forked from johannesgerer/jburkardt-m
-
Notifications
You must be signed in to change notification settings - Fork 0
/
stroud.html
1939 lines (1898 loc) · 65.5 KB
/
stroud.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
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
<html>
<head>
<title>
STROUD - Numerical Integration in M Dimensions
</title>
</head>
<body bgcolor="#EEEEEE" link="#CC0000" alink="#FF3300" vlink="#000055">
<h1 align = "center">
STROUD <br> Numerical Integration<br> in M Dimensions
</h1>
<hr>
<p>
<b>STROUD</b>
is a MATLAB library which
defines quadrature rules for a variety of M-dimensional regions,
including the interior of the square, cube and hypercube, the pyramid,
cone and ellipse, the hexagon, the M-dimensional octahedron,
the circle, sphere and hypersphere, the triangle, tetrahedron and simplex,
and the surface of the circle, sphere and hypersphere.
</p>
<p>
A few other rules have been collected as well,
particularly for quadrature over the interior of a triangle, which is
useful in finite element calculations.
</p>
<p>
Arthur Stroud published his vast collection of quadrature formulas
for multidimensional regions in 1971. In a few cases, he printed
sample FORTRAN77 programs to compute these integrals. Integration
regions included:
<ul>
<li>
<b>C2</b>, the interior of the square;
</li>
<li>
<b>C3</b>, the interior of the cube;
</li>
<li>
<b>CN</b>, the interior of the N-dimensional hypercube;
</li>
<li>
<b>CN:C2</b>, a 3-dimensional pyramid;
</li>
<li>
<b>CN:S2</b>, a 3-dimensional cone;
</li>
<li>
<b>CN_SHELL</b>, the region contained between two concentric
N-dimensional hypercubes;
</li>
<li>
<b>ELP</b>, the interior of the 2-dimensional ellipse with
weight function 1/sqrt((x-c)^2+y^2)/(sqrt((x+c)^2+y^2);
</li>
<li>
<b>EN_R</b>, all of N-dimensional space, with the weight function:<br>
w(x) = exp ( - sqrt ( sum ( 1 <= i < n ) x(i)^2 ) );
</li>
<li>
<b>EN_R2</b>, all of N-dimensional space, with the Hermite weight function:
w(x) = product ( 1 <= i <= n ) exp ( - x(i)^2 );
</li>
<li>
<b>GN</b>, the interior of the N-dimensional octahedron;
</li>
<li>
<b>H2</b>, the interior of the 2-dimensional hexagon;
</li>
<li>
<b>PAR</b>, the first parabolic region;
</li>
<li>
<b>PAR2</b>, the second parabolic region;
</li>
<li>
<b>PAR3</b>, the third parabolic region;
</li>
<li>
<b>S2</b>, the interior of the circle;
</li>
<li>
<b>S3</b>, the interior of the sphere;
</li>
<li>
<b>SN</b>, the interior of the N-dimensional hypersphere;
</li>
<li>
<b>SN_SHELL</b>, the region contained between two concentric N-dimensional hyperspheres;
</li>
<li>
<b>T2</b>, the interior of the triangle;
</li>
<li>
<b>T3</b>, the interior of the tetrahedron;
</li>
<li>
<b>TN</b>, the interior of the N-dimensional simplex;
</li>
<li>
<b>TOR3:S2</b>, the interior of a 3-dimensional torus with circular cross-section;
</li>
<li>
<b>TOR3:C2</b>, the interior of a 3-dimensional torus with square cross-section;
</li>
<li>
<b>U2</b>, the "surface" of the circle;
</li>
<li>
<b>U3</b>, the surface of the sphere;
</li>
<li>
<b>UN</b>, the surface of the N-dimensional sphere;
</li>
</ul>
</p>
<p>
We have added a few new terms for regions:
<ul>
<li>
<b>CN_GEG</b>, the N dimensional hypercube [-1,+1]^N, with the Gegenbauer
weight function:<br>
w(alpha;x) = product ( 1 <= i <= n ) ( 1 - x(i)^2 )^alpha;
</li>
<li>
<b>CN_JAC</b>, the N dimensional hypercube [-1,+1]^N, with the Beta or
Jacobi weight function:<br>
w(alpha,beta;x) = product ( 1 <= i <= n ) ( 1 - x(i) )^alpha * ( 1 + x(i) )^beta;
</li>
<li>
<b>CN_LEG</b>, the N dimensional hypercube [-1,+1]^N, with the Legendre
