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*> \date November 2011 93* 94*> \ingroup double_lin 95* 96* ===================================================================== 97 SUBROUTINE DEBCHVXX( THRESH, PATH ) 98 IMPLICIT NONE 99* .. Scalar Arguments .. 100 DOUBLE PRECISION THRESH 101 CHARACTER*3 PATH 102 103 INTEGER NMAX, NPARAMS, NERRBND, NTESTS, KL, KU 104 PARAMETER (NMAX = 10, NPARAMS = 2, NERRBND = 3, 105 $ NTESTS = 6) 106 107* .. Local Scalars .. 108 INTEGER N, NRHS, INFO, I ,J, k, NFAIL, LDA, 109 $ N_AUX_TESTS, LDAB, LDAFB 110 CHARACTER FACT, TRANS, UPLO, EQUED 111 CHARACTER*2 C2 112 CHARACTER(3) NGUAR, CGUAR 113 LOGICAL printed_guide 114 DOUBLE PRECISION NCOND, CCOND, M, NORMDIF, NORMT, RCOND, 115 $ RNORM, RINORM, SUMR, SUMRI, EPS, 116 $ BERR(NMAX), RPVGRW, ORCOND, 117 $ CWISE_ERR, NWISE_ERR, CWISE_BND, NWISE_BND, 118 $ CWISE_RCOND, NWISE_RCOND, 119 $ CONDTHRESH, ERRTHRESH 120 121* .. Local Arrays .. 122 DOUBLE PRECISION TSTRAT(NTESTS), RINV(NMAX), PARAMS(NPARAMS), 123 $ S(NMAX),R(NMAX),C(NMAX), DIFF(NMAX, NMAX), 124 $ ERRBND_N(NMAX*3), ERRBND_C(NMAX*3), 125 $ A(NMAX,NMAX),INVHILB(NMAX,NMAX),X(NMAX,NMAX), 126 $ AB( (NMAX-1)+(NMAX-1)+1, NMAX ), 127 $ ABCOPY( (NMAX-1)+(NMAX-1)+1, NMAX ), 128 $ AFB( 2*(NMAX-1)+(NMAX-1)+1, NMAX ), 129 $ WORK(NMAX*3*5), AF(NMAX, NMAX),B(NMAX, NMAX), 130 $ ACOPY(NMAX, NMAX) 131 INTEGER IPIV(NMAX), IWORK(3*NMAX) 132 133* .. External Functions .. 134 DOUBLE PRECISION DLAMCH 135 136* .. External Subroutines .. 137 EXTERNAL DLAHILB, DGESVXX, DPOSVXX, DSYSVXX, 138 $ DGBSVXX, DLACPY, LSAMEN 139 LOGICAL LSAMEN 140 141* .. Intrinsic Functions .. 142 INTRINSIC SQRT, MAX, ABS, DBLE 143 144* .. Parameters .. 145 INTEGER NWISE_I, CWISE_I 146 PARAMETER (NWISE_I = 1, CWISE_I = 1) 147 INTEGER BND_I, COND_I 148 PARAMETER (BND_I = 2, COND_I = 3) 149 150* Create the loop to test out the Hilbert matrices 151 152 FACT = 'E' 153 UPLO = 'U' 154 TRANS = 'N' 155 EQUED = 'N' 156 EPS = DLAMCH('Epsilon') 157 NFAIL = 0 158 N_AUX_TESTS = 0 159 LDA = NMAX 160 LDAB = (NMAX-1)+(NMAX-1)+1 161 LDAFB = 2*(NMAX-1)+(NMAX-1)+1 162 C2 = PATH( 2: 3 ) 163 164* Main loop to test the different Hilbert Matrices. 165 166 printed_guide = .false. 167 168 DO N = 1 , NMAX 169 PARAMS(1) = -1 170 PARAMS(2) = -1 171 172 KL = N-1 173 KU = N-1 174 NRHS = n 175 M = MAX(SQRT(DBLE(N)), 10.0D+0) 176 177* Generate the Hilbert matrix, its inverse, and the 178* right hand side, all scaled by the LCM(1,..,2N-1). 179 CALL DLAHILB(N, N, A, LDA, INVHILB, LDA, B, LDA, WORK, INFO) 180 181* Copy A into ACOPY. 182 CALL DLACPY('ALL', N, N, A, NMAX, ACOPY, NMAX) 183 184* Store A in band format for GB tests 185 DO J = 1, N 186 DO I = 1, KL+KU+1 187 AB( I, J ) = 0.0D+0 188 END DO 189 END DO 190 DO J = 1, N 191 DO I = MAX( 1, J-KU ), MIN( N, J+KL ) 192 AB( KU+1+I-J, J ) = A( I, J ) 193 END DO 194 END DO 195 196* Copy AB into ABCOPY. 197 DO J = 1, N 198 DO I = 1, KL+KU+1 199 ABCOPY( I, J ) = 0.0D+0 200 END DO 201 END DO 202 CALL DLACPY('ALL', KL+KU+1, N, AB, LDAB, ABCOPY, LDAB) 203 204* Call D**SVXX with default PARAMS and N_ERR_BND = 3. 