1 SUBROUTINE DEBCHVXX( THRESH, PATH )
2 IMPLICIT NONE
3 * .. Scalar Arguments ..
4 DOUBLE PRECISION THRESH
5 CHARACTER*3 PATH
6 *
7 * Purpose
8 * ======
9 *
10 * DEBCHVXX will run D**SVXX on a series of Hilbert matrices and then
11 * compare the error bounds returned by D**SVXX to see if the returned
12 * answer indeed falls within those bounds.
13 *
14 * Eight test ratios will be computed. The tests will pass if they are .LT.
15 * THRESH. There are two cases that are determined by 1 / (SQRT( N ) * EPS).
16 * If that value is .LE. to the component wise reciprocal condition number,
17 * it uses the guaranteed case, other wise it uses the unguaranteed case.
18 *
19 * Test ratios:
20 * Let Xc be X_computed and Xt be X_truth.
21 * The norm used is the infinity norm.
22
23 * Let A be the guaranteed case and B be the unguaranteed case.
24 *
25 * 1. Normwise guaranteed forward error bound.
26 * A: norm ( abs( Xc - Xt ) / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ) and
27 * ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N),10) * EPS.
28 * If these conditions are met, the test ratio is set to be
29 * ERRBND( *, nwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS.
30 * B: For this case, CGESVXX should just return 1. If it is less than
31 * one, treat it the same as in 1A. Otherwise it fails. (Set test
32 * ratio to ERRBND( *, nwise_i, bnd_i ) * THRESH?)
33 *
34 * 2. Componentwise guaranteed forward error bound.
35 * A: norm ( abs( Xc(j) - Xt(j) ) ) / norm (Xt(j)) .LE. ERRBND( *, cwise_i, bnd_i )
36 * for all j .AND. ERRBND( *, cwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS.
37 * If these conditions are met, the test ratio is set to be
38 * ERRBND( *, cwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS.
39 * B: Same as normwise test ratio.
40 *
41 * 3. Backwards error.
42 * A: The test ratio is set to BERR/EPS.
43 * B: Same test ratio.
44 *
45 * 4. Reciprocal condition number.
46 * A: A condition number is computed with Xt and compared with the one
47 * returned from CGESVXX. Let RCONDc be the RCOND returned by D**SVXX
48 * and RCONDt be the RCOND from the truth value. Test ratio is set to
49 * MAX(RCONDc/RCONDt, RCONDt/RCONDc).
50 * B: Test ratio is set to 1 / (EPS * RCONDc).
51 *
52 * 5. Reciprocal normwise condition number.
53 * A: The test ratio is set to
54 * MAX(ERRBND( *, nwise_i, cond_i ) / NCOND, NCOND / ERRBND( *, nwise_i, cond_i )).
55 * B: Test ratio is set to 1 / (EPS * ERRBND( *, nwise_i, cond_i )).
56 *
57 * 6. Reciprocal componentwise condition number.
58 * A: Test ratio is set to
59 * MAX(ERRBND( *, cwise_i, cond_i ) / CCOND, CCOND / ERRBND( *, cwise_i, cond_i )).
60 * B: Test ratio is set to 1 / (EPS * ERRBND( *, cwise_i, cond_i )).
61 *
62 * .. Parameters ..
63 * NMAX is determined by the largest number in the inverse of the hilbert
64 * matrix. Precision is exhausted when the largest entry in it is greater
65 * than 2 to the power of the number of bits in the fraction of the data
66 * type used plus one, which is 24 for single precision.
67 * NMAX should be 6 for single and 11 for double.
68
69 INTEGER NMAX, NPARAMS, NERRBND, NTESTS, KL, KU
70 PARAMETER (NMAX = 10, NPARAMS = 2, NERRBND = 3,
71 $ NTESTS = 6)
72
73 * .. Local Scalars ..
74 INTEGER N, NRHS, INFO, I ,J, k, NFAIL, LDA,
75 $ N_AUX_TESTS, LDAB, LDAFB
76 CHARACTER FACT, TRANS, UPLO, EQUED
77 CHARACTER*2 C2
78 CHARACTER(3) NGUAR, CGUAR
79 LOGICAL printed_guide
80 DOUBLE PRECISION NCOND, CCOND, M, NORMDIF, NORMT, RCOND,
81 $ RNORM, RINORM, SUMR, SUMRI, EPS,
82 $ BERR(NMAX), RPVGRW, ORCOND,
83 $ CWISE_ERR, NWISE_ERR, CWISE_BND, NWISE_BND,
84 $ CWISE_RCOND, NWISE_RCOND,
85 $ CONDTHRESH, ERRTHRESH
86
87 * .. Local Arrays ..
88 DOUBLE PRECISION TSTRAT(NTESTS), RINV(NMAX), PARAMS(NPARAMS),
89 $ S(NMAX),R(NMAX),C(NMAX), DIFF(NMAX, NMAX),
90 $ ERRBND_N(NMAX*3), ERRBND_C(NMAX*3),
91 $ A(NMAX,NMAX),INVHILB(NMAX,NMAX),X(NMAX,NMAX),
92 $ AB( (NMAX-1)+(NMAX-1)+1, NMAX ),
93 $ ABCOPY( (NMAX-1)+(NMAX-1)+1, NMAX ),
94 $ AFB( 2*(NMAX-1)+(NMAX-1)+1, NMAX ),
95 $ WORK(NMAX*3*5), AF(NMAX, NMAX),B(NMAX, NMAX),
96 $ ACOPY(NMAX, NMAX)
97 INTEGER IPIV(NMAX), IWORK(3*NMAX)
98
99 * .. External Functions ..
100 DOUBLE PRECISION DLAMCH
101
102 * .. External Subroutines ..