weight function:<br>
w(x) = 1;
</li>
<li>
<b>EPN_GLG</b>, the positive space [0,+oo)^N, with the generalized
Laguerre weight function:<br>
w(alpha;x) = product ( 1 <= i <= n ) x(i)^alpha exp ( - x(i) );
</li>
<li>
<b>EPN_LAG</b>, the positive space [0,+oo)^N, with the exponential or
Laguerre weight function:<br>
w(x) = product ( 1 <= i <= n ) exp ( - x(i) );
</li>
</ul>
</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>STROUD</b> is available in
<a href = "../../c_src/stroud/stroud.html">a C version</a> and
<a href = "../../cpp_src/stroud/stroud.html">a C++ version</a> and
<a href = "../../f77_src/stroud/stroud.html">a FORTRAN77 version</a> and
<a href = "../../f_src/stroud/stroud.html">a FORTRAN90 version</a> and
<a href = "../../m_src/stroud/stroud.html">a MATLAB version</a>.
</p>
<h3 align = "center">
Related Data and Programs:
</h3>
<p>
<a href = "../../m_src/disk_rule/disk_rule.html">
DISK_RULE</a>,
a MATLAB library which
computes quadrature rules for the unit disk in 2D, that is,
the interior of the circle of radius 1 and center (0,0).
</p>
<p>
<a href = "../../m_src/felippa/felippa.html">
FELIPPA</a>,
a MATLAB library which
defines quadrature rules for lines, triangles, quadrilaterals,
pyramids, wedges, tetrahedrons and hexahedrons.
</p>
<p>
<a href = "../../m_src/pyramid_rule/pyramid_rule.html">
PYRAMID_RULE</a>,
a MATLAB program which
computes a quadrature rule for a pyramid.
</p>
<p>
<a href = "../../m_src/simplex_gm_rule/simplex_gm_rule.html">
SIMPLEX_GM_RULE</a>,
a MATLAB library which
defines Grundmann-Moeller quadrature rules
over the interior of a triangle in 2D, a tetrahedron in 3D, or
over the interior of the simplex in M dimensions.
</p>
<p>
<a href = "../../m_src/sphere_lebedev_rule/sphere_lebedev_rule.html">
SPHERE_LEBEDEV_RULE</a>,
a MATLAB library which
computes Lebedev quadrature rules
on the surface of the unit sphere in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_arbq_rule/tetrahedron_arbq_rule.html">
TETRAHEDRON_ARBQ_RULE</a>,
a MATLAB library which
returns quadrature rules,
with exactness up to total degree 15,
over the interior of a tetrahedron in 3D,
by Hong Xiao and Zydrunas Gimbutas.
</p>
<p>
<a href = "../../m_src/tetrahedron_keast_rule/tetrahedron_keast_rule.html">
TETRAHEDRON_KEAST_RULE</a>,
a MATLAB library which
defines ten quadrature rules, with exactness degrees 0 through 8,
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_ncc_rule/tetrahedron_ncc_rule.html">
TETRAHEDRON_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/tetrahedron_nco_rule/tetrahedron_nco_rule.html">
TETRAHEDRON_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes Open (NCO) quadrature rules
over the interior of a tetrahedron in 3D.
</p>
<p>
<a href = "../../m_src/triangle_dunavant_rule/triangle_dunavant_rule.html">
TRIANGLE_DUNAVANT_RULE</a>,
a MATLAB library which
defines Dunavant rules for quadrature
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_fekete/triangle_fekete.html">
TRIANGLE_FEKETE</a>,
a MATLAB library which
defines Fekete rules for quadrature or interpolation
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../cpp_src/triangle_lyness_rule/triangle_lyness_rule.html">
TRIANGLE_LYNESS_RULE</a>,
a C++ library which
returns Lyness-Jespersen quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_ncc_rule/triangle_ncc_rule.html">
TRIANGLE_NCC_RULE</a>,
a MATLAB library which
defines Newton-Cotes Closed (NCC) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_nco_rule/triangle_nco_rule.html">
TRIANGLE_NCO_RULE</a>,
a MATLAB library which
defines Newton-Cotes Open (NCO) quadrature rules
over the interior of a triangle in 2D.
