/
integration.cs
executable file
·3777 lines (3372 loc) · 137 KB
/
integration.cs
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
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
/*************************************************************************
ALGLIB 4.00.0 (source code generated 2023-05-21)
Copyright (c) Sergey Bochkanov (ALGLIB project).
>>> SOURCE LICENSE >>>
This program is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation (www.fsf.org); either version 2 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU General Public License for more details.
A copy of the GNU General Public License is available at
http://www.fsf.org/licensing/licenses
>>> END OF LICENSE >>>
*************************************************************************/
#pragma warning disable 1691
#pragma warning disable 162
#pragma warning disable 164
#pragma warning disable 219
#pragma warning disable 8981
using System;
public partial class alglib
{
/*************************************************************************
Computation of nodes and weights for a Gauss quadrature formula
The algorithm generates the N-point Gauss quadrature formula with weight
function given by coefficients alpha and beta of a recurrence relation
which generates a system of orthogonal polynomials:
P-1(x) = 0
P0(x) = 1
Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
and zeroth moment Mu0
Mu0 = integral(W(x)dx,a,b)
INPUT PARAMETERS:
Alpha - array[0..N-1], alpha coefficients
Beta - array[0..N-1], beta coefficients
Zero-indexed element is not used and may be arbitrary.
Beta[I]>0.
Mu0 - zeroth moment of the weight function.
N - number of nodes of the quadrature formula, N>=1
OUTPUT PARAMETERS:
Info - error code:
* -3 internal eigenproblem solver hasn't converged
* -2 Beta[i]<=0
* -1 incorrect N was passed
* 1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
W - array[0..N-1] - array of quadrature weights.
-- ALGLIB --
Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
public static void gqgeneraterec(double[] alpha, double[] beta, double mu0, int n, out int info, out double[] x, out double[] w)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgeneraterec(alpha, beta, mu0, n, ref info, ref x, ref w, null);
}
public static void gqgeneraterec(double[] alpha, double[] beta, double mu0, int n, out int info, out double[] x, out double[] w, alglib.xparams _params)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgeneraterec(alpha, beta, mu0, n, ref info, ref x, ref w, _params);
}
/*************************************************************************
Computation of nodes and weights for a Gauss-Lobatto quadrature formula
The algorithm generates the N-point Gauss-Lobatto quadrature formula with
weight function given by coefficients alpha and beta of a recurrence which
generates a system of orthogonal polynomials.
P-1(x) = 0
P0(x) = 1
Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
and zeroth moment Mu0
Mu0 = integral(W(x)dx,a,b)
INPUT PARAMETERS:
Alpha - array[0..N-2], alpha coefficients
Beta - array[0..N-2], beta coefficients.
Zero-indexed element is not used, may be arbitrary.
Beta[I]>0
Mu0 - zeroth moment of the weighting function.
A - left boundary of the integration interval.
B - right boundary of the integration interval.
N - number of nodes of the quadrature formula, N>=3
(including the left and right boundary nodes).
OUTPUT PARAMETERS:
Info - error code:
* -3 internal eigenproblem solver hasn't converged
* -2 Beta[i]<=0
* -1 incorrect N was passed
* 1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
W - array[0..N-1] - array of quadrature weights.
-- ALGLIB --
Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
public static void gqgenerategausslobattorec(double[] alpha, double[] beta, double mu0, double a, double b, int n, out int info, out double[] x, out double[] w)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategausslobattorec(alpha, beta, mu0, a, b, n, ref info, ref x, ref w, null);
}
public static void gqgenerategausslobattorec(double[] alpha, double[] beta, double mu0, double a, double b, int n, out int info, out double[] x, out double[] w, alglib.xparams _params)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategausslobattorec(alpha, beta, mu0, a, b, n, ref info, ref x, ref w, _params);
}
/*************************************************************************
Computation of nodes and weights for a Gauss-Radau quadrature formula
The algorithm generates the N-point Gauss-Radau quadrature formula with
weight function given by the coefficients alpha and beta of a recurrence
which generates a system of orthogonal polynomials.
