forked from johannesgerer/jburkardt-f
-
Notifications
You must be signed in to change notification settings - Fork 1
/
test_nonlin.html
770 lines (741 loc) · 21.9 KB
/
test_nonlin.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
<html>
<head>
<title>
TEST_NONLIN - Nonlinear Equation Tests
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
TEST_NONLIN <br> Nonlinear Equation Tests
</h1>
<hr>
<p>
<b>TEST_NONLIN</b>
is a FORTRAN90 library which
defines a set of test problems for nonlinear
equation system solvers.
</p>
<p>
A few of the problems are small (2, 3, or 4 equations in 4
unknowns), but most of the problems may be set to any size whatsoever.
The software includes routines defining the initial approximation to the
solution of the system, the N function values at any point, and the N by
N jacobian matrix at any point.
</p>
<p>
The list of problems includes:
<OL>
<li>
Generalized Rosenbrock function, 1 < N.
</li>
<li>
Powell singular function, N = 4.
</li>
<li>
Powell badly scaled function, N = 2.
</li>
<li>
Wood function, N = 4.
</li>
<li>
Helical valley function, N = 3.
</li>
<li>
Watson function, 1 < N.
</li>
<li>
Chebyquad function, N arbitrary.
</li>
<li>
Brown almost linear function, N arbitrary.
</li>
<li>
Discrete boundary value function, N arbitrary.
</li>
<li>
Discrete integral equation function, N arbitrary.
</li>
<li>
Trigonometric function, N arbitrary.
</li>
<li>
Variably dimensioned function, N arbitrary.
</li>
<li>
Broyden tridiagonal function, N arbitrary.
</li>
<li>
Broyden banded function, N arbitrary.
</li>
<li>
Hammarling 2 by 2 matrix square root problem, N = 4.
</li>
<li>
Hammarling 3 by 3 matrix square root problem, N = 9.
</li>
<li>
Dennis and Schnabel example, N = 2.
</li>
<li>
Sample problem 18, N = 2.
</li>
<li>
Sample problem 19, N = 2.
</li>
<li>
Scalar problem, N = 1.
</li>
<li>
Freudenstein-Roth function, N = 2.
</li>
<li>
Boggs function, N = 2.
</li>
<li>
Chandrasekhar function, N arbitrary.
</li>
</ol>
</p>
<h3 align = "center">
Licensing:
</h3>
<p>
The computer code and data files described and made available on this web page
are distributed under
<a href = "../../txt/gnu_lgpl.txt">the GNU LGPL license.</a>
</p>
<h3 align = "center">
Languages:
</h3>
<p>
<b>TEST_NONLIN</b> is available in
<a href = "../../f77_src/test_nonlin/test_nonlin.html">a FORTRAN77 version</a> and
<a href = "../../f_src/test_nonlin/test_nonlin.html">a FORTRAN90 version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../cpp_src/gsl/gsl.html">
GSL</a>,
a C++ library which
can perform multidimensional root-finding.
</p>
<p>
<a href = "../../m_src/kelley/kelley.html">
KELLEY</a>,
a MATLAB library which
can seek solutions of systems of nonlinear equations.
</p>
<p>
<a href = "../../f_src/minpack/minpack.html">
MINPACK</a>,
a FORTRAN90 library which
is a minimization
package for which most of these problems were used as tests,
as part of ACM TOMS algorithm 566.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Subramanyan Chandrasekhar,<br>
Radiative Transfer,<br>
Dover, 1960,<br>
ISBN13: 978-0486605906,<br>
LC: QB461.C46.
</li>
<li>
John Dennis, David Gay, Phuong Vu,<br>
A new nonlinear equations test problem,<br>
Technical Report 83-16,<br>
Mathematical Sciences Department,<br>
Rice University, 1983.
</li>
<li>
John Dennis, Robert Schnabel,<br>
Numerical Methods for Unconstrained Optimization
and Nonlinear Equations,<br>
SIAM, 1996,<br>
ISBN13: 978-0-898713-64-0,<br>
LC: QA402.5.D44.