</p>
<p>
<a href = "../../m_src/triangle_wandzura_rule/triangle_wandzura_rule.html">
TRIANGLE_WANDZURA_RULE</a>,
a MATLAB library which
returns quadrature rules of exactness 5, 10, 15, 20, 25 and 30
over the interior of the triangle in 2D.
</p>
<h3 align = "center">
Reference:
</h3>
<p>
<ol>
<li>
Milton Abramowitz, Irene Stegun,<br>
Handbook of Mathematical Functions,<br>
National Bureau of Standards, 1964,<br>
ISBN: 0-486-61272-4,<br>
LC: QA47.A34.
</li>
<li>
Jarle Berntsen, Terje Espelid,<br>
Algorithm 706:
DCUTRI: an algorithm for adaptive cubature
over a collection of triangles,<br>
ACM Transactions on Mathematical Software,<br>
Volume 18, Number 3, September 1992, pages 329-342.
</li>
<li>
SF Bockman,<br>
Generalizing the Formula for Areas of Polygons to Moments,<br>
American Mathematical Society Monthly,<br>
Volume 96, Number 2, February 1989, pages 131-132.
</li>
<li>
Paul Bratley, Bennett Fox, Linus Schrage,<br>
A Guide to Simulation,<br>
Second Edition,<br>
Springer, 1987,<br>
ISBN: 0387964673,<br>
LC: QA76.9.C65.B73.
</li>
<li>
William Cody, Kenneth Hillstrom,<br>
Chebyshev Approximations for the Natural Logarithm of the
Gamma Function,
Mathematics of Computation,<br>
Volume 21, Number 98, April 1967, pages 198-203.
</li>
<li>
Philip Davis, Philip Rabinowitz,<br>
Methods of Numerical Integration,<br>
Second Edition,<br>
Dover, 2007,<br>
ISBN: 0486453391,<br>
LC: QA299.3.D28.
</li>
<li>
Elise deDoncker, Ian Robinson,<br>
Algorithm 612:
Integration over a Triangle Using Nonlinear Extrapolation,<br>
ACM Transactions on Mathematical Software,<br>
Volume 10, Number 1, March 1984, pages 17-22.
</li>
<li>
Hermann Engels,<br>
Numerical Quadrature and Cubature,<br>
Academic Press, 1980,<br>
ISBN: 012238850X,<br>
LC: QA299.3E5.
</li>
<li>
Thomas Ericson, Victor Zinoviev,<br>
Codes on Euclidean Spheres,<br>
Elsevier, 2001,<br>
ISBN: 0444503293,<br>
LC: QA166.7E75
</li>
<li>
Carlos Felippa,<br>
A compendium of FEM integration formulas for symbolic work,<br>
Engineering Computation,<br>
Volume 21, Number 8, 2004, pages 867-890.
</li>
<li>
Gerald Folland,<br>
How to Integrate a Polynomial Over a Sphere,<br>
American Mathematical Monthly,<br>
Volume 108, Number 5, May 2001, pages 446-448.
</li>
<li>
Bennett Fox,<br>
Algorithm 647:
Implementation and Relative Efficiency of Quasirandom
Sequence Generators,<br>
ACM Transactions on Mathematical Software,<br>
Volume 12, Number 4, December 1986, pages 362-376.
</li>
<li>
Axel Grundmann, Michael Moeller,<br>
Invariant Integration Formulas for the N-Simplex
by Combinatorial Methods,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 15, Number 2, April 1978, pages 282-290.
</li>
<li>
John Harris, Horst Stocker,<br>
Handbook of Mathematics and Computational Science,<br>
Springer, 1998,<br>
ISBN: 0-387-94746-9,<br>
LC: QA40.S76.
