1158:
3754:. This is equivalent to locating the peaks of the function from the projections of the integrand onto the coordinate axes. The efficiency of VEGAS depends on the validity of this assumption. It is most efficient when the peaks of the integrand are well-localized. If an integrand can be rewritten in a form which is approximately separable this will increase the efficiency of integration with VEGAS. VEGAS incorporates a number of additional features, and combines both stratified sampling and importance sampling.
31:
1145:. This result does not depend on the number of dimensions of the integral, which is the promised advantage of Monte Carlo integration against most deterministic methods that depend exponentially on the dimension. It is important to notice that, unlike in deterministic methods, the estimate of the error is not a strict error bound; random sampling may not uncover all the important features of the integrand that can result in an underestimate of the error.
1149:
integrands are very localized and only small subspace notably contributes to the integral. A large part of the Monte Carlo literature is dedicated in developing strategies to improve the error estimates. In particular, stratified samplingādividing the region in sub-domainsāand importance samplingāsampling from non-uniform distributionsāare two examples of such techniques.
2442:
899:
3269:
Consider the following example where one would like to numerically integrate a gaussian function, centered at 0, with Ļ = 1, from ā1000 to 1000. Naturally, if the samples are drawn uniformly on the interval , only a very small part of them would be significant to the integral. This can be improved by
3117:
possible bisections and selecting the one which will minimize the combined variance of the two sub-regions. The variance in the sub-regions is estimated by sampling with a fraction of the total number of points available to the current step. The same procedure is then repeated recursively for each of
1148:
While the naive Monte Carlo works for simple examples, an improvement over deterministic algorithms can only be accomplished with algorithms that use problem-specific sampling distributions. With an appropriate sample distribution it is possible to exploit the fact that almost all higher-dimensional
2583:
to multi-dimensional integrals. On each recursion step the integral and the error are estimated using a plain Monte Carlo algorithm. If the error estimate is larger than the required accuracy the integration volume is divided into sub-volumes and the procedure is recursively applied to sub-volumes.
707:
38:
is the inner circle and the domain E is the square. Because the square's area (4) can be easily calculated, the area of the circle (Ļ*1.0) can be estimated by the ratio (0.8) of the points inside the circle (40) to the total number of points (50), yielding an approximation for the circle's area of
3132:. This recursive allocation of integration points continues down to a user-specified depth where each sub-region is integrated using a plain Monte Carlo estimate. These individual values and their error estimates are then combined upwards to give an overall result and an estimate of its error.
2587:
The ordinary 'dividing by two' strategy does not work for multi-dimensions as the number of sub-volumes grows far too quickly to keep track. Instead one estimates along which dimension a subdivision should bring the most dividends and only subdivides the volume along this dimension.
3498:
3369:
462:
1303:
2571:
from the above illustration was integrated within a unit square using the suggested algorithm. The sampled points were recorded and plotted. Clearly stratified sampling algorithm concentrates the points in the regions where the variation of the function is
3006:
712:
1088:
555:
341:
2568:
3106:
185:
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The stratified sampling algorithm concentrates the sampling points in the regions where the variance of the function is largest thus reducing the grand variance and making the sampling more effective, as shown on the illustration.
985:
110:
approach: each realization provides a different outcome. In Monte Carlo, the final outcome is an approximation of the correct value with respective error bars, and the correct value is likely to be within those error bars.
3270:
choosing a different distribution from where the samples are chosen, for instance by sampling according to a gaussian distribution centered at 0, with Ļ = 1. Of course the "right" choice strongly depends on the integrand.
3378:
67:. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand is evaluated. This method is particularly useful for higher-dimensional integrals.
3276:
3748:
3502:
Intuitively, this says that if we pick a particular sample twice as much as other samples, we weight it half as much as the other samples. This estimator is naturally valid for uniform sampling, the case where
2873:
1488:
3633:. Each histogram is used to define a sampling distribution for the next pass. Asymptotically this procedure converges to the desired distribution. In order to avoid the number of histogram bins growing like
351:
262:
1198:
2190:
1386:
2878:
3613:
3539:
3257:
3219:
1001:
520:
1534:
3575:
3181:
273:
1569:
1190:
1119:
3013:
2774:
2733:
120:
214:
3154:
Importance sampling provides a very important tool to perform Monte-Carlo integration. The main result of importance sampling to this method is that the uniform sampling of
2222:
906:
3113:
The MISER algorithm proceeds by bisecting the integration region along one coordinate axis to give two sub-regions at each step. The direction is chosen by examining all
2692:
2656:
2448:
894:{\displaystyle \mathrm {Var} (Q_{N})={\frac {V^{2}}{N^{2}}}\sum _{i=1}^{N}\mathrm {Var} (f)=V^{2}{\frac {\mathrm {Var} (f)}{N}}=V^{2}{\frac {\sigma _{N}^{2}}{N}}.}
1143:
3629:
The VEGAS algorithm approximates the exact distribution by making a number of passes over the integration region which creates the histogram of the function
3640:
702:{\displaystyle \mathrm {Var} (f)\equiv \sigma _{N}^{2}={\frac {1}{N-1}}\sum _{i=1}^{N}\left(f({\overline {\mathbf {x} }}_{i})-\langle f\rangle \right)^{2}.}
2783:
1397:
3773:
3110:
Hence the smallest error estimate is obtained by allocating sample points in proportion to the standard deviation of the function in each sub-region.