</li>
<li>
Noel deVilliers, David Glasser,<br>
A continuation method for nonlinear regression,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 18, Number 6, December 1981, pages 1139-1154.
</li>
<li>
Chris Fraley, <br>
Solution of nonlinear least-squares problems,<br>
Technical Report STAN-CS-1165, <br>
Computer Science Department,<br>
Stanford University, 1987.
</li>
<li>
Chris Fraley,
Software performance on nonlinear least-squares problems,<br>
Technical Report SOL 88-17, <br>
Systems Optimization Laboratory,<br>
Department of Operations Research, <br>
Stanford University, 1988.
</li>
<li>
JJ McKeown,<br>
Specialized versus general-purpose algorithms for functions
that are sums of squared terms,<br>
Mathematical Programming,<br>
Volume 9, 1975, pages 57-68.
</li>
<li>
JJ McKeown,<br>
On algorithms for sums of squares problems,<br>
in Towards Global Optimisation,<br>
edited by Laurence Dixon, Gabor Szego,<br>
North-Holland, 1975, pages 229-257,<br>
ISBN: 0444109552,<br>
LC: QA402.5.T7.
</li>
<li>
Jorge More, Burton Garbow, Kenneth Hillstrom,<br>
Testing unconstrained optimization software,<br>
ACM Transactions on Mathematical Software,<br>
Volume 7, Number 1, March 1981, pages 17-41.
</li>
<li>
Jorge More, Burton Garbow, Kenneth Hillstrom,<br>
Algorithm 566:
FORTRAN Subroutines for Testing unconstrained optimization software,<br>
ACM Transactions on Mathematical Software,<br>
Volume 7, Number 1, March 1981, pages 136-140.
</li>
<li>
James Ortega, Werner Rheinboldt<br>
Iterative Solution of Nonlinear Equations in Several Variables,<br>
SIAM, 1987,<br>
ISBN13: 978-0898714616,<br>
LC: QA297.8.O77.
</li>
<li>
Werner Rheinboldt,<br>
Methods for Solving Systems of Nonlinear Equations,<br>
SIAM, 1998,<br>
ISBN: 089871415X,<br>
LC: QA214.R44.
</li>
<li>
Douglas Salane,<br>
A continuation approach for solving large residual nonlinear
least squares problems,<br>
SIAM Journal on Scientific and Statistical Computing,<br>
Volume 8, Number 4, July 1987, pages 655-671.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "test_nonlin.f90">test_nonlin.f90</a>, the source code;
</li>
<li>
<a href = "test_nonlin.sh">test_nonlin.sh</a>,
commands to compile the calling program;
</li>
</ul>
</p>
<h3 align = "center">
Examples and Tests:
</h3>
<p>
<ul>
<li>
<a href = "test_nonlin_prb.f90">test_nonlin_prb.f90</a>, the calling
program;
</li>
<li>
<a href = "test_nonlin_prb.sh">test_nonlin_prb.sh</a>,
commands to compile, link and run the calling program;
</li>
<li>
<a href = "test_nonlin_prb_output.txt">test_nonlin_prb_output.txt</a>,
the output file.
</li>
</ul>
</p>
<h3 align = "center">
List of Routines:
</h3>
<p>
<ul>
<li>
<b>P00_DIF</b> approximates the jacobian via finite differences.
</li>
<li>
<b>P00_FX</b> evaluates the function for any problem.
</li>
<li>
<b>P00_JAC</b> evaluates the jacobian for any problem.
</li>
<li>
<b>P00_N</b> returns the number of equations for a problem.
</li>
<li>
<b>P00_PROBLEM_NUM</b> returns the number of problems available.
</li>
<li>
<b>P00_SOL</b> returns the solution of any problem.
</li>
<li>
<b>P00_START</b> specifies a standard approximate solution.
</li>
<li>
<b>P00_TITLE</b> returns the title of the problem.
</li>
<li>
<b>P01_FX</b> evaluates the function for problem 1.
</li>
<li>
<b>P01_N</b> returns the number of equations for problem 1.