</li>
<li>
Patrick Keast,<br>
Moderate Degree Tetrahedral Quadrature Formulas,<br>
Computer Methods in Applied Mechanics and Engineering,<br>
Volume 55, Number 3, May 1986, pages 339-348.
</li>
<li>
Vladimir Krylov,<br>
Approximate Calculation of Integrals,<br>
Dover, 2006,<br>
ISBN: 0486445798,<br>
LC: QA311.K713.
</li>
<li>
Dirk Laurie,<br>
Algorithm 584:
CUBTRI, Automatic Cubature Over a Triangle,<br>
ACM Transactions on Mathematical Software,<br>
Volume 8, Number 2, 1982, pages 210-218.
</li>
<li>
Frank Lether,<br>
A Generalized Product Rule for the Circle,<br>
SIAM Journal on Numerical Analysis,<br>
Volume 8, Number 2, June 1971, pages 249-253.
</li>
<li>
James Lyness, Dennis Jespersen,<br>
Moderate Degree Symmetric Quadrature Rules for the Triangle,<br>
Journal of the Institute of Mathematics and its Applications,<br>
Volume 15, Number 1, February 1975, pages 19-32.
</li>
<li>
James Lyness, BJJ McHugh,<br>
Integration Over Multidimensional Hypercubes,
A Progressive Procedure,<br>
The Computer Journal,<br>
Volume 6, 1963, pages 264-270.
</li>
<li>
AD McLaren,<br>
Optimal Numerical Integration on a Sphere,<br>
Mathematics of Computation,<br>
Volume 17, Number 84, October 1963, pages 361-383.
</li>
<li>
Albert Nijenhuis, Herbert Wilf,<br>
Combinatorial Algorithms for Computers and Calculators,<br>
Second Edition,<br>
Academic Press, 1978,<br>
ISBN: 0-12-519260-6,<br>
LC: QA164.N54.
</li>
<li>
William Peirce,<br>
Numerical Integration Over the Planar Annulus,<br>
Journal of the Society for Industrial and Applied Mathematics,<br>
Volume 5, Number 2, June 1957, pages 66-73.
</li>
<li>
Hans Rudolf Schwarz,<br>
Finite Element Methods,<br>
Academic Press, 1988,<br>
ISBN: 0126330107,<br>
LC: TA347.F5.S3313.
</li>
<li>
Gilbert Strang, George Fix,<br>
An Analysis of the Finite Element Method,<br>
Cambridge, 1973,<br>
ISBN: 096140888X,<br>
LC: TA335.S77.
</li>
<li>
Arthur Stroud,<br>
Approximate Calculation of Multiple Integrals,<br>
Prentice Hall, 1971,<br>
ISBN: 0130438936,<br>
LC: QA311.S85.
</li>
<li>
Arthur Stroud, Don Secrest,<br>
Gaussian Quadrature Formulas,<br>
Prentice Hall, 1966,<br>
LC: QA299.4G3S7.
</li>
<li>
Stephen Wandzura, Hong Xiao,<br>
Symmetric Quadrature Rules on a Triangle,<br>
Computers and Mathematics with Applications,<br>
Volume 45, 2003, pages 1829-1840.
</li>
<li>
Stephen Wolfram,<br>
The Mathematica Book,<br>
Fourth Edition,<br>
Cambridge University Press, 1999,<br>
ISBN: 0-521-64314-7,<br>
LC: QA76.95.W65.
</li>
<li>
Dongbin Xiu,<br>
Numerical integration formulas of degree two,<br>
Applied Numerical Mathematics,<br>
Volume 58, 2008, pages 1515-1520.
</li>
<li>
Olgierd Zienkiewicz,<br>
The Finite Element Method,<br>
Sixth Edition,<br>
Butterworth-Heinemann, 2005,<br>
ISBN: 0750663200,<br>
LC: TA640.2.Z54
</li>
<li>
Daniel Zwillinger, editor,<br>
CRC Standard Mathematical Tables and Formulae,<br>
30th Edition,<br>
CRC Press, 1996,<br>
ISBN: 0-8493-2479-3,<br>
LC: QA47.M315.