103 EXTERNAL DLAHILB, DGESVXX, DPOSVXX, DSYSVXX,
104 $ DGBSVXX, DLACPY, LSAMEN
105 LOGICAL LSAMEN
106
107 * .. Intrinsic Functions ..
108 INTRINSIC SQRT, MAX, ABS, DBLE
109
110 * .. Parameters ..
111 INTEGER NWISE_I, CWISE_I
112 PARAMETER (NWISE_I = 1, CWISE_I = 1)
113 INTEGER BND_I, COND_I
114 PARAMETER (BND_I = 2, COND_I = 3)
115
116 * Create the loop to test out the Hilbert matrices
117
118 FACT = 'E'
119 UPLO = 'U'
120 TRANS = 'N'
121 EQUED = 'N'
122 EPS = DLAMCH('Epsilon')
123 NFAIL = 0
124 N_AUX_TESTS = 0
125 LDA = NMAX
126 LDAB = (NMAX-1)+(NMAX-1)+1
127 LDAFB = 2*(NMAX-1)+(NMAX-1)+1
128 C2 = PATH( 2: 3 )
129
130 * Main loop to test the different Hilbert Matrices.
131
132 printed_guide = .false.
133
134 DO N = 1 , NMAX
135 PARAMS(1) = -1
136 PARAMS(2) = -1
137
138 KL = N-1
139 KU = N-1
140 NRHS = n
141 M = MAX(SQRT(DBLE(N)), 10.0D+0)
142
143 * Generate the Hilbert matrix, its inverse, and the
144 * right hand side, all scaled by the LCM(1,..,2N-1).
145 CALL DLAHILB(N, N, A, LDA, INVHILB, LDA, B, LDA, WORK, INFO)
146
147 * Copy A into ACOPY.
148 CALL DLACPY('ALL', N, N, A, NMAX, ACOPY, NMAX)
149
150 * Store A in band format for GB tests
151 DO J = 1, N
152 DO I = 1, KL+KU+1
153 AB( I, J ) = 0.0D+0
154 END DO
155 END DO
156 DO J = 1, N
157 DO I = MAX( 1, J-KU ), MIN( N, J+KL )
158 AB( KU+1+I-J, J ) = A( I, J )
159 END DO
160 END DO
161
162 * Copy AB into ABCOPY.
163 DO J = 1, N
164 DO I = 1, KL+KU+1
165 ABCOPY( I, J ) = 0.0D+0
166 END DO
167 END DO
168 CALL DLACPY('ALL', KL+KU+1, N, AB, LDAB, ABCOPY, LDAB)
169
170 * Call D**SVXX with default PARAMS and N_ERR_BND = 3.
171 IF ( LSAMEN( 2, C2, 'SY' ) ) THEN
172 CALL DSYSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
173 $ IPIV, EQUED, S, B, LDA, X, LDA, ORCOND,
174 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
175 $ PARAMS, WORK, IWORK, INFO)
176 ELSE IF ( LSAMEN( 2, C2, 'PO' ) ) THEN
177 CALL DPOSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
178 $ EQUED, S, B, LDA, X, LDA, ORCOND,
179 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
180 $ PARAMS, WORK, IWORK, INFO)
181 ELSE IF ( LSAMEN( 2, C2, 'GB' ) ) THEN
182 CALL DGBSVXX(FACT, TRANS, N, KL, KU, NRHS, ABCOPY,
183 $ LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B,
184 $ LDA, X, LDA, ORCOND, RPVGRW, BERR, NERRBND,
185 $ ERRBND_N, ERRBND_C, NPARAMS, PARAMS, WORK, IWORK,
186 $ INFO)
187 ELSE
188 CALL DGESVXX(FACT, TRANS, N, NRHS, ACOPY, LDA, AF, LDA,
189 $ IPIV, EQUED, R, C, B, LDA, X, LDA, ORCOND,
190 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
191 $ PARAMS, WORK, IWORK, INFO)
192 END IF
193
194 N_AUX_TESTS = N_AUX_TESTS + 1
195 IF (ORCOND .LT. EPS) THEN
196 ! Either factorization failed or the matrix is flagged, and 1 <=
197 ! INFO <= N+1. We don't decide based on rcond anymore.
198 ! IF (INFO .EQ. 0 .OR. INFO .GT. N+1) THEN
199 ! NFAIL = NFAIL + 1
200 ! WRITE (*, FMT=8000) N, INFO, ORCOND, RCOND
201 ! END IF
202 ELSE
203 ! Either everything succeeded (INFO == 0) or some solution failed
204 ! to converge (INFO > N+1).
205 IF (INFO .GT. 0 .AND. INFO .LE. N+1) THEN
206 NFAIL = NFAIL + 1
207 WRITE (*, FMT=8000) C2, N, INFO, ORCOND, RCOND
208 END IF
209 END IF
210
211 * Calculating the difference between D**SVXX's X and the true X.