</li>
<li>
<b>P01_JAC</b> sets the jacobian for problem 1.
</li>
<li>
<b>P01_SOL</b> returns the solution of problem 1.
</li>
<li>
<b>P01_START</b> specifies a standard approximate solution for problem 1.
</li>
<li>
<b>P01_TITLE</b> returns the title of problem 1.
</li>
<li>
<b>P02_FX</b> evaluates the function for problem 2.
</li>
<li>
<b>P02_JAC</b> sets the jacobian for problem 2.
</li>
<li>
<b>P02_N</b> returns the number of equations for problem 2.
</li>
<li>
<b>P02_SOL</b> returns the solution of problem 2.
</li>
<li>
<b>P02_START</b> specifies a standard approximate solution for problem 2.
</li>
<li>
<b>P02_TITLE</b> returns the title of problem 2.
</li>
<li>
<b>P03_FX</b> evaluates the function for problem 3.
</li>
<li>
<b>P03_JAC</b> sets the jacobian for problem 3.
</li>
<li>
<b>P03_N</b> returns the number of equations for problem 3.
</li>
<li>
<b>P03_SOL</b> returns the solution of problem 3.
</li>
<li>
<b>P03_START</b> specifies a standard approximate solution for problem 3.
</li>
<li>
<b>P03_TITLE</b> returns the title of problem 3.
</li>
<li>
<b>P04_FX</b> evaluates the function for problem 4.
</li>
<li>
<b>P04_JAC</b> sets the jacobian for problem 4.
</li>
<li>
<b>P04_N</b> returns the number of equations for problem 4.
</li>
<li>
<b>P04_SOL</b> returns the solution of problem 4.
</li>
<li>
<b>P04_START</b> specifies a standard approximate solution for problem 4.
</li>
<li>
<b>P04_TITLE</b> returns the title of problem 4.
</li>
<li>
<b>P05_FX</b> evaluates the function for problem 4.
</li>
<li>
<b>P05_JAC</b> sets the jacobian for problem 5.
</li>
<li>
<b>P05_N</b> returns the number of equations for problem 5.
</li>
<li>
<b>P05_SOL</b> returns the solution of problem 5.
</li>
<li>
<b>P05_START</b> specifies a standard approximate solution for problem 5.
</li>
<li>
<b>P05_TITLE</b> returns the title of problem 5.
</li>
<li>
<b>P06_FX</b> evaluates the function for problem 6.
</li>
<li>
<b>P06_JAC</b> sets the jacobian for problem 6.
</li>
<li>
<b>P06_N</b> returns the number of equations for problem 6.
</li>
<li>
<b>P06_SOL</b> returns the solution of problem 6.
</li>
<li>
<b>P06_START</b> specifies a standard approximate solution for problem 6.
</li>
<li>
<b>P06_TITLE</b> returns the title of problem 6.
</li>
<li>
<b>P07_FX</b> evaluates the function for problem 7.
</li>
<li>
<b>P07_JAC</b> sets the jacobian for problem 7.
</li>
<li>
<b>P07_N</b> returns the number of equations for problem 7.
</li>
<li>
<b>P07_SOL</b> returns the solution of problem 7.
</li>
<li>
<b>P07_START</b> specifies a standard approximate solution for problem 7.
</li>
<li>
<b>P07_TITLE</b> returns the title of problem 7.
</li>
<li>
<b>P08_FX</b> evaluates the function for problem 8.
</li>
<li>
<b>P08_JAC</b> sets the jacobian for problem 8.
</li>
<li>
<b>P08_N</b> returns the number of equations for problem 8.
</li>
<li>
<b>P08_SOL</b> returns the solution of problem 8.
</li>
<li>
<b>P08_START</b> specifies a standard approximate solution for problem 8.
</li>
<li>
<b>P08_TITLE</b> returns the title of problem 8.
</li>
<li>
<b>P09_FX</b> evaluates the function for problem 9.
</li>
<li>
<b>P09_JAC</b> sets the jacobian for problem 9.
