5365:
Lyness (1969) is almost the current integrator. Created as a set of four modifications of McKeeman 1962, it replaces trisection with bisection to lower computational costs (Modifications 1+2, coinciding with the Kuncir integrator) and improves McKeeman's 1962/63 error estimates to the fifth order
1243:
1092:
39:. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. The technique is usually much more efficient than
1933:
at the parent level, will again be used for the subintervals. Doing so cuts down the cost of each recursive call from 6 to 2 evaluations of the input function. The size of the stack space used stays linear to the layer of recursions.
5224:
McKeeman's
Algorithm 145 (1962) is a similarly recursive integrator that splits the interval into three instead of two parts. The recursion is written in a more familiar manner. The 1962 algorithm, found to be over-cautious, uses
1346:
871:
2769:
Here is an implementation of the adaptive
Simpson's method in C99 that avoids redundant evaluations of f and quadrature computations. It includes all three "simple" defenses against numerical problems.
1893:
The epsilon-raising maneuver allows the routine to be used in a "best effort" mode: given a zero initial tolerance, the routine will try to get the most precise answer and return an actual error level.
5322:
5207:
5106:
Kuncir's
Algorithm 103 (1962) is the original recursive, bisecting, adaptive integrator. Algorithm 103 consists of a larger routine with a nested subroutine (loop AA), made recursive by the use of the
346:
760:
1097:
946:
1616:
5360:
5374:. Modification 4, not implemented here, contains provisions for roundoff error that allows for raising the ε to the minimum allowed by current precision and returning the new error.
27:
proposed by G.F. Kuncir in 1962. It is probably the first recursive adaptive algorithm for numerical integration to appear in print, although more modern adaptive methods based on
683:
632:
581:
421:
111:
703:
4297:, among other projects. The ORNL version has been enhanced with a call counter, templates for different datatypes, and wrappers for integrating over multiple dimensions.
1360:
and producing an infinite loop. Simple methods of guarding against this problem include adding a depth limitation (like in the C sample and in McKeeman), verifying that
495:
458:
941:
906:
530:
146:
5629:
5110:
statement. It guards against the underflowing of interval widths (loop BB), and aborts as soon as the user-specified eps is exceeded. The termination criteria is
237:
5522:
380:
5784:
5103:
Henriksson (1961) is a non-recursive variant of
Simpson's Rule. It "adapts" by integrating from left to right and adjusting the interval width as needed.
5455:
255:
5707:
28:
5602:
5622:
35:
are now generally preferred. Adaptive
Simpson's method uses an estimate of the error we get from calculating a definite integral using
1248:
773:
5228:
5113:
5748:
5722:
1792:
Some control of the adjustments is necessary. Significant increases and minor decreases of the tolerances should be inhibited.
1238:{\textstyle \left|S{\left(m,{\frac {b-m}{2}}\right)}+S{\left({\frac {b-m}{2}},b\right)}-S(m,b)\right|<15\varepsilon _{i+1}}
1087:{\textstyle \left|S{\left(a,{\frac {m-a}{2}}\right)}+S{\left({\frac {m-a}{2}},m\right)}-S(a,m)\right|<15\varepsilon _{i+1}}
32:
5615:
5717:
5712:
711:
4306:
4294:
1943:
5697:
5702:
4309:
with a behavioral software contract. The exported function computes the indeterminate integral for some given function
53:
is an interpolatory quadrature rule which is exact when the integrand is a polynomial of degree three or lower. Using
943:
to the sum of
Simpson's rules which are used to approximate the integral, otherwise, perform the same operation with
40:
2558:"""Integrate f from a to b using Adaptive Simpson's Rule with max error of eps."""
5738:
5646:
1422:
5327:
54:
5391:
The original 4i only mentions raising E'. However, later text mentioned that it can be lowered in large steps.
5638:
5460:
S. Henriksson (1961), "Contribution no. 2: Simpson numerical integration with variable length of step",
2008:"""Evaluates the Simpson's Rule, also returning m and f(m) to reuse"""
385:
24:
5541:
637:
586:
535:
5763:
1357:
1387:
Lyness's 1969 paper includes a "Modification 4" that addresses this problem in a more concrete way:
1370:
in floating-point arithmetics, or both (like Kuncir). The interval size may also approach the local
5743:
5689:
5679:
5371:
1356:
Some inputs will fail to converge in adaptive
Simpson's method quickly, resulting in the tolerance
60:
688:
5758:
463:
426:
252:
A criterion for determining when to stop subdividing an interval, suggested by J.N. Lyness, is
43:
since it uses fewer function evaluations in places where the function is well-approximated by a
5664:
5659:
50:
36:
5753:
5654:
5579:
5499:
5433:
911:
876:
500:
116:
5400:
This likely also applys to the tolerance/interval width underflows in the simplistic case.