205 IF ( LSAMEN( 2, C2, 'SY' ) ) THEN 206 CALL DSYSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA, 207 $ IPIV, EQUED, S, B, LDA, X, LDA, ORCOND, 208 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, 209 $ PARAMS, WORK, IWORK, INFO) 210 ELSE IF ( LSAMEN( 2, C2, 'PO' ) ) THEN 211 CALL DPOSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA, 212 $ EQUED, S, B, LDA, X, LDA, ORCOND, 213 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, 214 $ PARAMS, WORK, IWORK, INFO) 215 ELSE IF ( LSAMEN( 2, C2, 'GB' ) ) THEN 216 CALL DGBSVXX(FACT, TRANS, N, KL, KU, NRHS, ABCOPY, 217 $ LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B, 218 $ LDA, X, LDA, ORCOND, RPVGRW, BERR, NERRBND, 219 $ ERRBND_N, ERRBND_C, NPARAMS, PARAMS, WORK, IWORK, 220 $ INFO) 221 ELSE 222 CALL DGESVXX(FACT, TRANS, N, NRHS, ACOPY, LDA, AF, LDA, 223 $ IPIV, EQUED, R, C, B, LDA, X, LDA, ORCOND, 224 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS, 225 $ PARAMS, WORK, IWORK, INFO) 226 END IF 227 228 N_AUX_TESTS = N_AUX_TESTS + 1 229 IF (ORCOND .LT. EPS) THEN 230! Either factorization failed or the matrix is flagged, and 1 <= 231! INFO <= N+1. We don't decide based on rcond anymore. 232! IF (INFO .EQ. 0 .OR. INFO .GT. N+1) THEN 233! NFAIL = NFAIL + 1 234! WRITE (*, FMT=8000) N, INFO, ORCOND, RCOND 235! END IF 236 ELSE 237! Either everything succeeded (INFO == 0) or some solution failed 238! to converge (INFO > N+1). 239 IF (INFO .GT. 0 .AND. INFO .LE. N+1) THEN 240 NFAIL = NFAIL + 1 241 WRITE (*, FMT=8000) C2, N, INFO, ORCOND, RCOND 242 END IF 243 END IF 244 245* Calculating the difference between D**SVXX's X and the true X. 246 DO I = 1,N 247 DO J =1,NRHS 248 DIFF(I,J) = X(I,J) - INVHILB(I,J) 249 END DO 250 END DO 251 252* Calculating the RCOND 253 RNORM = 0.0D+0 254 RINORM = 0.0D+0 255 IF ( LSAMEN( 2, C2, 'PO' ) .OR. LSAMEN( 2, C2, 'SY' ) ) THEN 256 DO I = 1, N 257 SUMR = 0.0D+0 258 SUMRI = 0.0D+0 259 DO J = 1, N 260 SUMR = SUMR + S(I) * ABS(A(I,J)) * S(J) 261 SUMRI = SUMRI + ABS(INVHILB(I, J)) / (S(J) * S(I)) 262 263 END DO 264 RNORM = MAX(RNORM,SUMR) 265 RINORM = MAX(RINORM,SUMRI) 266 END DO 267 ELSE IF ( LSAMEN( 2, C2, 'GE' ) .OR. LSAMEN( 2, C2, 'GB' ) ) 268 $ THEN 269 DO I = 1, N 270 SUMR = 0.0D+0 271 SUMRI = 0.0D+0 272 DO J = 1, N 273 SUMR = SUMR + R(I) * ABS(A(I,J)) * C(J) 274 SUMRI = SUMRI + ABS(INVHILB(I, J)) / (R(J) * C(I)) 275 END DO 276 RNORM = MAX(RNORM,SUMR) 277 RINORM = MAX(RINORM,SUMRI) 278 END DO 279 END IF 280 281 RNORM = RNORM / ABS(A(1, 1)) 282 RCOND = 1.0D+0/(RNORM * RINORM) 283 284* Calculating the R for normwise rcond. 285 DO I = 1, N 286 RINV(I) = 0.0D+0 287 END DO 288 DO J = 1, N 289 DO I = 1, N 290 RINV(I) = RINV(I) + ABS(A(I,J)) 291 END DO 292 END DO 293 294* Calculating the Normwise rcond. 295 RINORM = 0.0D+0 296 DO I = 1, N 297 SUMRI = 0.0D+0 298 DO J = 1, N 299 SUMRI = SUMRI + ABS(INVHILB(I,J) * RINV(J)) 300 END DO 301 RINORM = MAX(RINORM, SUMRI) 302 END DO 303 304! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm 305! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix) 306 NCOND = ABS(A(1,1)) / RINORM 307 308 CONDTHRESH = M * EPS 309 ERRTHRESH = M * EPS 310 311 DO K = 1, NRHS 312 NORMT = 0.0D+0 313 NORMDIF = 0.0D+0 314 CWISE_ERR = 0.0D+0 315 DO I = 1, N 316 NORMT = MAX(ABS(INVHILB(I, K)), NORMT) 317 NORMDIF = MAX(ABS(X(I,K) - INVHILB(I,K)), NORMDIF) 318 IF (INVHILB(I,K) .NE. 0.0D+0) THEN 319 CWISE_ERR = MAX(ABS(X(I,K) - INVHILB(I,K)) 320 $ /ABS(INVHILB(I,K)), CWISE_ERR) 321 ELSE IF (X(I, K) .NE. 0.0D+0) THEN 322 CWISE_ERR = DLAMCH('OVERFLOW') 323 END IF 324 END DO 325 IF (NORMT .NE. 0.0D+0) THEN 326 NWISE_ERR = NORMDIF / NORMT 327 ELSE IF (NORMDIF .NE. 0.0D+0) THEN 328 NWISE_ERR = DLAMCH('OVERFLOW') 329 ELSE 330 NWISE_ERR = 0.0D+0 331 ENDIF 332 333 DO I = 1, N 334 RINV(I) = 0.0D+0 335 END DO 336 DO J = 1, N 337 DO I = 1, N 338 RINV(I) = RINV(I) + ABS(A(I, J) * INVHILB(J, K)) 339 END DO 340 END DO 341 RINORM = 0.0D+0 342 DO I = 1, N 343 SUMRI = 0.0D+0 344 DO J = 1, N 345 SUMRI = SUMRI 346 $ + ABS(INVHILB(I, J) * RINV(J) / INVHILB(I, K)) 347 END DO 348 RINORM = MAX(RINORM, SUMRI) 349 END DO 350! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm 351! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix) 352 CCOND = ABS(A(1,1))/RINORM 353 354! Forward error bound tests 355 NWISE_BND = ERRBND_N(K + (BND_I-1)*NRHS) 356 CWISE_BND = ERRBND_C(K + (BND_I-1)*NRHS) 357 NWISE_RCOND = ERRBND_N(K + (COND_I-1)*NRHS) 358 CWISE_RCOND = ERRBND_C(K + (COND_I-1)*NRHS) 359! write (*,*) 'nwise : ', n, k, ncond, nwise_rcond, 360! $ condthresh, ncond.ge.condthresh 361! write (*,*) 'nwise2: ', k, nwise_bnd, nwise_err, errthresh 362 IF (NCOND .GE. CONDTHRESH) THEN 363 NGUAR = 'YES' 364 IF (NWISE_BND .GT. ERRTHRESH) THEN 365 TSTRAT(1) = 1/(2.0D+0*EPS) 366 ELSE 367 IF (NWISE_BND .NE. 0.0D+0) THEN 368 TSTRAT(1) = NWISE_ERR / NWISE_BND 369 ELSE IF (NWISE_ERR .NE. 0.0D+0) THEN 370 TSTRAT(1) = 1/(16.0*EPS) 371 ELSE 372 TSTRAT(1) = 0.0D+0 373 END IF 374 IF (TSTRAT(1) .GT. 1.0D+0) THEN 375 TSTRAT(1) = 1/(4.0D+0*EPS) 376 END IF 377 END IF 378 ELSE 379 NGUAR = 'NO' 380 IF (NWISE_BND .LT. 1.0D+0) THEN 381 TSTRAT(1) = 1/(8.0D+0*EPS) 382 ELSE 383 TSTRAT(1) = 1.0D+0 384 END IF 385 END IF 386! write (*,*) 'cwise : ', n, k, ccond, cwise_rcond, 387! $ condthresh, ccond.ge.condthresh 388! write (*,*) 'cwise2: ', k, cwise_bnd, cwise_err, errthresh 389 IF (CCOND .GE. CONDTHRESH) THEN 390 CGUAR = 'YES' 391 IF (CWISE_BND .GT. ERRTHRESH) THEN 392 TSTRAT(2) = 1/(2.0D+0*EPS) 393 ELSE 394 IF (CWISE_BND .NE. 0.0D+0) THEN 395 TSTRAT(2) = CWISE_ERR / CWISE_BND 396 ELSE IF (CWISE_ERR .NE. 0.0D+0) THEN 397 TSTRAT(2) = 1/(16.0D+0*EPS) 398 ELSE 399 TSTRAT(2) = 0.0D+0 400 END IF 401 IF (TSTRAT(2) .GT. 1.0D+0) TSTRAT(2) = 1/(4.0D+0*EPS) 402 END IF 403 ELSE 404 CGUAR = 'NO' 405 IF (CWISE_BND .LT. 1.0D+0) THEN 406 TSTRAT(2) = 1/(8.0D+0*EPS) 407 ELSE 408 TSTRAT(2) = 1.0D+0 409 