P-1(x) = 0
P0(x) = 1
Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
and zeroth moment Mu0
Mu0 = integral(W(x)dx,a,b)
INPUT PARAMETERS:
Alpha - array[0..N-2], alpha coefficients.
Beta - array[0..N-1], beta coefficients
Zero-indexed element is not used.
Beta[I]>0
Mu0 - zeroth moment of the weighting function.
A - left boundary of the integration interval.
N - number of nodes of the quadrature formula, N>=2
(including the left boundary node).
OUTPUT PARAMETERS:
Info - error code:
* -3 internal eigenproblem solver hasn't converged
* -2 Beta[i]<=0
* -1 incorrect N was passed
* 1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
W - array[0..N-1] - array of quadrature weights.
-- ALGLIB --
Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
public static void gqgenerategaussradaurec(double[] alpha, double[] beta, double mu0, double a, int n, out int info, out double[] x, out double[] w)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategaussradaurec(alpha, beta, mu0, a, n, ref info, ref x, ref w, null);
}
public static void gqgenerategaussradaurec(double[] alpha, double[] beta, double mu0, double a, int n, out int info, out double[] x, out double[] w, alglib.xparams _params)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategaussradaurec(alpha, beta, mu0, a, n, ref info, ref x, ref w, _params);
}
/*************************************************************************
Returns nodes/weights for Gauss-Legendre quadrature on [-1,1] with N
nodes.
INPUT PARAMETERS:
N - number of nodes, >=1
OUTPUT PARAMETERS:
Info - error code:
* -4 an error was detected when calculating
weights/nodes. N is too large to obtain
weights/nodes with high enough accuracy.
Try to use multiple precision version.
* -3 internal eigenproblem solver hasn't converged
* -1 incorrect N was passed
* +1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
W - array[0..N-1] - array of quadrature weights.
-- ALGLIB --
Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gqgenerategausslegendre(int n, out int info, out double[] x, out double[] w)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategausslegendre(n, ref info, ref x, ref w, null);
}
public static void gqgenerategausslegendre(int n, out int info, out double[] x, out double[] w, alglib.xparams _params)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategausslegendre(n, ref info, ref x, ref w, _params);
}
/*************************************************************************
Returns nodes/weights for Gauss-Jacobi quadrature on [-1,1] with weight
function W(x)=Power(1-x,Alpha)*Power(1+x,Beta).
INPUT PARAMETERS:
N - number of nodes, >=1
Alpha - power-law coefficient, Alpha>-1
Beta - power-law coefficient, Beta>-1
OUTPUT PARAMETERS:
Info - error code:
* -4 an error was detected when calculating
weights/nodes. Alpha or Beta are too close
to -1 to obtain weights/nodes with high enough
accuracy, or, may be, N is too large. Try to
use multiple precision version.
* -3 internal eigenproblem solver hasn't converged
* -1 incorrect N/Alpha/Beta was passed
* +1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
W - array[0..N-1] - array of quadrature weights.
-- ALGLIB --
Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gqgenerategaussjacobi(int n, double alpha, double beta, out int info, out double[] x, out double[] w)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategaussjacobi(n, alpha, beta, ref info, ref x, ref w, null);
}
public static void gqgenerategaussjacobi(int n, double alpha, double beta, out int info, out double[] x, out double[] w, alglib.xparams _params)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategaussjacobi(n, alpha, beta, ref info, ref x, ref w, _params);
}
/*************************************************************************
Returns nodes/weights for Gauss-Laguerre quadrature on [0,+inf) with
weight function W(x)=Power(x,Alpha)*Exp(-x)
INPUT PARAMETERS:
N - number of nodes, >=1
Alpha - power-law coefficient, Alpha>-1
OUTPUT PARAMETERS:
Info - error code:
* -4 an error was detected when calculating
weights/nodes. Alpha is too close to -1 to
obtain weights/nodes with high enough accuracy
or, may be, N is too large. Try to use
multiple precision version.