212 DO I = 1,N
213 DO J =1,NRHS
214 DIFF(I,J) = X(I,J) - INVHILB(I,J)
215 END DO
216 END DO
217
218 * Calculating the RCOND
219 RNORM = 0.0D+0
220 RINORM = 0.0D+0
221 IF ( LSAMEN( 2, C2, 'PO' ) .OR. LSAMEN( 2, C2, 'SY' ) ) THEN
222 DO I = 1, N
223 SUMR = 0.0D+0
224 SUMRI = 0.0D+0
225 DO J = 1, N
226 SUMR = SUMR + S(I) * ABS(A(I,J)) * S(J)
227 SUMRI = SUMRI + ABS(INVHILB(I, J)) / (S(J) * S(I))
228
229 END DO
230 RNORM = MAX(RNORM,SUMR)
231 RINORM = MAX(RINORM,SUMRI)
232 END DO
233 ELSE IF ( LSAMEN( 2, C2, 'GE' ) .OR. LSAMEN( 2, C2, 'GB' ) )
234 $ THEN
235 DO I = 1, N
236 SUMR = 0.0D+0
237 SUMRI = 0.0D+0
238 DO J = 1, N
239 SUMR = SUMR + R(I) * ABS(A(I,J)) * C(J)
240 SUMRI = SUMRI + ABS(INVHILB(I, J)) / (R(J) * C(I))
241 END DO
242 RNORM = MAX(RNORM,SUMR)
243 RINORM = MAX(RINORM,SUMRI)
244 END DO
245 END IF
246
247 RNORM = RNORM / ABS(A(1, 1))
248 RCOND = 1.0D+0/(RNORM * RINORM)
249
250 * Calculating the R for normwise rcond.
251 DO I = 1, N
252 RINV(I) = 0.0D+0
253 END DO
254 DO J = 1, N
255 DO I = 1, N
256 RINV(I) = RINV(I) + ABS(A(I,J))
257 END DO
258 END DO
259
260 * Calculating the Normwise rcond.
261 RINORM = 0.0D+0
262 DO I = 1, N
263 SUMRI = 0.0D+0
264 DO J = 1, N
265 SUMRI = SUMRI + ABS(INVHILB(I,J) * RINV(J))
266 END DO
267 RINORM = MAX(RINORM, SUMRI)
268 END DO
269
270 ! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm
271 ! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix)
272 NCOND = ABS(A(1,1)) / RINORM
273
274 CONDTHRESH = M * EPS
275 ERRTHRESH = M * EPS
276
277 DO K = 1, NRHS
278 NORMT = 0.0D+0
279 NORMDIF = 0.0D+0
280 CWISE_ERR = 0.0D+0
281 DO I = 1, N
282 NORMT = MAX(ABS(INVHILB(I, K)), NORMT)
283 NORMDIF = MAX(ABS(X(I,K) - INVHILB(I,K)), NORMDIF)
284 IF (INVHILB(I,K) .NE. 0.0D+0) THEN
285 CWISE_ERR = MAX(ABS(X(I,K) - INVHILB(I,K))
286 $ /ABS(INVHILB(I,K)), CWISE_ERR)
287 ELSE IF (X(I, K) .NE. 0.0D+0) THEN
288 CWISE_ERR = DLAMCH('OVERFLOW')
289 END IF
290 END DO
291 IF (NORMT .NE. 0.0D+0) THEN
292 NWISE_ERR = NORMDIF / NORMT
293 ELSE IF (NORMDIF .NE. 0.0D+0) THEN
294 NWISE_ERR = DLAMCH('OVERFLOW')
295 ELSE
296 NWISE_ERR = 0.0D+0
297 ENDIF
298
299 DO I = 1, N
300 RINV(I) = 0.0D+0
301 END DO
302 DO J = 1, N
303 DO I = 1, N
304 RINV(I) = RINV(I) + ABS(A(I, J) * INVHILB(J, K))
305 END DO
306 END DO
307 RINORM = 0.0D+0
308 DO I = 1, N
309 SUMRI = 0.0D+0
310 DO J = 1, N
311 SUMRI = SUMRI
312 $ + ABS(INVHILB(I, J) * RINV(J) / INVHILB(I, K))
313 END DO
314 RINORM = MAX(RINORM, SUMRI)
315 END DO
316 ! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm
317 ! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix)
318 CCOND = ABS(A(1,1))/RINORM
319
320 ! Forward error bound tests
321 NWISE_BND = ERRBND_N(K + (BND_I-1)*NRHS)
322 CWISE_BND = ERRBND_C(K + (BND_I-1)*NRHS)
323 NWISE_RCOND = ERRBND_N(K + (COND_I-1)*NRHS)
324 CWISE_RCOND = ERRBND_C(K + (COND_I-1)*NRHS)
325 ! write (*,*) 'nwise : ', n, k, ncond, nwise_rcond,
326 ! $ condthresh, ncond.ge.condthresh
327 ! write (*,*) 'nwise2: ', k, nwise_bnd, nwise_err, errthresh
328 IF (NCOND .GE. CONDTHRESH) THEN
329 NGUAR = 'YES'
330 IF (NWISE_BND .GT. ERRTHRESH) THEN
331 TSTRAT(1) = 1/(2.0D+0*EPS)
332 ELSE
333 IF (NWISE_BND .NE. 0.0D+0) THEN
334 TSTRAT(1) = NWISE_ERR / NWISE_BND
335 ELSE IF (NWISE_ERR .NE. 0.0D+0) THEN
336 TSTRAT(1) = 1/(16.0*EPS)
337 ELSE
338 TSTRAT(1) = 0.0D+0
339 END IF
340 IF (TSTRAT(1) .GT. 1.0D+0) THEN
341 TSTRAT(1) = 1/(4.0D+0*EPS)
342 END IF
343 END IF
344 ELSE
345 NGUAR = 'NO'
346 IF (NWISE_BND .LT. 1.0D+0) THEN
347 TSTRAT(1) = 1/(8.0D+0*EPS)
348 ELSE
349 TSTRAT(1) = 1.0D+0
350 END IF
351 END IF
352 ! write (*,*) 'cwise : ', n, k, ccond, cwise_rcond,
353 ! $ condthresh, ccond.ge.condthresh
354 ! write (*,*) 'cwise2: ', k, cwise_bnd, cwise_err, errthresh
355 IF (CCOND .GE. CONDTHRESH) THEN
356 CGUAR = 'YES'
