to 1 / (EPS * ERRBND( *, nwise_i, cond_i )). 70*> 71*> 6. Reciprocal componentwise condition number. 72*> A: Test ratio is set to 73*> MAX(ERRBND( *, cwise_i, cond_i ) / CCOND, CCOND / ERRBND( *, cwise_i, cond_i )). 74*> B: Test ratio is set to 1 / (EPS * ERRBND( *, cwise_i, cond_i )). 75*> 76*> .. Parameters .. 77*> NMAX is determined by the largest number in the inverse of the hilbert 78*> matrix. Precision is exhausted when the largest entry in it is greater 79*> than 2 to the power of the number of bits in the fraction of the data 80*> type used plus one, which is 24 for single precision. 81*> NMAX should be 6 for single and 11 for double. 82*> \endverbatim 83* 84* Authors: 85* ======== 86* 87*> \author Univ. of Tennessee 88*> \author Univ. of California Berkeley 89*> \author Univ. of Colorado Denver 90*> \author NAG Ltd. 91* 92*> \ingroup double_lin 93* 94* ===================================================================== 95 SUBROUTINE DEBCHVXX( THRESH, PATH ) 96 IMPLICIT NONE 97* .. Scalar Arguments .. 98 DOUBLE PRECISION THRESH 99 CHARACTER*3 PATH 100 101 INTEGER NMAX, NPARAMS, NERRBND, NTESTS, KL, KU 102 PARAMETER (NMAX = 10, NPARAMS = 2, NERRBND = 3, 103 $ NTESTS = 6) 104 105* .. Local Scalars .. 106 INTEGER N, NRHS, INFO, I ,J, k, NFAIL, LDA, 107 $ N_AUX_TESTS, LDAB, LDAFB 108 CHARACTER FACT, TRANS, UPLO, EQUED 109 CHARACTER*2 C2 110 CHARACTER(3) NGUAR, CGUAR 111 LOGICAL printed_guide 112 DOUBLE PRECISION NCOND, CCOND, M, NORMDIF, NORMT, RCOND, 113 $ RNORM, RINORM, SUMR, SUMRI, EPS, 114 $ BERR(NMAX), RPVGRW, ORCOND, 115 $ CWISE_ERR, NWISE_ERR, CWISE_BND, NWISE_BND, 116 $ CWISE_RCOND, NWISE_RCOND, 117 $ CONDTHRESH, ERRTHRESH 118 119* .. Local Arrays .. 120 DOUBLE PRECISION TSTRAT(NTESTS), RINV(NMAX), PARAMS(NPARAMS), 121 $ S(NMAX),R(NMAX),C(NMAX), DIFF(NMAX, NMAX), 122 $ ERRBND_N(NMAX*3), ERRBND_C(NMAX*3), 123 $ A(NMAX,NMAX),INVHILB(NMAX,NMAX),X(NMAX,NMAX), 124 $ AB( (NMAX-1)+(NMAX-1)+1, NMAX ), 125 $ ABCOPY( (NMAX-1)+(NMAX-1)+1, NMAX ), 126 $ AFB( 2*(NMAX-1)+(NMAX-1)+1, NMAX ), 127 $ WORK(NMAX*3*5), AF(NMAX, NMAX),B(NMAX, NMAX), 128 $ ACOPY(NMAX, NMAX) 129 INTEGER IPIV(NMAX), IWORK(3*NMAX) 130 131* .. External Functions .. 132 DOUBLE PRECISION DLAMCH 133 134* .. External Subroutines .. 135 EXTERNAL DLAHILB, DGESVXX, DPOSVXX, DSYSVXX, 136 $ DGBSVXX, DLACPY, LSAMEN 137 LOGICAL LSAMEN 138 139* .. Intrinsic Functions .. 140 INTRINSIC SQRT, MAX, ABS, DBLE 141 142* .. Parameters .. 