* -3 internal eigenproblem solver hasn't converged
* -1 incorrect N/Alpha was passed
* +1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
W - array[0..N-1] - array of quadrature weights.
-- ALGLIB --
Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gqgenerategausslaguerre(int n, double alpha, out int info, out double[] x, out double[] w)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategausslaguerre(n, alpha, ref info, ref x, ref w, null);
}
public static void gqgenerategausslaguerre(int n, double alpha, out int info, out double[] x, out double[] w, alglib.xparams _params)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategausslaguerre(n, alpha, ref info, ref x, ref w, _params);
}
/*************************************************************************
Returns nodes/weights for Gauss-Hermite quadrature on (-inf,+inf) with
weight function W(x)=Exp(-x*x)
INPUT PARAMETERS:
N - number of nodes, >=1
OUTPUT PARAMETERS:
Info - error code:
* -4 an error was detected when calculating
weights/nodes. May be, N is too large. Try to
use multiple precision version.
* -3 internal eigenproblem solver hasn't converged
* -1 incorrect N/Alpha was passed
* +1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
W - array[0..N-1] - array of quadrature weights.
-- ALGLIB --
Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gqgenerategausshermite(int n, out int info, out double[] x, out double[] w)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategausshermite(n, ref info, ref x, ref w, null);
}
public static void gqgenerategausshermite(int n, out int info, out double[] x, out double[] w, alglib.xparams _params)
{
info = 0;
x = new double[0];
w = new double[0];
gq.gqgenerategausshermite(n, ref info, ref x, ref w, _params);
}
}
public partial class alglib
{
/*************************************************************************
Computation of nodes and weights of a Gauss-Kronrod quadrature formula
The algorithm generates the N-point Gauss-Kronrod quadrature formula with
weight function given by coefficients alpha and beta of a recurrence
relation which generates a system of orthogonal polynomials:
P-1(x) = 0
P0(x) = 1
Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
and zero moment Mu0
Mu0 = integral(W(x)dx,a,b)
INPUT PARAMETERS:
Alpha - alpha coefficients, array[0..floor(3*K/2)].
Beta - beta coefficients, array[0..ceil(3*K/2)].
Beta[0] is not used and may be arbitrary.
Beta[I]>0.
Mu0 - zeroth moment of the weight function.
N - number of nodes of the Gauss-Kronrod quadrature formula,
N >= 3,
N = 2*K+1.
OUTPUT PARAMETERS:
Info - error code:
* -5 no real and positive Gauss-Kronrod formula can
be created for such a weight function with a
given number of nodes.
* -4 N is too large, task may be ill conditioned -
x[i]=x[i+1] found.
* -3 internal eigenproblem solver hasn't converged
* -2 Beta[i]<=0
* -1 incorrect N was passed
* +1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
WKronrod - array[0..N-1] - Kronrod weights
WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
corresponding to extended Kronrod nodes).
-- ALGLIB --
Copyright 08.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gkqgeneraterec(double[] alpha, double[] beta, double mu0, int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss)
{
info = 0;
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
gkq.gkqgeneraterec(alpha, beta, mu0, n, ref info, ref x, ref wkronrod, ref wgauss, null);
}
public static void gkqgeneraterec(double[] alpha, double[] beta, double mu0, int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss, alglib.xparams _params)
{
info = 0;
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
gkq.gkqgeneraterec(alpha, beta, mu0, n, ref info, ref x, ref wkronrod, ref wgauss, _params);
}
/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes/weights for Gauss-Legendre
quadrature with N points.
GKQLegendreCalc (calculation) or GKQLegendreTbl (precomputed table) is
used depending on machine precision and number of nodes.
INPUT PARAMETERS:
N - number of Kronrod nodes, must be odd number, >=3.
OUTPUT PARAMETERS:
Info - error code:
* -4 an error was detected when calculating
weights/nodes. N is too large to obtain
weights/nodes with high enough accuracy.
Try to use multiple precision version.