357 IF (CWISE_BND .GT. ERRTHRESH) THEN
358 TSTRAT(2) = 1/(2.0D+0*EPS)
359 ELSE
360 IF (CWISE_BND .NE. 0.0D+0) THEN
361 TSTRAT(2) = CWISE_ERR / CWISE_BND
362 ELSE IF (CWISE_ERR .NE. 0.0D+0) THEN
363 TSTRAT(2) = 1/(16.0D+0*EPS)
364 ELSE
365 TSTRAT(2) = 0.0D+0
366 END IF
367 IF (TSTRAT(2) .GT. 1.0D+0) TSTRAT(2) = 1/(4.0D+0*EPS)
368 END IF
369 ELSE
370 CGUAR = 'NO'
371 IF (CWISE_BND .LT. 1.0D+0) THEN
372 TSTRAT(2) = 1/(8.0D+0*EPS)
373 ELSE
374 TSTRAT(2) = 1.0D+0
375 END IF
376 END IF
377
378 ! Backwards error test
379 TSTRAT(3) = BERR(K)/EPS
380
381 ! Condition number tests
382 TSTRAT(4) = RCOND / ORCOND
383 IF (RCOND .GE. CONDTHRESH .AND. TSTRAT(4) .LT. 1.0D+0)
384 $ TSTRAT(4) = 1.0D+0 / TSTRAT(4)
385
386 TSTRAT(5) = NCOND / NWISE_RCOND
387 IF (NCOND .GE. CONDTHRESH .AND. TSTRAT(5) .LT. 1.0D+0)
388 $ TSTRAT(5) = 1.0D+0 / TSTRAT(5)
389
390 TSTRAT(6) = CCOND / NWISE_RCOND
391 IF (CCOND .GE. CONDTHRESH .AND. TSTRAT(6) .LT. 1.0D+0)
392 $ TSTRAT(6) = 1.0D+0 / TSTRAT(6)
393
394 DO I = 1, NTESTS
395 IF (TSTRAT(I) .GT. THRESH) THEN
396 IF (.NOT.PRINTED_GUIDE) THEN
397 WRITE(*,*)
398 WRITE( *, 9996) 1
399 WRITE( *, 9995) 2
400 WRITE( *, 9994) 3
401 WRITE( *, 9993) 4
402 WRITE( *, 9992) 5
403 WRITE( *, 9991) 6
404 WRITE( *, 9990) 7
405 WRITE( *, 9989) 8
406 WRITE(*,*)
407 PRINTED_GUIDE = .TRUE.
408 END IF
409 WRITE( *, 9999) C2, N, K, NGUAR, CGUAR, I, TSTRAT(I)
410 NFAIL = NFAIL + 1
411 END IF
412 END DO
413 END DO
414
415 c$$$ WRITE(*,*)
416 c$$$ WRITE(*,*) 'Normwise Error Bounds'
417 c$$$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,nwise_i,bnd_i)
418 c$$$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,nwise_i,cond_i)
419 c$$$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,nwise_i,rawbnd_i)
420 c$$$ WRITE(*,*)
421 c$$$ WRITE(*,*) 'Componentwise Error Bounds'
422 c$$$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,cwise_i,bnd_i)
423 c$$$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,cwise_i,cond_i)
424 c$$$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,cwise_i,rawbnd_i)
425 c$$$ print *, 'Info: ', info
426 c$$$ WRITE(*,*)
427 * WRITE(*,*) 'TSTRAT: ',TSTRAT
428
429 END DO
430
431 WRITE(*,*)
432 IF( NFAIL .GT. 0 ) THEN
433 WRITE(*,9998) C2, NFAIL, NTESTS*N+N_AUX_TESTS
434 ELSE
435 WRITE(*,9997) C2
436 END IF
437 9999 FORMAT( ' D', A2, 'SVXX: N =', I2, ', RHS = ', I2,
438 $ ', NWISE GUAR. = ', A, ', CWISE GUAR. = ', A,
439 $ ' test(',I1,') =', G12.5 )
440 9998 FORMAT( ' D', A2, 'SVXX: ', I6, ' out of ', I6,
441 $ ' tests failed to pass the threshold' )
442 9997 FORMAT( ' D', A2, 'SVXX passed the tests of error bounds' )
443 * Test ratios.
444 9996 FORMAT( 3X, I2, ': Normwise guaranteed forward error', / 5X,
445 $ 'Guaranteed case: if norm ( abs( Xc - Xt )',
446 $ ' / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ), then',
447 $ / 5X,
448 $ 'ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS')
449 9995 FORMAT( 3X, I2, ': Componentwise guaranteed forward error' )
450 9994 FORMAT( 3X, I2, ': Backwards error' )
451 9993 FORMAT( 3X, I2, ': Reciprocal condition number' )
452 9992 FORMAT( 3X, I2, ': Reciprocal normwise condition number' )
453 9991 FORMAT( 3X, I2, ': Raw normwise error estimate' )
454 9990 FORMAT( 3X, I2, ': Reciprocal componentwise condition number' )
455 9989 FORMAT( 3X, I2, ': Raw componentwise error estimate' )
456
457 8000 FORMAT( ' D', A2, 'SVXX: N =', I2, ', INFO = ', I3,
458 $ ', ORCOND = ', G12.5, ', real RCOND = ', G12.5 )
459
460 END
2 IMPLICIT NONE
3 * .. Scalar Arguments ..
4 DOUBLE PRECISION THRESH
5 CHARACTER*3 PATH
6 *
7 * Purpose
8 * ======
9 *
10 * DEBCHVXX will run D**SVXX on a series of Hilbert matrices and then
11 * compare the error bounds returned by D**SVXX to see if the returned
12 * answer indeed falls within those bounds.