</li>
<li>
<b>P09_N</b> returns the number of equations for problem 9.
</li>
<li>
<b>P09_SOL</b> returns the solution of problem 9.
</li>
<li>
<b>P09_START</b> specifies a standard approximate solution for problem 9.
</li>
<li>
<b>P09_TITLE</b> returns the title of problem 9.
</li>
<li>
<b>P10_FX</b> evaluates the function for problem 10.
</li>
<li>
<b>P10_JAC</b> sets the jacobian for problem 10.
</li>
<li>
<b>P10_N</b> returns the number of equations for problem 10.
</li>
<li>
<b>P10_SOL</b> returns the solution of problem 10.
</li>
<li>
<b>P10_START</b> specifies a standard approximate solution for problem 10.
</li>
<li>
<b>P10_TITLE</b> returns the title of problem 10.
</li>
<li>
<b>P11_FX</b> evaluates the function for problem 11.
</li>
<li>
<b>P11_JAC</b> sets the jacobian for problem 11.
</li>
<li>
<b>P11_N</b> returns the number of equations for problem 11.
</li>
<li>
<b>P11_SOL</b> returns the solution of problem 11.
</li>
<li>
<b>P11_START</b> specifies a standard approximate solution for problem 11.
</li>
<li>
<b>P11_TITLE</b> returns the title of problem 11.
</li>
<li>
<b>P12_FX</b> evaluates the function for problem 12.
</li>
<li>
<b>P12_JAC</b> sets the jacobian for problem 12.
</li>
<li>
<b>P12_N</b> returns the number of equations for problem 12.
</li>
<li>
<b>P12_SOL</b> returns the solution of problem 12.
</li>
<li>
<b>P12_START</b> specifies a standard approximate solution for problem 12.
</li>
<li>
<b>P12_TITLE</b> returns the title of problem 12.
</li>
<li>
<b>P13_FX</b> evaluates the function for problem 13.
</li>
<li>
<b>P13_JAC</b> sets the jacobian for problem 13.
</li>
<li>
<b>P13_N</b> returns the number of equations for problem 13.
</li>
<li>
<b>P13_SOL</b> returns the solution of problem 13.
</li>
<li>
<b>P13_START</b> specifies a standard approximate solution for problem 13.
</li>
<li>
<b>P13_TITLE</b> returns the title of problem 13.
</li>
<li>
<b>P14_FX</b> evaluates the function for problem 14.
</li>
<li>
<b>P14_JAC</b> sets the jacobian for problem 14.
</li>
<li>
<b>P14_N</b> returns the number of equations for problem 14.
</li>
<li>
<b>P14_SOL</b> returns the solution of problem 14.
</li>
<li>
<b>P14_START</b> specifies a standard approximate solution for problem 14.
</li>
<li>
<b>P14_TITLE</b> returns the title of problem 14.
</li>
<li>
<b>P15_FX</b> evaluates the function for problem 15.
</li>
<li>
<b>P15_JAC</b> sets the jacobian for problem 15.
</li>
<li>
<b>P15_N</b> returns the number of equations for problem 15.
</li>
<li>
<b>P15_SOL</b> returns the solution of problem 15.
</li>
<li>
<b>P15_START</b> specifies a standard approximate solution for problem 15.
</li>
<li>
<b>P15_TITLE</b> returns the title of problem 15.
</li>
<li>
<b>P16_FX</b> evaluates the function for problem 16.
</li>
<li>
<b>P16_JAC</b> sets the jacobian for problem 16.
</li>
<li>
<b>P16_N</b> returns the number of equations for problem 16.
</li>
<li>
<b>P16_SOL</b> returns the solution of problem 16.
</li>
<li>
<b>P16_START</b> specifies a standard approximate solution for problem 16.
</li>
<li>
<b>P16_TITLE</b> returns the title of problem 16.
</li>
<li>
<b>P17_FX</b> evaluates the function for problem 17.
</li>
<li>
<b>P17_JAC</b> sets the jacobian for problem 17.