* -3 internal eigenproblem solver hasn't converged
* -1 incorrect N was passed
* +1 OK
X - array[0..N-1] - array of quadrature nodes, ordered in
ascending order.
WKronrod - array[0..N-1] - Kronrod weights
WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
corresponding to extended Kronrod nodes).
-- ALGLIB --
Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gkqgenerategausslegendre(int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss)
{
info = 0;
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
gkq.gkqgenerategausslegendre(n, ref info, ref x, ref wkronrod, ref wgauss, null);
}
public static void gkqgenerategausslegendre(int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss, alglib.xparams _params)
{
info = 0;
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
gkq.gkqgenerategausslegendre(n, ref info, ref x, ref wkronrod, ref wgauss, _params);
}
/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes/weights for Gauss-Jacobi
quadrature on [-1,1] with weight function
W(x)=Power(1-x,Alpha)*Power(1+x,Beta).
INPUT PARAMETERS:
N - number of Kronrod nodes, must be odd number, >=3.
Alpha - power-law coefficient, Alpha>-1
Beta - power-law coefficient, Beta>-1
OUTPUT PARAMETERS:
Info - error code:
* -5 no real and positive Gauss-Kronrod formula can
be created for such a weight function with a
given number of nodes.
* -4 an error was detected when calculating
weights/nodes. Alpha or Beta are too close
to -1 to obtain weights/nodes with high enough
accuracy, or, may be, N is too large. Try to
use multiple precision version.
* -3 internal eigenproblem solver hasn't converged
* -1 incorrect N was passed
* +1 OK
* +2 OK, but quadrature rule have exterior nodes,
x[0]<-1 or x[n-1]>+1
X - array[0..N-1] - array of quadrature nodes, ordered in
ascending order.
WKronrod - array[0..N-1] - Kronrod weights
WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
corresponding to extended Kronrod nodes).
-- ALGLIB --
Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gkqgenerategaussjacobi(int n, double alpha, double beta, out int info, out double[] x, out double[] wkronrod, out double[] wgauss)
{
info = 0;
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
gkq.gkqgenerategaussjacobi(n, alpha, beta, ref info, ref x, ref wkronrod, ref wgauss, null);
}
public static void gkqgenerategaussjacobi(int n, double alpha, double beta, out int info, out double[] x, out double[] wkronrod, out double[] wgauss, alglib.xparams _params)
{
info = 0;
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
gkq.gkqgenerategaussjacobi(n, alpha, beta, ref info, ref x, ref wkronrod, ref wgauss, _params);
}
/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes for quadrature with N points.
Reduction to tridiagonal eigenproblem is used.
INPUT PARAMETERS:
N - number of Kronrod nodes, must be odd number, >=3.
OUTPUT PARAMETERS:
Info - error code:
* -4 an error was detected when calculating
weights/nodes. N is too large to obtain
weights/nodes with high enough accuracy.
Try to use multiple precision version.
* -3 internal eigenproblem solver hasn't converged
* -1 incorrect N was passed
* +1 OK
X - array[0..N-1] - array of quadrature nodes, ordered in
ascending order.
WKronrod - array[0..N-1] - Kronrod weights
WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
corresponding to extended Kronrod nodes).
-- ALGLIB --
Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gkqlegendrecalc(int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss)
{
info = 0;
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
gkq.gkqlegendrecalc(n, ref info, ref x, ref wkronrod, ref wgauss, null);
}
public static void gkqlegendrecalc(int n, out int info, out double[] x, out double[] wkronrod, out double[] wgauss, alglib.xparams _params)
{
info = 0;
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
gkq.gkqlegendrecalc(n, ref info, ref x, ref wkronrod, ref wgauss, _params);
}
/*************************************************************************
Returns Gauss and Gauss-Kronrod nodes for quadrature with N points using
pre-calculated table. Nodes/weights were computed with accuracy up to
1.0E-32 (if MPFR version of ALGLIB is used). In standard double precision
accuracy reduces to something about 2.0E-16 (depending on your compiler's
handling of long floating point constants).