13 *
14 * Eight test ratios will be computed. The tests will pass if they are .LT.
15 * THRESH. There are two cases that are determined by 1 / (SQRT( N ) * EPS).
16 * If that value is .LE. to the component wise reciprocal condition number,
17 * it uses the guaranteed case, other wise it uses the unguaranteed case.
18 *
19 * Test ratios:
20 * Let Xc be X_computed and Xt be X_truth.
21 * The norm used is the infinity norm.
22
23 * Let A be the guaranteed case and B be the unguaranteed case.
24 *
25 * 1. Normwise guaranteed forward error bound.
26 * A: norm ( abs( Xc - Xt ) / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ) and
27 * ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N),10) * EPS.
28 * If these conditions are met, the test ratio is set to be
29 * ERRBND( *, nwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS.
30 * B: For this case, CGESVXX should just return 1. If it is less than
31 * one, treat it the same as in 1A. Otherwise it fails. (Set test
32 * ratio to ERRBND( *, nwise_i, bnd_i ) * THRESH?)
33 *
34 * 2. Componentwise guaranteed forward error bound.
35 * A: norm ( abs( Xc(j) - Xt(j) ) ) / norm (Xt(j)) .LE. ERRBND( *, cwise_i, bnd_i )
36 * for all j .AND. ERRBND( *, cwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS.
37 * If these conditions are met, the test ratio is set to be
38 * ERRBND( *, cwise_i, bnd_i ) / MAX(SQRT(N), 10). Otherwise it is 1/EPS.
39 * B: Same as normwise test ratio.
40 *
41 * 3. Backwards error.
42 * A: The test ratio is set to BERR/EPS.
43 * B: Same test ratio.
44 *
45 * 4. Reciprocal condition number.
46 * A: A condition number is computed with Xt and compared with the one
47 * returned from CGESVXX. Let RCONDc be the RCOND returned by D**SVXX
48 * and RCONDt be the RCOND from the truth value. Test ratio is set to
49 * MAX(RCONDc/RCONDt, RCONDt/RCONDc).
50 * B: Test ratio is set to 1 / (EPS * RCONDc).
51 *
52 * 5. Reciprocal normwise condition number.
53 * A: The test ratio is set to
54 * MAX(ERRBND( *, nwise_i, cond_i ) / NCOND, NCOND / ERRBND( *, nwise_i, cond_i )).
55 * B: Test ratio is set to 1 / (EPS * ERRBND( *, nwise_i, cond_i )).
56 *
57 * 6. Reciprocal componentwise condition number.
58 * A: Test ratio is set to
59 * MAX(ERRBND( *, cwise_i, cond_i ) / CCOND, CCOND / ERRBND( *, cwise_i, cond_i )).
60 * B: Test ratio is set to 1 / (EPS * ERRBND( *, cwise_i, cond_i )).
61 *
62 * .. Parameters ..
63 * NMAX is determined by the largest number in the inverse of the hilbert
64 * matrix. Precision is exhausted when the largest entry in it is greater
65 * than 2 to the power of the number of bits in the fraction of the data
66 * type used plus one, which is 24 for single precision.
67 * NMAX should be 6 for single and 11 for double.
68
69 INTEGER NMAX, NPARAMS, NERRBND, NTESTS, KL, KU
70 PARAMETER (NMAX = 10, NPARAMS = 2, NERRBND = 3,
71 $ NTESTS = 6)
72
73 * .. Local Scalars ..
74 INTEGER N, NRHS, INFO, I ,J, k, NFAIL, LDA,
75 $ N_AUX_TESTS, LDAB, LDAFB
76 CHARACTER FACT, TRANS, UPLO, EQUED
77 CHARACTER*2 C2
78 CHARACTER(3) NGUAR, CGUAR
79 LOGICAL printed_guide
80 DOUBLE PRECISION NCOND, CCOND, M, NORMDIF, NORMT, RCOND,
81 $ RNORM, RINORM, SUMR, SUMRI, EPS,
82 $ BERR(NMAX), RPVGRW, ORCOND,
83 $ CWISE_ERR, NWISE_ERR, CWISE_BND, NWISE_BND,
84 $ CWISE_RCOND, NWISE_RCOND,
85 $ CONDTHRESH, ERRTHRESH
86
87 * .. Local Arrays ..
88 DOUBLE PRECISION TSTRAT(NTESTS), RINV(NMAX), PARAMS(NPARAMS),
89 $ S(NMAX),R(NMAX),C(NMAX), DIFF(NMAX, NMAX),
90 $ ERRBND_N(NMAX*3), ERRBND_C(NMAX*3),
91 $ A(NMAX,NMAX),INVHILB(NMAX,NMAX),X(NMAX,NMAX),
92 $ AB( (NMAX-1)+(NMAX-1)+1, NMAX ),
93 $ ABCOPY( (NMAX-1)+(NMAX-1)+1, NMAX ),
94 $ AFB( 2*(NMAX-1)+(NMAX-1)+1, NMAX ),
95 $ WORK(NMAX*3*5), AF(NMAX, NMAX),B(NMAX, NMAX),
96 $ ACOPY(NMAX, NMAX)
97 INTEGER IPIV(NMAX), IWORK(3*NMAX)
98
99 * .. External Functions ..
100 DOUBLE PRECISION DLAMCH
101
102 * .. External Subroutines ..
103 EXTERNAL DLAHILB, DGESVXX, DPOSVXX, DSYSVXX,
104 $ DGBSVXX, DLACPY, LSAMEN
105 LOGICAL LSAMEN
106
107 * .. Intrinsic Functions ..
108 INTRINSIC SQRT, MAX, ABS, DBLE
109
110 * .. Parameters ..