</li>
</ol>
</p>
<h3 align = "center">
Source Code:
</h3>
<p>
<ul>
<li>
<a href = "ball_f1_nd.m">ball_f1_nd.m</a>,
approximates an integral in a ball in ND;
</li>
<li>
<a href = "ball_f3_nd.m">ball_f3_nd.m</a>,
approximates an integral in a ball in ND;
</li>
<li>
<a href = "ball_monomial_nd.m">ball_monomial_nd.m</a>,
approximates the integral of a monomial in a ball in ND;
</li>
<li>
<a href = "ball_unit_07_3d.m">ball_unit_07_3d.m</a>,
approximates an integral in the unit ball in 3D;
</li>
<li>
<a href = "ball_unit_14_3d.m">ball_unit_14_3d.m</a>,
approximates an integral in the unit ball in 3D;
</li>
<li>
<a href = "ball_unit_15_3d.m">ball_unit_15_3d.m</a>,
approximates an integral in the unit ball in 3D;
</li>
<li>
<a href = "ball_unit_f1_nd.m">ball_unit_f1_nd.m</a>,
approximates an integral in the unit ball in ND;
</li>
<li>
<a href = "ball_unit_f3_nd.m">ball_unit_f3_nd.m</a>,
approximates an integral in the unit ball in ND;
</li>
<li>
<a href = "ball_unit_volume_3d.m">ball_unit_volume_3d.m</a>,
returns the volume of the unit ball in 3D;
</li>
<li>
<a href = "ball_unit_volume_nd.m">ball_unit_volume_nd.m</a>,
returns the volume of the unit ball in ND;
</li>
<li>
<a href = "ball_volume_3d.m">ball_volume_3d.m</a>,
returns the volume of a ball in 3D;
</li>
<li>
<a href = "ball_volume_nd.m">ball_volume_nd.m</a>,
returns the volume of a ball in ND;
</li>
<li>
<a href = "c1_geg_monomial_integral.m">c1_geg_monomial_integral.m</a>
integral of a monomial with Gegenbauer weight over C1.
</li>
<li>
<a href = "c1_jac_monomial_integral.m">c1_jac_monomial_integral.m</a>
integral of a monomial with Jacobi weight over C1.
</li>
<li>
<a href = "c1_leg_monomial_integral.m">c1_leg_monomial_integral.m</a>
integral of a monomial with Legendre weight over C1.
</li>
<li>
<a href = "circle_annulus.m">
circle_annulus.m</a>,
approximates an integral inside a circular annulus in 2D;
</li>
<li>
<a href = "circle_annulus_area_2d.m">
circle_annulus_area_2d.m</a>,
returns the area of a circular annulus in 2D;
</li>
<li>
<a href = "circle_annulus_sector.m">
circle_annulus_sector.m</a>,
approximates an integral in a sector of a circular annulus in 2D;
</li>
<li>
<a href = "circle_annulus_sector_area_2d.m">
circle_annulus_sector_area_2d.m</a>,
returns the area of a sector of a circular annulus in 2D;
</li>
<li>
<a href = "circle_area_2d.m">
circle_area_2d.m</a>,
returns the area of a circle in 2D;
</li>
<li>
<a href = "circle_cap_area_2d.m">
circle_cap_area_2d.m</a>,
computes the area of a circle cap in 2D.