143 INTEGER NWISE_I, CWISE_I 144 PARAMETER (NWISE_I = 1, CWISE_I = 1) 145 INTEGER BND_I, COND_I 146 PARAMETER (BND_I = 2, COND_I = 3) 147 148* Create the loop to test out the Hilbert matrices 149 150 FACT = 'E' 151 UPLO = 'U' 152 TRANS = 'N' 153 EQUED = 'N' 154 EPS = DLAMCH('Epsilon') 155 NFAIL = 0 156 N_AUX_TESTS = 0 157 LDA = NMAX 158 LDAB = (NMAX-1)+(NMAX-1)+1 159 LDAFB = 2*(NMAX-1)+(NMAX-1)+1 160 C2 = PATH( 2: 3 ) 161 162* Main loop to test the different Hilbert Matrices. 163 164 printed_guide = .false. 165 166 DO N = 1 , NMAX 167 PARAMS(1) = -1 168 PARAMS(2) = -1 169 170 KL = N-1 171 KU = N-1 172 NRHS = n 173 M = MAX(SQRT(DBLE(N)), 10.0D+0) 174 175* Generate the Hilbert matrix, its inverse, and the 176* right hand side, all scaled by the LCM(1,..,2N-1). 177 CALL DLAHILB(N, N, A, LDA, INVHILB, LDA, B, LDA, WORK, INFO) 178 179* Copy A into ACOPY. 180 CALL DLACPY('ALL', N, N, A, NMAX, ACOPY, NMAX) 181 182* Store A in band format for GB tests 183 DO J = 1, N 184 DO I = 1, KL+KU+1 185 AB( I, J ) = 0.0D+0 186 END DO 187 END DO 188 DO J = 1, N 189 DO I = MAX( 1, J-KU ), MIN( N, J+KL ) 190 AB( KU+1+I-J, J ) = A( I, J ) 191 END DO 192 END DO 193 194* Copy AB into ABCOPY. 195 DO J = 1, N 196 DO I = 1, KL+KU+1 197 ABCOPY( I, J ) = 0.0D+0 198 END DO 199 END DO 200 CALL DLACPY('ALL', KL+KU+1, N, AB, LDAB, ABCOPY, LDAB) 201 202* Call D**SVXX with default PARAMS and N_ERR_BND = 3. 203 IF ( LSAMEN( 2, C2, 'SY' ) ) THEN 204 CALL DSYSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA, 205 $ IPIV, EQUED, S, B, LDA, X, LDA, ORCOND, 206 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, 207 $ PARAMS, WORK, IWORK, INFO) 208 ELSE IF ( LSAMEN( 2, C2, 'PO' ) ) THEN 209 CALL DPOSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA, 210 $ EQUED, S, B, LDA, X, LDA, ORCOND, 211 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, 212 $ PARAMS, WORK, IWORK, INFO) 213 ELSE IF ( LSAMEN( 2, C2, 'GB' ) ) THEN 214 CALL DGBSVXX(FACT, TRANS, N, KL, KU, NRHS, ABCOPY, 215 $ LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, 216 $ LDA, X, LDA, ORCOND, RPVGRW, BERR, NERRBND, 217 $ ERRBND_N, ERRBND_C, NPARAMS, PARAMS, WORK, IWORK, 218 $ INFO) 219 ELSE 220 CALL DGESVXX(FACT, TRANS, N, NRHS, ACOPY, LDA, AF, LDA, 221 $ IPIV, EQUED, R, C, B, LDA, X, LDA, ORCOND, 222 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, 223 $ PARAMS, WORK, IWORK, INFO) 224 END IF 225 226 N_AUX_TESTS = N_AUX_TESTS + 1 227 IF (ORCOND .LT. EPS) THEN 228! Either factorization failed or the matrix is flagged, and 1 <= 229! INFO <= N+1. We don't decide based on rcond anymore. 