111 INTEGER NWISE_I, CWISE_I
112 PARAMETER (NWISE_I = 1, CWISE_I = 1)
113 INTEGER BND_I, COND_I
114 PARAMETER (BND_I = 2, COND_I = 3)
115
116 * Create the loop to test out the Hilbert matrices
117
118 FACT = 'E'
119 UPLO = 'U'
120 TRANS = 'N'
121 EQUED = 'N'
122 EPS = DLAMCH('Epsilon')
123 NFAIL = 0
124 N_AUX_TESTS = 0
125 LDA = NMAX
126 LDAB = (NMAX-1)+(NMAX-1)+1
127 LDAFB = 2*(NMAX-1)+(NMAX-1)+1
128 C2 = PATH( 2: 3 )
129
130 * Main loop to test the different Hilbert Matrices.
131
132 printed_guide = .false.
133
134 DO N = 1 , NMAX
135 PARAMS(1) = -1
136 PARAMS(2) = -1
137
138 KL = N-1
139 KU = N-1
140 NRHS = n
141 M = MAX(SQRT(DBLE(N)), 10.0D+0)
142
143 * Generate the Hilbert matrix, its inverse, and the
144 * right hand side, all scaled by the LCM(1,..,2N-1).
145 CALL DLAHILB(N, N, A, LDA, INVHILB, LDA, B, LDA, WORK, INFO)
146
147 * Copy A into ACOPY.
148 CALL DLACPY('ALL', N, N, A, NMAX, ACOPY, NMAX)
149
150 * Store A in band format for GB tests
151 DO J = 1, N
152 DO I = 1, KL+KU+1
153 AB( I, J ) = 0.0D+0
154 END DO
155 END DO
156 DO J = 1, N
157 DO I = MAX( 1, J-KU ), MIN( N, J+KL )
158 AB( KU+1+I-J, J ) = A( I, J )
159 END DO
160 END DO
161
162 * Copy AB into ABCOPY.
163 DO J = 1, N
164 DO I = 1, KL+KU+1
165 ABCOPY( I, J ) = 0.0D+0
166 END DO
167 END DO
168 CALL DLACPY('ALL', KL+KU+1, N, AB, LDAB, ABCOPY, LDAB)
169
170 * Call D**SVXX with default PARAMS and N_ERR_BND = 3.
171 IF ( LSAMEN( 2, C2, 'SY' ) ) THEN
172 CALL DSYSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
173 $ IPIV, EQUED, S, B, LDA, X, LDA, ORCOND,
174 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
175 $ PARAMS, WORK, IWORK, INFO)
176 ELSE IF ( LSAMEN( 2, C2, 'PO' ) ) THEN
177 CALL DPOSVXX(FACT, UPLO, N, NRHS, ACOPY, LDA, AF, LDA,
178 $ EQUED, S, B, LDA, X, LDA, ORCOND,
179 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
180 $ PARAMS, WORK, IWORK, INFO)
181 ELSE IF ( LSAMEN( 2, C2, 'GB' ) ) THEN
182 CALL DGBSVXX(FACT, TRANS, N, KL, KU, NRHS, ABCOPY,
183 $ LDAB, AFB, LDAFB, IPIV, EQUED, R, C, B,
184 $ LDA, X, LDA, ORCOND, RPVGRW, BERR, NERRBND,
185 $ ERRBND_N, ERRBND_C, NPARAMS, PARAMS, WORK, IWORK,
186 $ INFO)
187 ELSE
188 CALL DGESVXX(FACT, TRANS, N, NRHS, ACOPY, LDA, AF, LDA,
189 $ IPIV, EQUED, R, C, B, LDA, X, LDA, ORCOND,
190 $ RPVGRW, BERR, NERRBND, ERRBND_N, ERRBND_C, NPARAMS,
191 $ PARAMS, WORK, IWORK, INFO)
192 END IF
193
194 N_AUX_TESTS = N_AUX_TESTS + 1
195 IF (ORCOND .LT. EPS) THEN
196 ! Either factorization failed or the matrix is flagged, and 1 <=
197 ! INFO <= N+1. We don't decide based on rcond anymore.
198 ! IF (INFO .EQ. 0 .OR. INFO .GT. N+1) THEN
199 ! NFAIL = NFAIL + 1
200 ! WRITE (*, FMT=8000) N, INFO, ORCOND, RCOND
201 ! END IF
202 ELSE
203 ! Either everything succeeded (INFO == 0) or some solution failed
204 ! to converge (INFO > N+1).
205 IF (INFO .GT. 0 .AND. INFO .LE. N+1) THEN
206 NFAIL = NFAIL + 1
207 WRITE (*, FMT=8000) C2, N, INFO, ORCOND, RCOND
208 END IF
209 END IF
210
211 * Calculating the difference between D**SVXX's X and the true X.
212 DO I = 1,N
213 DO J =1,NRHS
214 DIFF(I,J) = X(I,J) - INVHILB(I,J)
215 END DO
216 END DO
217
218 * Calculating the RCOND
219 RNORM = 0.0D+0
220 RINORM = 0.0D+0
221 IF ( LSAMEN( 2, C2, 'PO' ) .OR. LSAMEN( 2, C2, 'SY' ) ) THEN
222 DO I = 1, N
223 SUMR = 0.0D+0
224 SUMRI = 0.0D+0
225 DO J = 1, N
226 SUMR = SUMR + S(I) * ABS(A(I,J)) * S(J)
227 SUMRI = SUMRI + ABS(INVHILB(I, J)) / (S(J) * S(I))
228
229 END DO
230 RNORM = MAX(RNORM,SUMR)
231 RINORM = MAX(RINORM,SUMRI)
232 END DO
233 ELSE IF ( LSAMEN( 2, C2, 'GE' ) .OR. LSAMEN( 2, C2, 'GB' ) )
234 $ THEN
235 DO I = 1, N
236 SUMR = 0.0D+0
237 SUMRI = 0.0D+0
238 DO J = 1, N
239 SUMR = SUMR + R(I) * ABS(A(I,J)) * C(J)
240 SUMRI = SUMRI + ABS(INVHILB(I, J)) / (R(J) * C(I))
241 END DO
242 RNORM = MAX(RNORM,SUMR)
243 RINORM = MAX(RINORM,SUMRI)
244 END DO
245 END IF
246
247 RNORM = RNORM / ABS(A(1, 1))
248 RCOND = 1.0D+0/(RNORM * RINORM)
249
250 * Calculating the R for normwise rcond.