INPUT PARAMETERS:
N - number of Kronrod nodes.
N can be 15, 21, 31, 41, 51, 61.
OUTPUT PARAMETERS:
X - array[0..N-1] - array of quadrature nodes, ordered in
ascending order.
WKronrod - array[0..N-1] - Kronrod weights
WGauss - array[0..N-1] - Gauss weights (interleaved with zeros
corresponding to extended Kronrod nodes).
-- ALGLIB --
Copyright 12.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void gkqlegendretbl(int n, out double[] x, out double[] wkronrod, out double[] wgauss, out double eps)
{
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
eps = 0;
gkq.gkqlegendretbl(n, ref x, ref wkronrod, ref wgauss, ref eps, null);
}
public static void gkqlegendretbl(int n, out double[] x, out double[] wkronrod, out double[] wgauss, out double eps, alglib.xparams _params)
{
x = new double[0];
wkronrod = new double[0];
wgauss = new double[0];
eps = 0;
gkq.gkqlegendretbl(n, ref x, ref wkronrod, ref wgauss, ref eps, _params);
}
}
public partial class alglib
{
/*************************************************************************
Integration report:
* TerminationType = completetion code:
* -5 non-convergence of Gauss-Kronrod nodes
calculation subroutine.
* -1 incorrect parameters were specified
* 1 OK
* Rep.NFEV countains number of function calculations
* Rep.NIntervals contains number of intervals [a,b]
was partitioned into.
*************************************************************************/
public class autogkreport : alglibobject
{
//
// Public declarations
//
public int terminationtype { get { return _innerobj.terminationtype; } set { _innerobj.terminationtype = value; } }
public int nfev { get { return _innerobj.nfev; } set { _innerobj.nfev = value; } }
public int nintervals { get { return _innerobj.nintervals; } set { _innerobj.nintervals = value; } }
public autogkreport()
{
_innerobj = new autogk.autogkreport();
}
public override alglib.alglibobject make_copy()
{
return new autogkreport((autogk.autogkreport)_innerobj.make_copy());
}
//
// Although some of declarations below are public, you should not use them
// They are intended for internal use only
//
private autogk.autogkreport _innerobj;
public autogk.autogkreport innerobj { get { return _innerobj; } }
public autogkreport(autogk.autogkreport obj)
{
_innerobj = obj;
}
}
/*************************************************************************
This structure stores state of the integration algorithm.
Although this class has public fields, they are not intended for external
use. You should use ALGLIB functions to work with this class:
* autogksmooth()/AutoGKSmoothW()/... to create objects
* autogkintegrate() to begin integration
* autogkresults() to get results
*************************************************************************/
public class autogkstate : alglibobject
{
//
// Public declarations
//
public bool needf { get { return _innerobj.needf; } set { _innerobj.needf = value; } }
public double x { get { return _innerobj.x; } set { _innerobj.x = value; } }
public double xminusa { get { return _innerobj.xminusa; } set { _innerobj.xminusa = value; } }
public double bminusx { get { return _innerobj.bminusx; } set { _innerobj.bminusx = value; } }
public double f { get { return _innerobj.f; } set { _innerobj.f = value; } }
public autogkstate()
{
_innerobj = new autogk.autogkstate();
}
public override alglib.alglibobject make_copy()
{
return new autogkstate((autogk.autogkstate)_innerobj.make_copy());
}
//
// Although some of declarations below are public, you should not use them
// They are intended for internal use only
//
private autogk.autogkstate _innerobj;
public autogk.autogkstate innerobj { get { return _innerobj; } }
public autogkstate(autogk.autogkstate obj)
{
_innerobj = obj;
}
}
/*************************************************************************
Integration of a smooth function F(x) on a finite interval [a,b].
Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
is calculated with accuracy close to the machine precision.
Algorithm works well only with smooth integrands. It may be used with
continuous non-smooth integrands, but with less performance.