230! IF (INFO .EQ. 0 .OR. INFO .GT. N+1) THEN 231! NFAIL = NFAIL + 1 232! WRITE (*, FMT=8000) N, INFO, ORCOND, RCOND 233! END IF 234 ELSE 235! Either everything succeeded (INFO == 0) or some solution failed 236! to converge (INFO > N+1). 237 IF (INFO .GT. 0 .AND. INFO .LE. N+1) THEN 238 NFAIL = NFAIL + 1 239 WRITE (*, FMT=8000) C2, N, INFO, ORCOND, RCOND 240 END IF 241 END IF 242 243* Calculating the difference between D**SVXX's X and the true X. 244 DO I = 1,N 245 DO J =1,NRHS 246 DIFF(I,J) = X(I,J) - INVHILB(I,J) 247 END DO 248 END DO 249 250* Calculating the RCOND 251 RNORM = 0.0D+0 252 RINORM = 0.0D+0 253 IF ( LSAMEN( 2, C2, 'PO' ) .OR. LSAMEN( 2, C2, 'SY' ) ) THEN 254 DO I = 1, N 255 SUMR = 0.0D+0 256 SUMRI = 0.0D+0 257 DO J = 1, N 258 SUMR = SUMR + S(I) * ABS(A(I,J)) * S(J) 259 SUMRI = SUMRI + ABS(INVHILB(I, J)) / (S(J) * S(I)) 260 261 END DO 262 RNORM = MAX(RNORM,SUMR) 263 RINORM = MAX(RINORM,SUMRI) 264 END DO 265 ELSE IF ( LSAMEN( 2, C2, 'GE' ) .OR. LSAMEN( 2, C2, 'GB' ) ) 266 $ THEN 267 DO I = 1, N 268 SUMR = 0.0D+0 269 SUMRI = 0.0D+0 270 DO J = 1, N 271 SUMR = SUMR + R(I) * ABS(A(I,J)) * C(J) 272 SUMRI = SUMRI + ABS(INVHILB(I, J)) / (R(J) * C(I)) 273 END DO 274 RNORM = MAX(RNORM,SUMR) 275 RINORM = MAX(RINORM,SUMRI) 276 END DO 277 END IF 278 279 RNORM = RNORM / ABS(A(1, 1)) 280 RCOND = 1.0D+0/(RNORM * RINORM) 281 282* Calculating the R for normwise rcond. 283 DO I = 1, N 284 RINV(I) = 0.0D+0 285 END DO 286 DO J = 1, N 287 DO I = 1, N 288 RINV(I) = RINV(I) + ABS(A(I,J)) 289 END DO 290 END DO 291 292* Calculating the Normwise rcond. 293 RINORM = 0.0D+0 294 DO I = 1, N 295 SUMRI = 0.0D+0 296 DO J = 1, N 297 SUMRI = SUMRI + ABS(INVHILB(I,J) * RINV(J)) 298 END DO 299 RINORM = MAX(RINORM, SUMRI) 300 END DO 301 302! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm 303! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix) 304 NCOND = ABS(A(1,1)) / RINORM 305 306 CONDTHRESH = M * EPS 307 ERRTHRESH = M * EPS 308 309 DO K = 1, NRHS 310 NORMT = 0.0D+0 311 NORMDIF = 0.0D+0 312 CWISE_ERR = 0.0D+0 313 DO I = 1, N 314 NORMT = MAX(ABS(INVHILB(I, K)), NORMT) 315 NORMDIF = MAX(ABS(X(I,K) - INVHILB(I,K)), NORMDIF) 316 IF (INVHILB(I,K) .NE. 0.0D+0) THEN 317 CWISE_ERR = MAX(ABS(X(I,K) - INVHILB(I,K)) 318 $ /ABS(INVHILB(I,K)), CWISE_ERR) 319 ELSE IF (X(I, K) .NE. 0.0D+0) THEN 320 CWISE_ERR = DLAMCH('OVERFLOW') 321 END IF 322 END DO 323 IF (NORMT .NE. 0.0D+0) THEN 324 NWISE_ERR = NORMDIF / NORMT 325 ELSE IF (NORMDIF .NE. 0.0D+0) THEN 326 NWISE_ERR = DLAMCH('OVERFLOW') 327 ELSE 328 NWISE_ERR = 0.0D+0 329 ENDIF 330 331 DO I = 1, N 332 RINV(I) = 0.0D+0 333 END DO 334 DO J = 1, N 335 DO I = 1, N 336 RINV(I) = RINV(I) + ABS(A(I, J) * INVHILB(J, K)) 337 END DO 338 END DO 339 RINORM = 0.0D+0 340 DO I = 1, N 341 SUMRI = 0.0D+0 342 DO J = 1, N 343 SUMRI = SUMRI 344 $ + ABS(INVHILB(I, J) * RINV(J) / INVHILB(I, K)) 345 END DO 346 RINORM = MAX(RINORM, SUMRI) 347 END DO 348! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm 349! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix) 350 CCOND = ABS(A(1,1))/RINORM 351 352! Forward error bound tests 353 NWISE_BND = ERRBND_N(K + (BND_I-1)*NRHS) 354 CWISE_BND = ERRBND_C(K + (BND_I-1)*NRHS) 355 NWISE_RCOND = ERRBND_N(K + (COND_I-1)*NRHS) 356 CWISE_RCOND = ERRBND_C(K + (COND_I-1)*NRHS) 357! write (*,*) 'nwise : ', n, k, ncond, nwise_rcond, 358! $ condthresh, ncond.ge.condthresh 359! write (*,*) 'nwise2: ', k, nwise_bnd, nwise_err, errthresh 360 IF (NCOND .GE. CONDTHRESH) THEN 361 NGUAR = 'YES' 362 IF (NWISE_BND .GT. ERRTHRESH) THEN 363 TSTRAT(1) = 1/(2.0D+0*EPS) 364 ELSE 365 IF (NWISE_BND .NE. 0.0D+0) THEN 366 TSTRAT(1) = NWISE_ERR / NWISE_BND 367 ELSE IF (NWISE_ERR .NE. 0.0D+0) THEN 368 TSTRAT(1) = 1/(16.0*EPS) 369 ELSE 370 TSTRAT(1) = 0.0D+0 371 END IF 372 IF (TSTRAT(1) .GT. 1.0D+0) THEN 373 TSTRAT(1) = 1/(4.0D+0*EPS) 374 END IF 375 END IF 376 ELSE 377 NGUAR = 'NO' 378 IF (NWISE_BND .LT. 1.0D+0) THEN 379 TSTRAT(1) = 1/(8.0D+0*EPS) 380 ELSE 381 TSTRAT(1) = 1.0D+0 382 END IF 383 END IF 384! write (*,*) 'cwise : ', n, k, ccond, cwise_rcond, 385! $ condthresh, ccond.ge.condthresh 386! write (*,*) 'cwise2: ', k, cwise_bnd, cwise_err, errthresh 387 IF (CCOND .GE. CONDTHRESH) THEN 388 CGUAR = 'YES' 389 IF (CWISE_BND .GT. ERRTHRESH) THEN 390 TSTRAT(2) = 1/(2.0D+0*EPS) 391 ELSE 392 IF (CWISE_BND .NE. 0.0D+0) THEN 393 TSTRAT(2) = CWISE_ERR / CWISE_BND 394 ELSE IF (CWISE_ERR .NE. 0.0D+0) THEN 395 TSTRAT(2) = 1/(16.0D+0*EPS) 396 ELSE 397 TSTRAT(2) = 0.0D+0 398 END IF 399 IF (TSTRAT(2) .GT. 1.0D+0) TSTRAT(2) = 1/(4.0D+0*EPS) 400 END IF 401 ELSE 402 CGUAR = 'NO' 403 IF (CWISE_BND .LT. 1.0D+0) THEN 404 TSTRAT(2) = 1/(8.0D+0*EPS) 405 ELSE 406 TSTRAT(2) = 1.0D+0 407 END IF 408 END IF 409 410! Backwards error test 411 TSTRAT(3) = BERR(K)/EPS 412 413! Condition number tests 414 TSTRAT(4) = RCOND / ORCOND 415 IF (RCOND .GE. CONDTHRESH .AND. TSTRAT(4) .LT. 1.0D+0) 416 $ TSTRAT(4) = 1.0D+0 / TSTRAT(4) 417 418 TSTRAT(5) = NCOND / NWISE_RCOND 419 IF (NCOND .GE. CONDTHRESH .AND. TSTRAT(5) .LT. 1.0D+0) 420 $ TSTRAT(5) = 1.0D+0 / TSTRAT(5) 421 422 TSTRAT(6) = CCOND / NWISE_RCOND 423 IF (CCOND .GE. CONDTHRESH .AND. TSTRAT(6) .LT. 