1371:
5674:
5523:"RayTrace-miniapp: src/AtomicModel/interp.hpp · de5e8229bccf60ae5c1c5bab14f861dc0326d5f9"
5367:
151:
44:
353:
5778:
4444:;; -----------------------------------------------------------------------------
4318:;; -----------------------------------------------------------------------------
4290:
239:. So, the obtained estimate is exact for polynomials of degree five or less.
5584:
5567:
5504:
5438:
5607:
5490:
J.N. Lyness (1969), "Notes on the adaptive
Simpson quadrature routine",
2197:
Function values at the start, middle, end of the intervals are retained.
5454:
For an earlier, non-recursive adaptive integrator more reminiscent of
1738:(Modification 4 i, ii) If further recursion is used on an interval:
113:
for six function values is combined with the less accurate estimate
2194:
Efficient recursive implementation of adaptive
Simpson's rule.
5568:"Algorithm 145: Adaptive numerical integration by Simpson's rule"
1966:# "structured" adaptive version, translated from Racket
5424:
G.F. Kuncir (1962), "Algorithm 103: Simpson's rule integrator",
5107:
1881:, then the fifth-order acceleration/correction would not apply:
5611:
1902:
A common implementation technique shown below is passing down
1682:, either the round-off level have been reached or a zero for
4305:
Here is an implementation of the adaptive
Simpson method in
1918:
along with the interval . These values, used for evaluating
1391:
Let the initial interval be . Let the original tolerance be
1341:{\displaystyle |S(a,m)+S(m,b)-S(a,b)|<15\varepsilon _{i}}
866:{\displaystyle |S(a,m)+S(m,b)-S(a,b)|<15\varepsilon _{i}}
770:
To perform adaptive Simpson's method, do the following: if
5317:{\displaystyle |S^{(3)}(a,b)-S(a,b)|<3^{-n}\epsilon \,}
5202:{\displaystyle |S^{(2)}(a,b)-S(a,b)|<2^{-n}\epsilon \,}
1942:
Here is an implementation of adaptive Simpson's method in
1709:
level for the current interval. A routine-static variable
705:
is the desired maximum error tolerance for the interval.
1884:
The "15" factor in the termination criteria is disabled.
341:{\displaystyle |S(a,m)+S(m,b)-S(a,b)|<15\varepsilon }
3324:// Lyness 1969 + Richardson extrapolation; see article
1741:
If round-off appears to have been reached, change the
1642:. The original termination criteria would then become
1425:
1100:
949:
640:
589:
538:
5330:
5231:
5116:
4289:
This implementation has been incorporated into a C++
1251:
914:
879:
776:
755:{\displaystyle \varepsilon _{i+1}=\varepsilon _{i}/2}
714:
691:
503:
466:
429:
388:
356:
258:
154:
148:
for three function values by applying the correction
119:
63:
1688:
is found in the interval. A change in the tolerance
5731:
5688:
5645:
3033:// serious numerical trouble: it won't converge
5354:
5316:
5201:
2799:/** Adaptive Simpson's Rule, Recursive Core */
1610:
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1237:
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935:
900:
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677:
626:
575:
524:
489:
452:
415:
374:
340:
231:
140:
105:
486:
449:
5542:"[racket] adaptive simpson integration"
5566:McKeeman, William Marshall (1 December 1962).
4116:// Was it successful? (depth=1 is too shallow)
4029:// Let I be the integral of sin(x) from 0 to 2
5623:
1611:{\textstyle {\frac {12}{a-b}}={\frac {1}{4}}}
532:given by Simpson's rule are the estimates of
8:
5450:
5448:
5324:for termination, so a 1963 improvement uses
3555:* (fills in cached function evaluations) */
1705:The recursive routines now need to return a
5355:{\displaystyle {\sqrt {3}}^{\,-n}\epsilon }
4600:;; the "efficient" implementation
5630:
5616:
5608:
4888:;; evaluate half an interval (1 func eval)
3900:// for the hostile example (rand function)
5583:
5503:
5437:
5340:
5339:
5332:
5329:
5313:
5301:
5289:
5241:
5232:
5230:
5198:
5186:
5174:
5126:
5117:
5115:
1868:, weighted by the width of the intervals.
1505:
1451:
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1317:
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1223:
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1006:
1000:
970:
958:
948:
913:
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777:
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738:
719:
713:
690:
668:
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617:
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566:
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502:
485:
465:
448:
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395:
387:
355:
324:
259:
257:
221:
153:
118:
62:
5485:
5483:
5481:
5479:
5477:
5475:
5419:
5417:
5413:
5384:
4293:intended for X-Ray Laser simulation at
4218:"integrate(frand48, 0, 0.25) = %lf
3552:/** Adaptive Simpson's Rule Wrapper
1887:The correction term should not be used.