It should never be used with integrands which have integrable singularities
at lower or upper limits - algorithm may crash. Use AutoGKSingular in such
cases.
INPUT PARAMETERS:
A, B - interval boundaries (A<B, A=B or A>B)
OUTPUT PARAMETERS
State - structure which stores algorithm state
SEE ALSO
AutoGKSmoothW, AutoGKSingular, AutoGKResults.
-- ALGLIB --
Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void autogksmooth(double a, double b, out autogkstate state)
{
state = new autogkstate();
autogk.autogksmooth(a, b, state.innerobj, null);
}
public static void autogksmooth(double a, double b, out autogkstate state, alglib.xparams _params)
{
state = new autogkstate();
autogk.autogksmooth(a, b, state.innerobj, _params);
}
/*************************************************************************
Integration of a smooth function F(x) on a finite interval [a,b].
This subroutine is same as AutoGKSmooth(), but it guarantees that interval
[a,b] is partitioned into subintervals which have width at most XWidth.
Subroutine can be used when integrating nearly-constant function with
narrow "bumps" (about XWidth wide). If "bumps" are too narrow, AutoGKSmooth
subroutine can overlook them.
INPUT PARAMETERS:
A, B - interval boundaries (A<B, A=B or A>B)
OUTPUT PARAMETERS
State - structure which stores algorithm state
SEE ALSO
AutoGKSmooth, AutoGKSingular, AutoGKResults.
-- ALGLIB --
Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void autogksmoothw(double a, double b, double xwidth, out autogkstate state)
{
state = new autogkstate();
autogk.autogksmoothw(a, b, xwidth, state.innerobj, null);
}
public static void autogksmoothw(double a, double b, double xwidth, out autogkstate state, alglib.xparams _params)
{
state = new autogkstate();
autogk.autogksmoothw(a, b, xwidth, state.innerobj, _params);
}
/*************************************************************************
Integration on a finite interval [A,B].
Integrand have integrable singularities at A/B.
F(X) must diverge as "(x-A)^alpha" at A, as "(B-x)^beta" at B, with known
alpha/beta (alpha>-1, beta>-1). If alpha/beta are not known, estimates
from below can be used (but these estimates should be greater than -1 too).
One of alpha/beta variables (or even both alpha/beta) may be equal to 0,
which means than function F(x) is non-singular at A/B. Anyway (singular at
bounds or not), function F(x) is supposed to be continuous on (A,B).
Fast-convergent algorithm based on a Gauss-Kronrod formula is used. Result
is calculated with accuracy close to the machine precision.
INPUT PARAMETERS:
A, B - interval boundaries (A<B, A=B or A>B)
Alpha - power-law coefficient of the F(x) at A,
Alpha>-1
Beta - power-law coefficient of the F(x) at B,
Beta>-1
OUTPUT PARAMETERS
State - structure which stores algorithm state
SEE ALSO
AutoGKSmooth, AutoGKSmoothW, AutoGKResults.
-- ALGLIB --
Copyright 06.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void autogksingular(double a, double b, double alpha, double beta, out autogkstate state)
{
state = new autogkstate();
autogk.autogksingular(a, b, alpha, beta, state.innerobj, null);
}
public static void autogksingular(double a, double b, double alpha, double beta, out autogkstate state, alglib.xparams _params)
{
state = new autogkstate();
autogk.autogksingular(a, b, alpha, beta, state.innerobj, _params);
}
/*************************************************************************
This function provides reverse communication interface
Reverse communication interface is not documented or recommended to use.