251 DO I = 1, N
252 RINV(I) = 0.0D+0
253 END DO
254 DO J = 1, N
255 DO I = 1, N
256 RINV(I) = RINV(I) + ABS(A(I,J))
257 END DO
258 END DO
259
260 * Calculating the Normwise rcond.
261 RINORM = 0.0D+0
262 DO I = 1, N
263 SUMRI = 0.0D+0
264 DO J = 1, N
265 SUMRI = SUMRI + ABS(INVHILB(I,J) * RINV(J))
266 END DO
267 RINORM = MAX(RINORM, SUMRI)
268 END DO
269
270 ! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm
271 ! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix)
272 NCOND = ABS(A(1,1)) / RINORM
273
274 CONDTHRESH = M * EPS
275 ERRTHRESH = M * EPS
276
277 DO K = 1, NRHS
278 NORMT = 0.0D+0
279 NORMDIF = 0.0D+0
280 CWISE_ERR = 0.0D+0
281 DO I = 1, N
282 NORMT = MAX(ABS(INVHILB(I, K)), NORMT)
283 NORMDIF = MAX(ABS(X(I,K) - INVHILB(I,K)), NORMDIF)
284 IF (INVHILB(I,K) .NE. 0.0D+0) THEN
285 CWISE_ERR = MAX(ABS(X(I,K) - INVHILB(I,K))
286 $ /ABS(INVHILB(I,K)), CWISE_ERR)
287 ELSE IF (X(I, K) .NE. 0.0D+0) THEN
288 CWISE_ERR = DLAMCH('OVERFLOW')
289 END IF
290 END DO
291 IF (NORMT .NE. 0.0D+0) THEN
292 NWISE_ERR = NORMDIF / NORMT
293 ELSE IF (NORMDIF .NE. 0.0D+0) THEN
294 NWISE_ERR = DLAMCH('OVERFLOW')
295 ELSE
296 NWISE_ERR = 0.0D+0
297 ENDIF
298
299 DO I = 1, N
300 RINV(I) = 0.0D+0
301 END DO
302 DO J = 1, N
303 DO I = 1, N
304 RINV(I) = RINV(I) + ABS(A(I, J) * INVHILB(J, K))
305 END DO
306 END DO
307 RINORM = 0.0D+0
308 DO I = 1, N
309 SUMRI = 0.0D+0
310 DO J = 1, N
311 SUMRI = SUMRI
312 $ + ABS(INVHILB(I, J) * RINV(J) / INVHILB(I, K))
313 END DO
314 RINORM = MAX(RINORM, SUMRI)
315 END DO
316 ! invhilb is the inverse *unscaled* Hilbert matrix, so scale its norm
317 ! by 1/A(1,1) to make the scaling match A (the scaled Hilbert matrix)
318 CCOND = ABS(A(1,1))/RINORM
319
320 ! Forward error bound tests
321 NWISE_BND = ERRBND_N(K + (BND_I-1)*NRHS)
322 CWISE_BND = ERRBND_C(K + (BND_I-1)*NRHS)
323 NWISE_RCOND = ERRBND_N(K + (COND_I-1)*NRHS)
324 CWISE_RCOND = ERRBND_C(K + (COND_I-1)*NRHS)
325 ! write (*,*) 'nwise : ', n, k, ncond, nwise_rcond,
326 ! $ condthresh, ncond.ge.condthresh
327 ! write (*,*) 'nwise2: ', k, nwise_bnd, nwise_err, errthresh
328 IF (NCOND .GE. CONDTHRESH) THEN
329 NGUAR = 'YES'
330 IF (NWISE_BND .GT. ERRTHRESH) THEN
331 TSTRAT(1) = 1/(2.0D+0*EPS)
332 ELSE
333 IF (NWISE_BND .NE. 0.0D+0) THEN
334 TSTRAT(1) = NWISE_ERR / NWISE_BND
335 ELSE IF (NWISE_ERR .NE. 0.0D+0) THEN
336 TSTRAT(1) = 1/(16.0*EPS)
337 ELSE
338 TSTRAT(1) = 0.0D+0
339 END IF
340 IF (TSTRAT(1) .GT. 1.0D+0) THEN
341 TSTRAT(1) = 1/(4.0D+0*EPS)
342 END IF
343 END IF
344 ELSE
345 NGUAR = 'NO'
346 IF (NWISE_BND .LT. 1.0D+0) THEN
347 TSTRAT(1) = 1/(8.0D+0*EPS)
348 ELSE
349 TSTRAT(1) = 1.0D+0
350 END IF
351 END IF
352 ! write (*,*) 'cwise : ', n, k, ccond, cwise_rcond,
353 ! $ condthresh, ccond.ge.condthresh
354 ! write (*,*) 'cwise2: ', k, cwise_bnd, cwise_err, errthresh
355 IF (CCOND .GE. CONDTHRESH) THEN
356 CGUAR = 'YES'
357 IF (CWISE_BND .GT. ERRTHRESH) THEN
358 TSTRAT(2) = 1/(2.0D+0*EPS)
359 ELSE
360 IF (CWISE_BND .NE. 0.0D+0) THEN
361 TSTRAT(2) = CWISE_ERR / CWISE_BND
362 ELSE IF (CWISE_ERR .NE. 0.0D+0) THEN
363 TSTRAT(2) = 1/(16.0D+0*EPS)
364 ELSE
365 TSTRAT(2) = 0.0D+0
366 END IF