5516:
5514:
5366:(Modification 3), in a way related to
5215:is the current level of recursion and
2790:// include file for printf and perror
57:, the more accurate Simpson estimate
7:
5785:Numerical integration (quadrature)
5603:Module for Adaptive Simpson's Rule
678:{\textstyle \int _{m}^{b}f(x)\,dx}
627:{\textstyle \int _{a}^{m}f(x)\,dx}
576:{\textstyle \int _{a}^{b}f(x)\,dx}
416:{\displaystyle m={\frac {a+b}{2}}}
14:
4083:"integrate(sinf, 0, 2) = %lf
1836:pairs. The first entry should be
5708:Gauss–Kronrod quadrature formula
2781:// include file for fabs and sin
1877:for an interval is higher than
5290:
5286:
5274:
5265:
5253:
5248:
5242:
5233:
5221:is the more accurate estimate.
5175:
5171:
5159:
5150:
5138:
5133:
5127:
5118:
1861:is the arithmetic mean of all
1731:is defined and initialized to
1605:
1602:
1596:
1587:
1578:
1566:
1560:
1548:
1539:
1527:
1521:
1515:
1499:
1496:
1484:
1475:
1463:
1458:
1452:
1444:
1403:For each subinterval , define
1318:
1314:
1302:
1293:
1281:
1272:
1260:
1253:
1205:
1193:
1054:
1042:
930:
918:
895:
883:
843:
839:
827:
818:
806:
797:
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778:
665:
659:
614:
608:
563:
557:
519:
507:
482:
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433:
369:
357:
325:
321:
309:
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288:
279:
267:
260:
218:
215:
203:
194:
182:
173:
161:
155:
135:
123:
100:
88:
79:
67:
1:
4295:Oak Ridge National Laboratory
382:is an interval with midpoint
106:{\displaystyle S(a,m)+S(m,b)}
4321:;; interface, with contract
4242:"adaptiveSimpsons"
4110:"adaptiveSimpsons"
3588:// function ptr to integrate
1820:is changed into an array of
698:{\displaystyle \varepsilon }
5546:Racket Mailing-list "users"
1898:Sample code implementations
1797:To calculate the effective
5801:
5749:Clenshaw–Curtis quadrature
5723:Chebyshev–Gauss quadrature
3321:// depth limit too shallow
1804:over the entire interval:
490:{\displaystyle S(a,m)\,\!}
453:{\displaystyle S(a,b)\,\!}
33:Clenshaw–Curtis quadrature
5718:Gauss–Legendre quadrature
5713:Gauss–Laguerre quadrature
5670:Adaptive Simpson's method
5572:Communications of the ACM
5462:BIT Numerical Mathematics
5426:Communications of the ACM
1418:, the error estimate, as
17:Adaptive Simpson's method
5698:Gauss–Hermite quadrature
4315:
2772:
1948:
55:Richardson extrapolation
41:composite Simpson's rule
29:Gauss–Kronrod quadrature
5703:Gauss–Jacobi quadrature
1352:Numerical consideration
21:adaptive Simpson's rule
5356:
5318:
5203:
4257:"(%d evaluations)
4125:"(%d evaluations)
1612:
1342:
1239:
1088:
937:
936:{\displaystyle S(m,b)}
902:
901:{\displaystyle S(a,m)}
867:
756:
699:
679:
628:
577:
526:
525:{\displaystyle S(m,b)}
491:
454:
417:
376:
342:
243:Mathematical procedure
233:
142:
141:{\displaystyle S(a,b)}
107:
5739:Barnes–Hut simulation
5647:Newton–Cotes formulas
5639:Numerical integration
5585:10.1145/355580.369102
5540:Felleisen, Matthias.
5505:10.1145/321526.321537
5439:10.1145/367766.368179
5357:
5319:
5204:
4248:// won't converge
1613:
1343:
1240:
1089:
938:
903:
868:
757:
700:
680:
629:
578:
527:
492:
455:
418:
377:
343:
234:
143:
108:
25:numerical integration
5764:Tanh-sinh quadrature
5328:
5229:
5114:
4209:// a random function
3891:/** usage example */
1423:
1249:
1098:
947:
912:
877:
774:
712:
689:
638:
587:
536:
501:
464:
427:
386:
354:
256:
152:
117:
61:
5744:Bayesian quadrature
5690:Gaussian quadrature
4101:// print the result
3828:adaptiveSimpsonsAux
3480:adaptiveSimpsonsAux
3408:adaptiveSimpsonsAux
2805:adaptiveSimpsonsAux
655:
604:
553:
232:{\displaystyle /15}
5759:Lebedev quadrature
5665:Simpson's 3/8 rule
5492:Journal of the ACM
5352:
5314:
5199:
5098:Related algorithms
4900:simpson-1call-to-f
4714:simpson-1call-to-f
4669:simpson-1call-to-f
4543:simpson-1call-to-f
4447:;; implementation
3621:// error tolerance
2615:_quad_simpsons_mem
2275:_quad_simpsons_mem
2221:_quad_simpsons_mem
2200:"""
2191:"""
1972:_quad_simpsons_mem
1763:Otherwise, adjust
1608:
1338:
1235:
1084:
933:
898:
863:
752:
695:
685:respectively, and
675:
641:
624:
590:
573:
539:
522:
487:
450:
413:
372:
338:
229:
138:
103:
5772:
5771:
5521:Berrill, Mark A.