END IF 410 END IF 411 412! Backwards error test 413 TSTRAT(3) = BERR(K)/EPS 414 415! Condition number tests 416 TSTRAT(4) = RCOND / ORCOND 417 IF (RCOND .GE. CONDTHRESH .AND. TSTRAT(4) .LT. 1.0D+0) 418 $ TSTRAT(4) = 1.0D+0 / TSTRAT(4) 419 420 TSTRAT(5) = NCOND / NWISE_RCOND 421 IF (NCOND .GE. CONDTHRESH .AND. TSTRAT(5) .LT. 1.0D+0) 422 $ TSTRAT(5) = 1.0D+0 / TSTRAT(5) 423 424 TSTRAT(6) = CCOND / NWISE_RCOND 425 IF (CCOND .GE. CONDTHRESH .AND. TSTRAT(6) .LT. 1.0D+0) 426 $ TSTRAT(6) = 1.0D+0 / TSTRAT(6) 427 428 DO I = 1, NTESTS 429 IF (TSTRAT(I) .GT. THRESH) THEN 430 IF (.NOT.PRINTED_GUIDE) THEN 431 WRITE(*,*) 432 WRITE( *, 9996) 1 433 WRITE( *, 9995) 2 434 WRITE( *, 9994) 3 435 WRITE( *, 9993) 4 436 WRITE( *, 9992) 5 437 WRITE( *, 9991) 6 438 WRITE( *, 9990) 7 439 WRITE( *, 9989) 8 440 WRITE(*,*) 441 PRINTED_GUIDE = .TRUE. 442 END IF 443 WRITE( *, 9999) C2, N, K, NGUAR, CGUAR, I, TSTRAT(I) 444 NFAIL = NFAIL + 1 445 END IF 446 END DO 447 END DO 448 449c$$$ WRITE(*,*) 450c$$$ WRITE(*,*) 'Normwise Error Bounds' 451c$$$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,nwise_i,bnd_i) 452c$$$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,nwise_i,cond_i) 453c$$$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,nwise_i,rawbnd_i) 454c$$$ WRITE(*,*) 455c$$$ WRITE(*,*) 'Componentwise Error Bounds' 456c$$$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,cwise_i,bnd_i) 457c$$$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,cwise_i,cond_i) 458c$$$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,cwise_i,rawbnd_i) 459c$$$ print *, 'Info: ', info 460c$$$ WRITE(*,*) 461* WRITE(*,*) 'TSTRAT: ',TSTRAT 462 463 END DO 464 465 WRITE(*,*) 466 IF( NFAIL .GT. 0 ) THEN 467 WRITE(*,9998) C2, NFAIL, NTESTS*N+N_AUX_TESTS 468 ELSE 469 WRITE(*,9997) C2 470 END IF 471 9999 FORMAT( ' D', A2, 'SVXX: N =', I2, ', RHS = ', I2, 472 $ ', NWISE GUAR. = ', A, ', CWISE GUAR. = ', A, 473 $ ' test(',I1,') =', G12.5 ) 474 9998 FORMAT( ' D', A2, 'SVXX: ', I6, ' out of ', I6, 475 $ ' tests failed to pass the threshold' ) 476 9997 FORMAT( ' D', A2, 'SVXX passed the tests of error bounds' ) 477* Test ratios. 478 9996 FORMAT( 3X, I2, ': Normwise guaranteed forward error', / 5X, 479 $ 'Guaranteed case: if norm ( abs( Xc - Xt )', 480 $ ' / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ), then', 481 $ / 5X, 482 $ 'ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS') 483 9995 FORMAT( 3X, I2, ': Componentwise guaranteed forward error' ) 484 9994 FORMAT( 3X, I2, ': Backwards error' ) 485 9993 FORMAT( 3X, I2, ': Reciprocal condition number' ) 486 9992 FORMAT( 3X, I2, ': Reciprocal normwise condition number' ) 487 9991 FORMAT( 3X, I2, ': Raw normwise error estimate' ) 488 9990 FORMAT( 3X, I2, ': Reciprocal componentwise condition number' ) 489 9989 FORMAT( 3X, I2, ': Raw componentwise error estimate' ) 490 491 8000 FORMAT( ' D', A2, 'SVXX: N =', I2, ', INFO = ', I3, 492 $ ', ORCOND = ', G12.5, ', real RCOND = ', G12.5 ) 493 494 END 495