</li>
<li>
<a href = "circle_cum.m">
circle_cum.m</a>,
computes an integral on the circumference of a circle in 2D;
</li>
<li>
<a href = "circle_rt_set.m">
circle_rt_set.m</a>,
sets an R-Theta product quadrature rule for inside a circle in 2D;
</li>
<li>
<a href = "circle_rt_size.m">
circle_rt_size.m</a>,
sizes an R-Theta product quadrature rule for inside a circle in 2D;
</li>
<li>
<a href = "circle_rt_sum.m">
circle_rt_sum.m</a>,
applies an R-Theta product quadrature rule inside a circle in 2D;
</li>
<li>
<a href = "circle_sector.m">
circle_sector.m</a>,
approximates an integral in a circular sector in 2D;
</li>
<li>
<a href = "circle_sector_area_2d.m">
circle_sector_area_2d.m</a>,
returns the area of a circular sector in 2D;
</li>
<li>
<a href = "circle_triangle_area_2d.m">
circle_triangle_area_2d.m</a>,
returns the area of a circular triangle in 2D;
</li>
<li>
<a href = "circle_xy_set.m">
circle_xy_set.m</a>,
sets an XY quadrature rule for inside a circle in 2D;
</li>
<li>
<a href = "circle_xy_size.m">
circle_xy_size.m</a>,
sizes an XY product quadrature rule for inside a circle in 2D;
</li>
<li>
<a href = "circle_xy_sum.m">
circle_xy_sum.m</a>,
applies an XY quadrature rule inside a circle in 2D;
</li>
<li>
<a href = "cn_geg_00_1.m">cn_geg_00_1.m</a>
implements the midpoint rule for region CN_GEG.
</li>
<li>
<a href = "cn_geg_01_1.m">cn_geg_01_1.m</a>
implements a precision 1 rule for region CN_GEG.
</li>
<li>
<a href = "cn_geg_02_xiu.m">cn_geg_02_xiu.m</a>
implements the Xiu rule for region CN_GEG.
</li>
<li>
<a href = "cn_geg_03_xiu.m">cn_geg_03_xiu.m</a>
implements the Xiu rule for region CN_GEG.
</li>
<li>
<a href = "cn_geg_monomial_integral.m">cn_geg_monoial_integral.m</a>
integral of a monomial with Gegenbauer weight over CN.
</li>
<li>
<a href = "cn_jac_00_1.m">cn_jac_00_1.m</a>
implements the midpoint rule for region CN_JAC.
</li>
<li>
<a href = "cn_jac_01_1.m">cn_jac_01_1.m</a>
implements a precision 1 rule for region CN_JAC.
</li>
<li>
<a href = "cn_jac_02_xiu.m">cn_jac_02_xiu.m</a>
implements the Xiu rule for region CN_JAC.
</li>
<li>
<a href = "cn_jac_monomial_integral.m">cn_jac_monoial_integral.m</a>
integral of a monomial with Jacobi weight over CN.
</li>
<li>
<a href = "cn_leg_01_1.m">cn_leg_01_1.m</a>
implements the midpoint rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_02_xiu.m">cn_leg_02_xiu.m</a>
implements the Xiu rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_03_1.m">cn_leg_03_1.m</a>
implements the Stroud CN:3-1 rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_03_xiu.m">cn_leg_03_xiu.m</a>
implements the Xiu rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_05_1.m">cn_leg_05_1.m</a>
implements the Stroud CN:5-1 rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_05_2.m">cn_leg_05_2.m</a>
implements the Stroud CN:5-2 rule for region CN_LEG.
</li>
<li>
<a href = "cn_leg_monomial_integral.m">cn_leg_monoial_integral.m</a>
integral of a monomial with Legendre weight over CN.
</li>
<li>
<a href = "cone_unit_3d.m">
cone_unit_3d.m</a>,
approximates an integral inside a cone in 3D;
</li>
<li>
<a href = "cone_volume_3d.m">
cone_volume_3d.m</a>,
returns the volume of a cone in 3D;
</li>
<li>
<a href = "cube_shell_nd.m">
cube_shell_nd.m</a>,
aproximates an integral inside a cubic shell in ND;
</li>
<li>
<a href = "cube_shell_volume_nd.m">
cube_shell_volume_nd.m</a>,
returns the volume of a cubic shell in ND;
</li>
<li>
<a href = "cube_unit_3d.m">cube_unit_3d.m</a>,
approximates an integral inside the unit cube in 3D;
</li>
<li>
<a href = "cube_unit_nd.m">cube_unit_nd.m</a>,
approximates an integral inside the unit cube in ND;
</li>
<li>
<a href = "cube_unit_volume_nd.m">cube_unit_volume_nd.m</a>,
returns the volume of the unit cube in ND;
</li>
<li>
<a href = "ellipse_area_2d.m">ellipse_area_2d.m</a>,
computes the area of an ellipse in 2D;
</li>
<li>
<a href = "ellipse_circumference_2d.m">ellipse_circumference_2d.m</a>,
computes the circumference of an ellipse in 2D;
</li>
<li>
<a href = "ellipse_eccentricity_2d.m">ellipse_eccentricity_2d.m</a>,
computes the eccentricity of an ellipse in 2D;
</li>
<li>
<a href = "ellipsoid_volume_3d.m">ellipsoid_volume_3d.m</a>,
computes the volume of an ellipsoid in 3D;
</li>
<li>
<a href = "en_r2_01_1.m">en_r2_01_1.m</a>
a precision 1 rule for EN_R2.