5337:
4330:[adaptive-simpson
1963:# python 2 compat
1513:
1442:
1173:
1137:
1022:
986:
411:
23:, is a method of
5792:
5754:Filon quadrature
5680:Romberg's method
5655:Trapezoidal rule
5632:
5625:
5618:
5609:
5590:
5589:
5587:
5563:
5557:
5556:
5554:
5552:
5537:
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5509:
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5470:
5469:
5452:
5443:
5442:
5441:
5421:
5401:
5398:
5392:
5389:
5372:Romberg's method
5361:
5359:
5358:
5353:
5348:
5347:
5338:
5333:
5323:
5321:
5320:
5315:
5309:
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5236:
5220:
5214:
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4459:adaptive-simpson
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4327:provide/contract
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4173:adaptiveSimpsons
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4041:adaptiveSimpsons
4039:
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3897:<stdlib.h>
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3661:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3636:// recursion cap
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3561:adaptiveSimpsons
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3538:
3535:
3532:
3529:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3475:
3472:
3469:
3466:
3463:
3460:
3457:
3454:
3451:
3448:
3445:
3442:
3439:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3400:
3397:
3394:
3391:
3388:
3385:
3382:
3379:
3376:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3301:
3298:
3295:
3292:
3289:
3286:
3283:
3280:
3277:
3274:
3271:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3235:
3232:
3229:
3226:
3223:
3220:
3217:
3214:
3211:
3208:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3154:
3151:
3148:
3145:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3097:
3094:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3061:
3058:
3055:
3052:
3049:
3046:
3043:
3040:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3007:
3004:
3001:
2998:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2953:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2923:
2920:
2917:
2914:
2911:
2908:
2905:
2902:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2878:
2875:
2872:
2869:
2866:
2863:
2860:
2857:
2854:
2851:
2848:
2845:
2842:
2839:
2836:
2833:
2830:
2827:
2824:
2821:
2818:
2815:
2812:
2809:
2806:
2803:
2800:
2797:
2794:
2791:
2788:
2785:
2782:
2779:
2776:
2760:
2757:
2754:
2751:
2748:
2745:
2742:
2739:
2736:
2733:
2730:
2727:
2724:
2721:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2667:
2664:
2661:
2658:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2628:
2625:
2622:
2619:
2616:
2613:
2610:
2607:
2604:
2601:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2538:
2535:
2532:
2529:
2526:
2523:
2520:
2517:
2514:
2511:
2508:
2505:
2502:
2499:
2496:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2472:
2469:
2466:
2463:
2460:
2456:
2453:
2450:
2447:
2444:
2441:
2438:
2435:
2432:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2387:
2384:
2381:
2378:
2375:
2372:
2369:
2366:
2363:
2360:
2357:
2354:
2351:
2348:
2345:
2342:
2339:
2336:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2294:
2291:
2288:
2285:
2282:
2279:
2276:
2273:
2270:
2267:
2264:
2261:
2258:
2255:
2252:
2249:
2246:
2243:
2240:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2210:
2207:
2204:
2201:
2198:
2195:
2192:
2189:
2186:
2183:
2180:
2177:
2174:
2171:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2132:
2129:
2126:
2123:
2120:
2117:
2114:
2111:
2108:
2105:
2102:
2099:
2096:
2093:
2090:
2087:
2084:
2081:
2078:
2075:
2072:
2069:
2066:
2063:
2060:
2057:
2054:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
2012:
2009:
2006:
2003:
2000:
1997:
1994:
1991:
1988:
1985:
1982:
1979:
1976:
1973:
1970:
1967:
1964:
1961:
1958:
1955:
1952:
1932:
1917:
1850:
1835:
1815:
1786:
1759:
1730:
1687:
1681:
1651:
1641:
1619:
1617:
1615:
1614:
1609:
1514:
1506:
1462:
1461:
1443:
1441:
1427:
1417:
1399:
1383:
1369:
1347:
1345:
1344:
1339:
1337:
1336:
1321:
1256:
1244:
1242:
1241:
1236:
1234:
1233:
1212:
1208:
1186:
1185:
1181:
1174:
1169:
1158:
1144:
1143:
1139:
1138:
1133:
1122:
1093:
1091:
1090:
1085:
1083:
1082:
1061:
1057:
1035:
1034:
1030:
1023:
1018:
1007:
993:
992:
988:
987:
982:
971:
942:
940:
939:
934:
907:
905:
904:
899:
872:
870:
869:
864:
862:
861:
846:
781:
766:Procedural steps
761:
759:
758:
753:
748:
743:
742:
730:
729:
704:
702:
701:
696:
684:
682:
681:
676:
654:
649:
633:
631:
630:
625:
603:
598:
582:
580:
579:
574:
552:
547:
531:
529:
528:
523:
496:
494:
493:
488:
459:
457:
456:
451:
422:
420:
419:
414:
412:
407:
396:
381:
379:
378:
375:{\displaystyle }
373:
347:
345:
344:
339:
328:
263:
238:
236:
235:
230:
225:
147:
145:
144:
139:
112:
110:
109:
104:
5800:
5799:
5795:
5794:
5793:
5791:
5790:
5789:
5775:
5774:
5773:
5768:
5727:
5684:
5641:
5636:
5599:
5594:
5593:
5565:
5564:
5560:
5550:
5548:
5539:
5538:
5534:
5520:
5519:
5512:
5489:
5488:
5473:
5459:
5453:
5446:
5423:
5422:
5415:
5410:
5405:
5404:
5399:
5395:
5390:
5386:
5381:
5331:
5326:
5325:
5297:
5237:
5227:
5226:
5216:
5210:
5182:
5122:
5112:
5111:
5100:
5095:
5094:
5091:
5088:
5085:
5082:
5079:
5076:
5073:
5070:
5067:
5064:
5061:
5058:
5055:
5052:
5049:
5046:
5043:
5040:
5037:
5034:
5031:
5028:
5025:
5022:
5019:
5016:
5013:
5010:
5007:
5004:
5001:
4998:
4995:
4992:
4989:
4986:
4983:
4980:
4977:
4974:
4971:
4968:
4965:
4962:
4959:
4956:
4953:
4950:
4947:
4944:
4941:
4938:
4935:
4932:
4929:
4926:
4923:
4920:
4917:
4914:
4911:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4812:
4809:
4806:
4803:
4800:
4797:
4794:
4791:
4788:
4785:
4782:
4779:
4776:
4773:
4770:
4767:
4764:
4761:
4758:
4755:
4752:
4749:
4746:
4743:
4740:
4737:
4734:
4731:
4728:
4725:
4722:
4719:
4716:
4713:
4710:
4707:
4704:
4701:
4698:
4695:
4692:
4689:
4686:
4683:
4680:
4677:
4674:
4671:
4668:
4665:
4662:
4659:
4656:
4653:
4650:
4647:
4644:
4641:
4638:
4635:
4632:
4629:
4626:
4623:
4620:
4617:
4614:
4611:
4608:
4605:
4602:
4599:
4596:
4593:
4590:
4587:
4584:
4581:
4578:
4575:
4572:
4569:
4566:
4563:
4560:
4557:
4554:
4551:
4548:
4545:
4542:
4539:
4536:
4533:
4530:
4527:
4524:
4521:
4518:
4515:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4485:
4482:
4479:
4476:
4473:
4470:
4467:
4464:
4461:
4458:
4455:
4452:
4449:
4446:
4443:
4440:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4404:
4401:
4398:
4395:
4392:
4389:
4386:
4383:
4380:
4377:
4374:
4371:
4368:
4365:
4362:
4359:
4356:
4353:
4350:
4347:
4344:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4303:
4287:
4286:
4283:
4280:
4277:
4274:
4271:
4268:
4265:
4262:
4259:
4256:
4253:
4250:
4247:
4244:
4241:
4238:
4235:
4232:
4229:
4226:
4223:
4220:
4217:
4214:
4211:
4208:
4205:
4202:
4199:
4196:
4193:
4190:
4187:
4184:
4181:
4178:
4175:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4151:
4148:
4145:
4142:
4139:
4136:
4133:
4130:
4127:
4124:
4121:
4118:
4115:
4112:
4109:
4106:
4103:
4100:
4097:
4094:
4091:
4088:
4085:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4061:
4058:
4055:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4028:
4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
3998:
3995:
3992:
3989:
3986:
3983:
3980:
3977:
3974:
3971:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
3944:
3941:
3938:
3935:
3932:
3929:
3926:
3923:
3920:
3917:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3893:
3890:
3887:
3884:
3881:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3857:
3854:
3851:
3848:
3845:
3842:
3839:
3836:
3833:
3830:
3827:
3824:
3821:
3818:
3815:
3812:
3809:
3806:
3803:
3800:
3797:
3794:
3791:
3788:
3785:
3782:
3779:
3776:
3773:
3770:
3767:
3764:
3761:
3758:
3755:
3752:
3749:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
3722:
3719:
3716:
3713:
3710:
3707:
3704:
3701:
3698:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3578:
3575:
3572:
3569:
3566:
3563:
3560:
3557:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3440:
3437:
3434:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3404:
3401:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3275:
3272:
3269:
3266:
3263:
3260:
3257:
3254:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3227:
3224:
3221:
3218:
3215:
3212:
3209:
3206:
3203:
3200:
3197:
3194:
3191:
3188:
3185:
3182:
3179:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3062:
3059:
3056:
3053:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3008:
3005:
3002:
2999:
2996:
2993:
2990:
2987:
2984:
2981:
2978:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2954:
2951:
2948:
2945:
2942:
2939:
2936:
2933:
2930:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2906:
2903:
2900:
2897:
2894:
2891:
2888:
2885:
2882:
2879:
2876:
2873:
2870:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2828:
2825:
2822:
2819:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2796:<errno.h>
2795:
2792:
2789:
2787:<stdio.h>
2786:
2783:
2780:
2777:
2774:
2767:
2762:
2761:
2758:
2755:
2752:
2749:
2746:
2743:
2740:
2737:
2734:
2731:
2728:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2677:
2674:
2671:
2668:
2665:
2662:
2659:
2656:
2653:
2650:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2620:
2617:
2614:
2611:
2608:
2605:
2602:
2599:
2596:
2593:
2590:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2497:
2494:
2491:
2488:
2485:
2482:
2479:
2476:
2473:
2470:
2467:
2464:
2461:
2458:
2454:
2451:
2448:
2445:
2442:
2439:
2436:
2433:
2430:
2427:
2424:
2421:
2418:
2415:
2412:
2409:
2406:
2403:
2400:
2397:
2394:
2391:
2388:
2385:
2382:
2379:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2349:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2313:
2310:
2307:
2304:
2301:
2298:
2295:
2292:
2289:
2286:
2283:
2280:
2277:
2274:
2271:
2268:
2265:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2235:
2232:
2229:
2226:
2223:
2220:
2217:
2214:
2211:
2208:
2205:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2157:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2088:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2061:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2028:
2025:
2022:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1956:
1953:
1950:
1940:
1919:
1903:
1900:
1873:If the current
1867:
1859:
1848:
1837:
1831:
1827:
1821:
1813:
1808:
1803:
1768:
1746:
1720:
1710:
1701:
1694:
1683:
1656:
1643:
1631:
1621:
1447:
1431:
1421:
1420:
1419:
1404:
1398:
1392:
1375:
1372:machine epsilon
1361:
1354:
1328:
1247:
1246:
1219:
1159:
1156:
1152:
1123:
1114:
1110:
1105:
1101:
1096:
1095:
1068:
1008:
1005:
1001:
972:
963:
959:
954:
950:
945:
944:
910:
909:
875:
874:
853:
772:
771:
768:
734:
715:
710:
709:
687:
686:
636:
635:
585:
584:
534:
533:
499:
498:
462:
461:
425:
424:
397:
384:
383:
352:
351:
254:
253:
250:
245:
150:
149:
115:
114:
59:
58:
12:
11:
5:
5798:
5796:
5788:
5787:
5777:
5776:
5770:
5769:
5767:
5766:
5761:
5756:
5751:
5746:
5741:
5735:
5733:
5729:
5728:
5726:
5725:
5720:
5715:
5710:
5705:
5700:
5694:
5692:
5686:
5685:
5683:
5682:
5677:
5672:
5667:
5662:
5660:Simpson's rule
5657:
5651:
5649:
5643:
5642:
5637:
5635:
5634:
5627:
5620:
5612:
5606:
5605:
5598:
5597:External links
5595:
5592:
5591:
5558:
5532:
5510:
5498:(3): 483–495,
5471:
5444:
5412:
5411:
5409:
5406:
5403:
5402:
5393:
5383:
5382:
5380:
5377:
5376:
5375:
5363:
5351:
5346:
5343:
5336:
5312:
5307:
5304:
5300:
5296:
5292:
5288:
5285:
5282:
5279:
5276:
5273:
5270:
5267:
5264:
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5258:
5255:
5250:
5247:
5244:
5240:
5235:
5222:
5197:
5192:
5189:
5185:
5181:
5177:
5173:
5170:
5167:
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5161:
5158:
5155:
5152:
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5146:
5143:
5140:
5135:
5132:
5129:
5125:
5120:
5104:
5099:
5096:
4316:
4302:
4299:
2778:<math.h>
2773:
2766:
2763:
1949:
1939:
1936:
1899:
1896:
1891:
1890:
1889:
1888:
1885:
1871:
1870:
1869:
1865:
1857:
1852:
1846:
1829:
1825:
1811:
1801:
1795:
1794:
1793:
1790:
1789:
1788:
1761:
1736:
1718:
1702:is necessary.