See below for functions which provide better documented API
*************************************************************************/
public static bool autogkiteration(autogkstate state)
{
return autogk.autogkiteration(state.innerobj, null);
}
public static bool autogkiteration(autogkstate state, alglib.xparams _params)
{
return autogk.autogkiteration(state.innerobj, _params);
}
/*************************************************************************
This function is used to launcn iterations of ODE solver
It accepts following parameters:
diff - callback which calculates dy/dx for given y and x
obj - optional object which is passed to diff; can be NULL
-- ALGLIB --
Copyright 07.05.2009 by Bochkanov Sergey
*************************************************************************/
public static void autogkintegrate(autogkstate state, integrator1_func func, object obj)
{
autogkintegrate(state, func, obj, null);
}
public static void autogkintegrate(autogkstate state, integrator1_func func, object obj, alglib.xparams _params)
{
if( func==null )
throw new alglibexception("ALGLIB: error in 'autogkintegrate()' (func is null)");
while( alglib.autogkiteration(state, _params) )
{
if( state.needf )
{
func(state.innerobj.x, state.innerobj.xminusa, state.innerobj.bminusx, ref state.innerobj.f, obj);
continue;
}
throw new alglibexception("ALGLIB: unexpected error in 'autogksolve'");
}
}
/*************************************************************************
Adaptive integration results
Called after AutoGKIteration returned False.
Input parameters:
State - algorithm state (used by AutoGKIteration).
Output parameters:
V - integral(f(x)dx,a,b)
Rep - optimization report (see AutoGKReport description)
-- ALGLIB --
Copyright 14.11.2007 by Bochkanov Sergey
*************************************************************************/
public static void autogkresults(autogkstate state, out double v, out autogkreport rep)
{
v = 0;
rep = new autogkreport();
autogk.autogkresults(state.innerobj, ref v, rep.innerobj, null);
}
public static void autogkresults(autogkstate state, out double v, out autogkreport rep, alglib.xparams _params)
{
v = 0;
rep = new autogkreport();
autogk.autogkresults(state.innerobj, ref v, rep.innerobj, _params);
}
}
public partial class alglib
{
public class gq
{
/*************************************************************************
Computation of nodes and weights for a Gauss quadrature formula
The algorithm generates the N-point Gauss quadrature formula with weight
function given by coefficients alpha and beta of a recurrence relation
which generates a system of orthogonal polynomials:
P-1(x) = 0
P0(x) = 1
Pn+1(x) = (x-alpha(n))*Pn(x) - beta(n)*Pn-1(x)
and zeroth moment Mu0
Mu0 = integral(W(x)dx,a,b)
INPUT PARAMETERS:
Alpha - array[0..N-1], alpha coefficients
Beta - array[0..N-1], beta coefficients
Zero-indexed element is not used and may be arbitrary.
Beta[I]>0.
Mu0 - zeroth moment of the weight function.
N - number of nodes of the quadrature formula, N>=1
OUTPUT PARAMETERS:
Info - error code:
* -3 internal eigenproblem solver hasn't converged
* -2 Beta[i]<=0
* -1 incorrect N was passed
* 1 OK
X - array[0..N-1] - array of quadrature nodes,
in ascending order.
W - array[0..N-1] - array of quadrature weights.
-- ALGLIB --
Copyright 2005-2009 by Bochkanov Sergey
*************************************************************************/
public static void gqgeneraterec(double[] alpha,
double[] beta,
double mu0,
int n,
ref int info,
ref double[] x,
ref double[] w,
alglib.xparams _params)
{
int i = 0;
double[] d = new double[0];
double[] e = new double[0];
double[,] z = new double[0,0];
info = 0;
x = new double[0];
w = new double[0];
if( n<1 )
{
info = -1;
return;
}
info = 1;
//
// Initialize
//
d = new double[n];
e = new double[n];
for(i=1; i<=n-1; i++)
{
d[i-1] = alpha[i-1];
if( (double)(beta[i])<=(double)(0) )
{
info = -2;
return;
}
e[i-1] = Math.Sqrt(beta[i]);
}
d[n-1] = alpha[n-1];
//
// EVD
//
if( !evd.smatrixtdevd(d, e, n, 3, ref z, _params) )
{
info = -3;
return;
}
//
// Generate
//
x = new double[n];
w = new double[n];
for(i=1; i<=n; i++)
{
x[i-1] = d[i-1];
w[i-1] = mu0*math.sqr(z[0,i-1]);
}
}