367 IF (TSTRAT(2) .GT. 1.0D+0) TSTRAT(2) = 1/(4.0D+0*EPS)
368 END IF
369 ELSE
370 CGUAR = 'NO'
371 IF (CWISE_BND .LT. 1.0D+0) THEN
372 TSTRAT(2) = 1/(8.0D+0*EPS)
373 ELSE
374 TSTRAT(2) = 1.0D+0
375 END IF
376 END IF
377
378 ! Backwards error test
379 TSTRAT(3) = BERR(K)/EPS
380
381 ! Condition number tests
382 TSTRAT(4) = RCOND / ORCOND
383 IF (RCOND .GE. CONDTHRESH .AND. TSTRAT(4) .LT. 1.0D+0)
384 $ TSTRAT(4) = 1.0D+0 / TSTRAT(4)
385
386 TSTRAT(5) = NCOND / NWISE_RCOND
387 IF (NCOND .GE. CONDTHRESH .AND. TSTRAT(5) .LT. 1.0D+0)
388 $ TSTRAT(5) = 1.0D+0 / TSTRAT(5)
389
390 TSTRAT(6) = CCOND / NWISE_RCOND
391 IF (CCOND .GE. CONDTHRESH .AND. TSTRAT(6) .LT. 1.0D+0)
392 $ TSTRAT(6) = 1.0D+0 / TSTRAT(6)
393
394 DO I = 1, NTESTS
395 IF (TSTRAT(I) .GT. THRESH) THEN
396 IF (.NOT.PRINTED_GUIDE) THEN
397 WRITE(*,*)
398 WRITE( *, 9996) 1
399 WRITE( *, 9995) 2
400 WRITE( *, 9994) 3
401 WRITE( *, 9993) 4
402 WRITE( *, 9992) 5
403 WRITE( *, 9991) 6
404 WRITE( *, 9990) 7
405 WRITE( *, 9989) 8
406 WRITE(*,*)
407 PRINTED_GUIDE = .TRUE.
408 END IF
409 WRITE( *, 9999) C2, N, K, NGUAR, CGUAR, I, TSTRAT(I)
410 NFAIL = NFAIL + 1
411 END IF
412 END DO
413 END DO
414
415 c$$$ WRITE(*,*)
416 c$$$ WRITE(*,*) 'Normwise Error Bounds'
417 c$$$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,nwise_i,bnd_i)
418 c$$$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,nwise_i,cond_i)
419 c$$$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,nwise_i,rawbnd_i)
420 c$$$ WRITE(*,*)
421 c$$$ WRITE(*,*) 'Componentwise Error Bounds'
422 c$$$ WRITE(*,*) 'Guaranteed error bound: ',ERRBND(NRHS,cwise_i,bnd_i)
423 c$$$ WRITE(*,*) 'Reciprocal condition number: ',ERRBND(NRHS,cwise_i,cond_i)
424 c$$$ WRITE(*,*) 'Raw error estimate: ',ERRBND(NRHS,cwise_i,rawbnd_i)
425 c$$$ print *, 'Info: ', info
426 c$$$ WRITE(*,*)
427 * WRITE(*,*) 'TSTRAT: ',TSTRAT
428
429 END DO
430
431 WRITE(*,*)
432 IF( NFAIL .GT. 0 ) THEN
433 WRITE(*,9998) C2, NFAIL, NTESTS*N+N_AUX_TESTS
434 ELSE
435 WRITE(*,9997) C2
436 END IF
437 9999 FORMAT( ' D', A2, 'SVXX: N =', I2, ', RHS = ', I2,
438 $ ', NWISE GUAR. = ', A, ', CWISE GUAR. = ', A,
439 $ ' test(',I1,') =', G12.5 )
440 9998 FORMAT( ' D', A2, 'SVXX: ', I6, ' out of ', I6,
441 $ ' tests failed to pass the threshold' )
442 9997 FORMAT( ' D', A2, 'SVXX passed the tests of error bounds' )
443 * Test ratios.
444 9996 FORMAT( 3X, I2, ': Normwise guaranteed forward error', / 5X,
445 $ 'Guaranteed case: if norm ( abs( Xc - Xt )',
446 $ ' / norm ( Xt ) .LE. ERRBND( *, nwise_i, bnd_i ), then',
447 $ / 5X,
448 $ 'ERRBND( *, nwise_i, bnd_i ) .LE. MAX(SQRT(N), 10) * EPS')
449 9995 FORMAT( 3X, I2, ': Componentwise guaranteed forward error' )
450 9994 FORMAT( 3X, I2, ': Backwards error' )
451 9993 FORMAT( 3X, I2, ': Reciprocal condition number' )
452 9992 FORMAT( 3X, I2, ': Reciprocal normwise condition number' )
453 9991 FORMAT( 3X, I2, ': Raw normwise error estimate' )
454 9990 FORMAT( 3X, I2, ': Reciprocal componentwise condition number' )
455 9989 FORMAT( 3X, I2, ': Raw componentwise error estimate' )
456
457 8000 FORMAT( ' D', A2, 'SVXX: N =', I2, ', INFO = ', I3,
458 $ ', ORCOND = ', G12.5, ', real RCOND = ', G12.5 )
459
460 END