1699:
1692:
1653:
1629:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1580:
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1471:
1468:
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1350:
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1320:
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1229:
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1192:
1189:
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1177:
1172:
1168:
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1162:
1155:
1150:
1147:
1142:
1136:
1132:
1129:
1126:
1120:
1117:
1113:
1108:
1104:
1081:
1078:
1075:
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1060:
1056:
1053:
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1041:
1038:
1033:
1029:
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1021:
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1014:
1011:
1004:
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996:
991:
985:
981:
978:
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966:
962:
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953:
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929:
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923:
920:
917:
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888:
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856:
852:
849:
845:
841:
838:
835:
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829:
826:
823:
820:
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814:
811:
808:
805:
802:
799:
796:
793:
790:
787:
784:
780:
767:
764:
751:
747:
741:
737:
733:
728:
725:
722:
718:
694:
674:
671:
667:
664:
661:
658:
653:
648:
644:
623:
620:
616:
613:
610:
607:
602:
597:
593:
572:
569:
565:
562:
559:
556:
551:
546:
542:
521:
518:
515:
512:
509:
506:
484:
481:
478:
475:
472:
469:
447:
444:
441:
438:
435:
432:
410:
406:
403:
400:
394:
391:
371:
368:
365:
362:
359:
337:
334:
331:
327:
323:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
262:
249:
248:Defining terms
246:
244:
241:
228:
224:
220:
217:
214:
211:
208:
205:
202:
199:
196:
193:
190:
187:
184:
181:
178:
175:
172:
169:
166:
163:
160:
157:
137:
134:
131:
128:
125:
122:
102:
99:
96:
93:
90:
87:
84:
81:
78:
75:
72:
69:
66:
51:Simpson's rule
45:cubic function
37:Simpson's rule
19:, also called
13:
10:
9:
6:
4:
3:
2:
5797:
5786:
5783:
5782:
5780:
5765:
5762:
5760:
5757:
5755:
5752:
5750:
5747:
5745:
5742:
5740:
5737:
5736:
5734:
5730:
5724:
5721:
5719:
5716:
5714:
5711:
5709:
5706:
5704:
5701:
5699:
5696:
5695:
5693:
5691:
5687:
5681:
5678:
5676:
5673:
5671:
5668:
5666:
5663:
5661:
5658:
5656:
5653:
5652:
5650:
5648:
5644:
5640:
5633:
5628:
5626:
5621:
5619:
5614:
5613:
5610:
5604:
5601:
5600:
5596:
5586:
5581:
5577:
5573:
5569:
5562:
5559:
5547:
5543:
5536:
5533:
5528:
5524:
5517:
5515:
5511:
5506:
5501:
5497:
5493:
5486:
5484:
5482:
5480:
5478:
5476:
5472:
5467:
5463:
5457:
5451:
5449:
5445:
5440:
5435:
5431:
5427:
5420:
5418:
5414:
5407:
5397:
5394:
5388:
5385:
5378:
5373:
5369:
5364:
5349:
5344:
5341:
5334:
5310:
5305:
5302:
5298:
5294:
5283:
5280:
5277:
5271:
5268:
5262:
5259:
5256:
5245:
5238:
5223:
5219:
5213:
5195:
5190:
5187:
5183:
5179:
5168:
5165:
5162:
5156:
5153:
5147:
5144:
5141:
5130:
5123:
5109:
5105:
5102:
5101:
5097:
4693:define-values
4648:define-values
4522:define-values
4314:
4312:
4308:
4300:
4298:
4296:
4292:
2771:
2764:
2457:\
1947:
1945:
1937:
1935:
1930:
1926:
1922:
1915:
1911:
1907:
1897:
1895:
1886:
1883:
1882:
1880:
1876:
1872:
1864:
1860:
1853:
1845:
1841:
1833:
1819:
1816:at which the
1814:
1806:
1805:
1800:
1796:
1791:
1784:
1780:
1776:
1772:
1766:
1762:
1757:
1753:
1749:
1744:
1740:
1739:
1737:
1734:
1728:
1724:
1717:
1713:
1708:
1704:
1703:
1698:
1691:
1686:
1679:
1675:
1671:
1667:
1663:
1659:
1654:
1650:
1646:
1639:
1635:
1628:
1624:
1599:
1593:
1590:
1584:
1581:
1575:
1572:
1569:
1563:
1557:
1554:
1551:
1545:
1542:
1536:
1533:
1530:
1524:
1518:
1510:
1507:
1502:
1493:
1490:
1487:
1481:
1478:
1472:
1469:
1466:
1455:
1448:
1438:
1435:
1432:
1428:
1415:
1411:
1407:
1402:
1395:
1390:
1389:
1388:
1385:
1382:
1378:
1373:
1368:
1364:
1359:
1351:
1349:
1333:
1329:
1325:
1322:
1311:
1308:
1305:
1299:
1296:
1290:
1287:
1284:
1278:
1275:
1269:
1266:
1263:
1257:
1230:
1227:
1224:
1220:
1216:
1213:
1209:
1202:
1199:
1196:
1190:
1187:
1182:
1178:
1175:
1170:
1166:
1163:
1160:
1153:
1148:
1145:
1140:
1134:
1130:
1127:
1124:
1118:
1115:
1111:
1106:
1102:
1079:
1076:
1073:
1069:
1065:
1062:
1058:
1051:
1048:
1045:
1039:
1036:
1031:
1027:
1024:
1019:
1015:
1012:
1009:
1002:
997:
994:
989:
983:
979:
976:
973:
967:
964:
960:
955:
951:
927:
924:
921:
915:
892:
889:
886:
880:
858:
854:
850:
847:
836:
833:
830:
824:
821:
815:
812:
809:
803:
800:
794:
791:
788:
782:
765:
763:
749:
745:
739:
735:
731:
726:
723:
720:
716:
706:
692:
672:
669:
662:
656:
651:
646:
642:
621:
618:
611:
605:
600:
595:
591:
570:
567:
560:
554:
549:
544:
540:
516:
513:
510:
504:
479:
476:
473:
467:
442:
439:
436:
430:
408:
404:
401:
398:
392:
389:
366:
363:
360:
348:
335:
332:
329:
318:
315:
312:
306:
303:
297:
294:
291:
285:
282:
276:
273:
270:
264:
247:
242:
240:
226:
222:
212:
209:
206:
200:
197:
191:
188:
185:
179:
176:
170:
167:
164:
158:
132:
129:
126:
120:
97:
94:
91:
85:
82:
76:
73:
70:
64:
56:
52:
48:
46:
42:
38:
34:
30:
26:
22:
18:
5675:Boole's rule
5669:
5575:
5571:
5561:
5551:26 September
5549:. Retrieved
5545:
5535:
5526:
5495:
5491:
5465:
5461:
5429:
5425:
5408:Bibliography
5396:
5387:
5368:Boole's rule
5217:
5211:
4310:
4304:
4288:
3609:// interval
2768:
1941:
1928:
1924:
1920:
1913:
1909:
1905:
1901:
1892:
1878:
1874:
1862:
1855:
1843:
1839:
1823:
1817:
1809:
1798:
1782:
1778:
1774:
1770:
1764:
1755:
1751:
1747:
1742:
1732:
1726:
1722:
1715:
1711:
1706:
1696:
1689:
1684:
1677:
1673:
1669:
1665:
1661:
1657:
1648:
1644:
1637:
1633:
1626:
1622:
1413:
1409:
1405:
1393:
1386:
1380:
1376:
1366:
1362:
1358:underflowing
1355:
769:
707:
349:
251:
49:
20:
16:
15:
5578:(12): 604.
5527:ORNL GitLab
5456:ODE solvers
3882:maxRecDepth
3627:maxRecDepth
1854:The actual
1245:instead of
5432:(6): 347,
4291:ray tracer
3294:&&
1954:__future__
1668:) ≥
5350:ϵ
5342:−
5311:ϵ
5303:−
5269:−
5196:ϵ
5188:−
5154:−
4471:#:epsilon
4411:#:epsilon
2654:_quad_asr
2459:_quad_asr
2389:_quad_asr
2131:_quad_asr
1807:Log each
1570:−
1531:−
1479:−
1436:−
1374:, giving
1330:ε
1297:−
1221:ε
1188:−
1164:−
1128:−
1070:ε
1037:−
1013:−
977:−
855:ε
822:−
736:ε
717:ε
693:ε
643:∫
592:∫
541:∫
336:ε
304:−
198:−
5779:Category
5362:instead.
5209:, where
4876:f@rightM
4867:epsilon1
4831:epsilon1
4786:epsilon1
4702:f@rightM
4179:frand48c
3978:frand48c
3894:#include
2793:#include
2784:#include
2775:#include
2732:quad_asr
2528:quad_asr
1960:division
1863:ε′
1844:ε′
1799:ε′
1697:ε′
1647:≤
423:, while
4840:f@leftM
4657:f@leftM
4269:callcnt
4155:srand48
4143:callcnt
4137:callcnt
4008:drand48
3996:callcnt
3945:callcnt
3909:callcnt
3852:epsilon
3615:epsilon
1655:If the
1620:Define
5458:, see
5050:define
4969:values
4948:define
4924:define
4894:define
4873:rightM
4870:right*
4783:define
4759:right*
4741:delta*
4738:define
4705:right*
4699:rightM
4606:define
4501:define
4480:define
4453:define
4399:>/c
4336:->i
4307:Racket
4301:Racket
4275:return
4263:"
4251:printf
4236:perror
4224:"
4212:printf
4131:"
4119:printf
4104:perror
4089:"
4077:printf
4005:return
3972:static
3954:return
3921:static
3903:static
3825:return
3690:return
3405:return
3372:return
3315:ERANGE
3093:return
2720:import
2651:return
2386:return
2362:return
2056:return
1957:import
1944:Python
1938:Python
1714:= 180
1625:= 180
873:, add
708:Note,
634:, and
497:, and
350:where
5732:Other
5468:: 290
5379:Notes
4837:leftM
4834:left*
4777:[else
4765:whole
4756:left*
4660:left*
4654:leftM
4633:whole
4588:whole
4534:whole
4432:real?
4420:real?
4393:real?
4390:and/c
4366:real?
4354:real?
4351:real?
4348:->
4047:sinfc
4032:float
3984:float
3975:float
3933:float
3927:sinfc
3924:float
3771:float
3699:float
3651:float
3639:errno
3612:float
3600:float
3591:float
3582:float
3567:float
3558:float
3516:right
3390:delta
3381:right
3357:<=
3351:delta
3336:<=
3309:errno
3297:errno
3288:<=
3273:whole
3267:right
3255:delta
3252:float
3201:right
3198:float
3144:float
3105:float
3096:whole
3081:errno
2970:float
2907:float
2886:float
2877:float
2868:float
2862:whole
2859:float
2850:float
2841:float
2832:float
2826:float
2811:float
2802:float
2756:1e-09
2726:print
2696:whole
2609:whole
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