</li>
<li>
<a href = "en_r2_02_xiu.m">en_r2_02_xiu.m</a>
a precision 2 rule for EN_R2.
</li>
<li>
<a href = "en_r2_03_1.m">en_r2_03_1.m</a>
a precision 3 rule for EN_R2.
</li>
<li>
<a href = "en_r2_03_2.m">en_r2_03_2.m</a>
a precision 3 rule for EN_R2.
</li>
<li>
<a href = "en_r2_03_xiu.m">en_r2_03_xiu.m</a>
a precision 3 rule for EN_R2.
</li>
<li>
<a href = "en_r2_05_1.m">en_r2_05_1.m</a>
a precision 5 rule for EN_R2.
</li>
<li>
<a href = "en_r2_05_2.m">en_r2_05_2.m</a>
a precision 5 rule for EN_R2.
</li>
<li>
<a href = "en_r2_05_3.m">en_r2_05_3.m</a>
a precision 5 rule for EN_R2.
</li>
<li>
<a href = "en_r2_05_4.m">en_r2_05_4.m</a>
a precision 5 rule for EN_R2.
</li>
<li>
<a href = "en_r2_05_5.m">en_r2_05_5.m</a>
a precision 5 rule for EN_R2.
</li>
<li>
<a href = "en_r2_05_6.m">en_r2_05_6.m</a>
a precision 5 rule for EN_R2.
</li>
<li>
<a href = "en_r2_07_1.m">en_r2_07_1.m</a>
a precision 7 rule for EN_R2.
</li>
<li>
<a href = "en_r2_07_2.m">en_r2_07_2.m</a>
a precision 7 rule for EN_R2.
</li>
<li>
<a href = "en_r2_07_3.m">en_r2_07_3.m</a>
a precision 7 rule for EN_R2.
</li>
<li>
<a href = "en_r2_09_1.m">en_r2_09_1.m</a>
a precision 9 rule for EN_R2.
</li>
<li>
<a href = "en_r2_11_1.m">en_r2_11_1.m</a>
a precision 11 rule for EN_R2.
</li>
<li>
<a href = "en_r2_monomial_integral.m">en_r2_monomial_integral.m</a>
evaluates the exact integral of a monomial in Stroud's EN_R2 region.
</li>
<li>
<a href = "ep1_glg_monomial_integral.m">ep1_glg_monomial_integral.m</a>
integral of monomial with generalized Laguerre weight on EP1.
</li>
<li>
<a href = "ep1_lag_monomial_integral.m">ep1_lag_monomial_integral.m</a>
integral of monomial with Laguerre weight on EP1.
</li>
<li>
<a href = "epn_glg_00_1.m">epn_glg_001.m</a>
implements the midpoint rule for region EPN_GLG.
</li>
<li>
<a href = "epn_glg_01_1.m">epn_glg_01_1.m</a>
implements a precision 1 rule for region EPN_GLG.
</li>
<li>
<a href = "epn_glg_02_xiu.m">epn_glg_02_xiu.m</a>
implements the Xiu rule for region EPN_GLG.
</li>
<li>
<a href = "epn_glg_monomial_integral.m">epn_glg_monomial_integral.m</a>
integral of monomial with generalized Laguerre weight on EPN.
</li>
<li>
<a href = "epn_lag_00_1.m">epn_lag_001.m</a>
implements the midpoint rule for region EPN_LAG.