</li>
<li>
<b>P17_N</b> returns the number of equations for problem 17.
</li>
<li>
<b>P17_SOL</b> returns the solution of problem 17.
</li>
<li>
<b>P17_START</b> specifies a standard approximate solution for problem 17.
</li>
<li>
<b>P17_TITLE</b> returns the title of problem 17.
</li>
<li>
<b>P18_FX</b> evaluates the function for problem 18.
</li>
<li>
<b>P18_JAC</b> sets the jacobian for problem 18.
</li>
<li>
<b>P18_N</b> returns the number of equations for problem 18.
</li>
<li>
<b>P18_SOL</b> returns the solution of problem 18.
</li>
<li>
<b>P18_START</b> specifies a standard approximate solution for problem 18.
</li>
<li>
<b>P18_TITLE</b> returns the title of problem 18.
</li>
<li>
<b>P19_FX</b> evaluates the function for problem 19.
</li>
<li>
<b>P19_JAC</b> sets the jacobian for problem 19.
</li>
<li>
<b>P19_N</b> returns the number of equations for problem 19.
</li>
<li>
<b>P19_SOL</b> returns the solution of problem 19.
</li>
<li>
<b>P19_START</b> specifies a standard approximate solution for problem 19.
</li>
<li>
<b>P19_TITLE</b> returns the title of problem 19.
</li>
<li>
<b>P20_FX</b> evaluates the function for problem 20.
</li>
<li>
<b>P20_JAC</b> sets the jacobian for problem 20.
</li>
<li>
<b>P20_N</b> returns the number of equations for problem 20.
</li>
<li>
<b>P20_SOL</b> returns the solution of problem 20.
</li>
<li>
<b>P20_START</b> specifies a standard approximate solution for problem 20.
</li>
<li>
<b>P20_TITLE</b> returns the title of problem 20.
</li>
<li>
<b>P21_FX</b> evaluates the function for problem 21.
</li>
<li>
<b>P21_JAC</b> sets the jacobian for problem 21
</li>
<li>
<b>P21_N</b> returns the number of equations for problem 21.
</li>
<li>
<b>P21_SOL</b> returns the solution of problem 21.
</li>
<li>
<b>P21_START</b> specifies a standard approximate solution for problem 21.
</li>
<li>
<b>P21_TITLE</b> returns the title of problem 21.
</li>
<li>
<b>P22_FX</b> evaluates the function for problem 22.
</li>
<li>
<b>P22_JAC</b> sets the jacobian for problem 22.
</li>
<li>
<b>P22_N</b> returns the number of equations for problem 22.
</li>
<li>
<b>P22_SOL</b> returns the solution of problem 22.
</li>
<li>
<b>P22_START</b> specifies a standard approximate solution for problem 22.
</li>
<li>
<b>P22_TITLE</b> returns the title of problem 22.
</li>
<li>
<b>P23_FX</b> evaluates the function for problem 23.
</li>
<li>
<b>P23_N</b> returns the number of equations for problem 23.
</li>
<li>
<b>P23_JAC</b> sets the jacobian for problem 23.
</li>
<li>
<b>P23_SOL</b> returns the solution of problem 23.
</li>
<li>
<b>P23_START</b> specifies a standard approximate solution for problem 23.
</li>
<li>
<b>P23_TITLE</b> returns the title of problem 23.
</li>
<li>
<b>R8_SWAP</b> switches two R8's.
</li>
<li>
<b>R8VEC_NORM2</b> returns the 2-norm of a vector.
</li>
<li>
<b>R8GE_FA</b> factors a general matrix.
</li>
<li>
<b>R8GE_SL</b> solves a system factored by SGE_FA.
</li>
<li>
<b>TIMESTAMP</b> prints the current YMDHMS date as a time stamp.
</li>
</ul>
</p>
<p>
You can go up one level to <a href = "../f_src.html">
the FORTRAN90 source codes</a>.
</p>
<hr>
<i>
Last revised on 25 June 2007.
</i>
<!-- John Burkardt -->
</body>
</html>