1.0D+0) 424 $ TSTRAT(6) = 1.0D+0 / TSTRAT(6) 425 426 DO I = 1, NTESTS 427 IF (TSTRAT(I) .GT. THRESH) THEN 428 IF (.NOT.PRINTED_GUIDE) THEN 429 WRITE(*,*) 430 WRITE( *, 9996) 1 431 WRITE( *, 9995) 2 432 WRITE( *, 9994) 3 433 WRITE( *, 9993) 4 434 WRITE( *, 9992) 5 435 WRITE( *, 9991) 6 436 WRITE( *, 9990) 7 437 WRITE( *, 9989) 8 438 WRITE(*,*) 439 PRINTED_GUIDE = .TRUE. 440 END IF 441 WRITE( *, 9999) C2, N, K, NGUAR, CGUAR, I, TSTRAT(I) 442 NFAIL = NFAIL + 1 443 END IF 444 END DO 445 END DO 446 447c$$$ WRITE(*,*) 448c$$$ WRITE(*,*) 'Normwise Error Bounds' 449c$$$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,nwise_i,bnd_i) 450c$$$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,nwise_i,cond_i) 451c$$$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,nwise_i,rawbnd_i) 452c$$$ WRITE(*,*) 453c$$$ WRITE(*,*) 'Componentwise Error Bounds' 454c$$$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,cwise_i,bnd_i) 455c$$$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,cwise_i,cond_i) 456c$$$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,cwise_i,rawbnd_i) 457c$$$ print *, 'Info: ', info 458c$$$ WRITE(*,*) 459* WRITE(*,*) 'TSTRAT: ',TSTRAT 460 461 END DO 462 463 WRITE(*,*) 464 IF( NFAIL .GT. 0 ) THEN 465 WRITE(*,9998) C2, NFAIL, NTESTS*N+N_AUX_TESTS 466 ELSE 467 WRITE(*,9997) C2 468 END IF 469 9999 FORMAT( ' D', A2, 'SVXX: N =', I2, ', RHS = ', I2, 470 $ ', NWISE GUAR. = ', A, ', CWISE GUAR. = ', A, 471 $ ' test(',I1,') =', G12.5 ) 472 9998 FORMAT( ' D', A2, 'SVXX: ', I6, ' out of ', I6, 473 $ ' tests failed to pass the threshold' ) 474 9997 FORMAT( ' D', A2, 'SVXX passed the tests of error bounds' ) 475* Test ratios. 476 9996 FORMAT( 3X, I2, ': Normwise guaranteed forward error', / 5X, 477 $ 'Guaranteed case: if norm ( abs( Xc - Xt )', 478 $ ' / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ), then', 479 $ / 5X, 480 $ 'ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS') 481 9995 FORMAT( 3X, I2, ': Componentwise guaranteed forward error' ) 482 9994 FORMAT( 3X, I2, ': Backwards error' ) 483 9993 FORMAT( 3X, I2, ': Reciprocal condition number' ) 484 9992 FORMAT( 3X, I2, ': Reciprocal normwise condition number' ) 485 9991 FORMAT( 3X, I2, ': Raw normwise error estimate' ) 486 9990 FORMAT( 3X, I2, ': Reciprocal componentwise condition number' ) 487 9989 FORMAT( 3X, I2, ': Raw componentwise error estimate' ) 488 489 8000 FORMAT( ' D', A2, 'SVXX: N =', I2, ', INFO = ', I3, 490 $ ', ORCOND = ', G12.5, ', real RCOND = ', G12.5 ) 491* 492* End of DEBCHVXX 493* 494 END 495