</li>
<li>
<a href = "epn_lag_01_1.m">epn_lag_01_1.m</a>
implements a precision 1 rule for region EPN_LAG.
</li>
<li>
<a href = "epn_lag_02_xiu.m">epn_lag_02_xiu.m</a>
implements the Xiu rule for region EPN_LAG.
</li>
<li>
<a href = "epn_lag_monomial_integral.m">epn_lag_monomial_integral.m</a>
integral of monomial with Laguerre weight on EPN.
</li>
<li>
<a href = "gw_02_xiu.m">gw_02_xiu.m</a>
implements the Golub-Welsch version of the Xiu rule.
</li>
<li>
<a href = "hexagon_area_2d.m">hexagon_area_2d.m</a>,
returns the area of a regular hexagon in 2D;
</li>
<li>
<a href = "hexagon_sum.m">hexagon_sum.m</a>,
applies a quadrature rule for a hexagon in 2D;
</li>
<li>
<a href = "hexagon_unit_area_2d.m">hexagon_unit_area_2d.m</a>,
returns the area of the unit hexagon in 2D;
</li>
<li>
<a href = "hexagon_unit_set.m">hexagon_unit_set.m</a>,
sets a quadrature rule for the unit hexagon in 2D;
</li>
<li>
<a href = "hexagon_unit_size.m">hexagon_unit_size.m</a>,
sizes a quadrature rule for the unit hexagon in 2D;
</li>
<li>
<a href = "i4_factorial.m">i4_factorial.m</a>,
computes N! for small values of N;
</li>
<li>
<a href = "i4_factorial2.m">i4_factorial2.m</a>,
computes N!!, the double factorial function;
</li>
<li>
<a href = "ksub_next2.m">ksub_next2.m</a>,
computes the next K subset of an N set;
</li>
<li>
<a href = "legendre_set.m">legendre_set.m</a>,
sets a Gauss-Legendre rule to integrate F(X)
on [-1,1];
</li>
<li>
<a href = "legendre_set_x1.m">legendre_set_x1.m</a>,
sets a Gauss-Legendre rule to integrate ( 1 + X ) * F(X)
on [-1,1];
</li>
<li>
<a href = "legendre_set_x2.m">legendre_set_x2.m</a>,
sets a Gauss-Legendre rule to integrate ( 1 + X )**2 * F(X)
on [-1,1];
</li>
<li>
<a href = "lens_half_area_2d.m">
lens_half_area_2d.m</a>,
returns the area of a circular half lens in 2D;
</li>
<li>
<a href = "lens_half_h_area_2d.m">
lens_half_area_h_2d.m</a>,
returns the area of a circular half lens in 2D;
</li>
<li>
<a href = "lens_half_w_area_2d.m">
lens_half_area_w_2d.m</a>,
returns the area of a circular half lens in 2D;
</li>
<li>
<a href = "monomial_value.m">
monomial_value.m</a>,
evaluates a monomial given the exponents.
</li>
<li>
<a href = "octahedron_unit_nd.m">octahedron_volume_nd.m</a>,
approximates an integral in the unit octahedron in ND;
</li>
<li>
<a href = "octahedron_unit_volume_nd.m">octahedron_unit_volume_nd.m</a>,
returns the volume of the unit octahedron in ND;
</li>
<li>
<a href = "parallelipiped_volume_3d.m">parallelipiped_volume_3d.m</a>,
returns the volume of a parallelipiped in 3D;
</li>
<li>
<a href = "parallelipiped_volume_nd.m">parallelipiped_volume_nd.m</a>,
returns the volume of a parallelipiped in ND;
</li>
<li>
<a href = "polygon_1_2d.m">polygon_1_2d.m</a>,
integrates the function 1 over a polygon in 2D;
</li>
<li>
<a href = "polygon_x_2d.m">polygon_x_2d.m</a>,
integrates the function X over a polygon in 2D;
</li>
<li>
<a href = "polygon_xx_2d.m">polygon_xx_2d.m</a>,
integrates the function X^2 over a polygon in 2D;
</li>
<li>
<a href = "polygon_xy_2d.m">polygon_xy_2d.m</a>,