219:
2108:
1312:
4177:
4160:
4124:
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4062:
1157:
3493:{\displaystyle Q_{N}\equiv {\frac {1}{N}}\sum _{i=1}^{N}{\frac {f({\overline {\mathbf {x} }}_{i})}{p({\overline {\mathbf {x} }}_{i})}}}
2607:. This technique aims to reduce the overall integration error by concentrating integration points in the regions of highest variance.
3545:
71:
473:
4015:
3364:{\displaystyle p({\overline {\mathbf {x} }}):\qquad {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in V,}
3118:
the two half-spaces from the best bisection. The remaining sample points are allocated to the sub-regions using the formula for
4183:
1838:
4203:
457:{\displaystyle I\approx Q_{N}\equiv V{\frac {1}{N}}\sum _{i=1}^{N}f({\overline {\mathbf {x} }}_{i})=V\langle f\rangle .}
3768:
87:
1298:{\displaystyle H\left(x,y\right)={\begin{cases}1&{\text{if }}x^{2}+y^{2}\leq 1\\0&{\text{else}}\end{cases}}}
3580:
3506:
3224:
3186:
1122:
1495:
3551:
3157:
4103:
3763:
1543:
1164:
1093:
103:
3001:{\displaystyle \mathrm {Var} (f)={\frac {\sigma _{a}^{2}(f)}{4N_{a}}}+{\frac {\sigma _{b}^{2}(f)}{4N_{b}}}}
2738:
2697:
4180: : A blog article describing Monte Carlo integration (principle, hypothesis, confidence interval)
1083:{\displaystyle \delta Q_{N}\approx {\sqrt {\mathrm {Var} (Q_{N})}}=V{\frac {\sigma _{N}}{\sqrt {N}}},}
190:
4184:
Boost.Math : Naive Monte Carlo integration: Documentation for the C++ naive Monte-Carlo routines
4024:
3991:
3935:
547:
467:
52:
3183:
is a particular case of a more generic choice, on which the samples are drawn from any distribution
2478:
1232:
4188:
3141:
2604:
2580:
2436:
2195:
79:
75:
1195:
A paradigmatic example of a Monte Carlo integration is the estimation of Ļ. Consider the function
3964:
3951:
3921:
3783:
3778:
336:{\displaystyle {\overline {\mathbf {x} }}_{1},\cdots ,{\overline {\mathbf {x} }}_{N}\in \Omega ,}
60:
2563:{\displaystyle f(x,y)={\begin{cases}1&x^{2}+y^{2}<1\\0&x^{2}+y^{2}\geq 1\end{cases}}}
4156:
4120:
4089:
4058:
2661:
2625:
115:
64:
30:
4114:
4032:
3999:
3943:
99:
56:
4134:
3101:{\displaystyle {\frac {N_{a}}{N_{a}+N_{b}}}={\frac {\sigma _{a}}{\sigma _{a}+\sigma _{b}}}}
180:{\displaystyle I=\int _{\Omega }f({\overline {\mathbf {x} }})\,d{\overline {\mathbf {x} }}}
4130:
3624:
543:
83:
4028:
3995:
3939:
1390:
Thus, a crude way of calculating the value of Ļ with Monte Carlo integration is to pick
1128:
17:
3010:
It can be shown that this variance is minimized by distributing the points such that,
980:{\displaystyle \left\{\sigma _{1}^{2},\sigma _{2}^{2},\sigma _{3}^{2},\ldots \right\}}
4197:
4036:
3926:
2611:
4013:
Lepage, G. P. (1978). "A New
Algorithm for Adaptive Multidimensional Integration".
3955:
2038:# Calculate area and print; should be closer to Pi with increasing number of throws
4044:
Lepage, G. P. (1980). "VEGAS: An
Adaptive Multi-dimensional Integration Program".
2445:
An illustration of
Recursive Stratified Sampling. In this example, the function:
4072:
2225:
44:
4107:
3947:
107:
3980:"Recursive Stratified Sampling for Multidimensional Monte Carlo Integration"
1161:
Relative error as a function of the number of samples, showing the scaling
70:
There are different methods to perform a Monte Carlo integration, such as
2610:
The idea of stratified sampling begins with the observation that for two
266:
The naive Monte Carlo approach is to sample points uniformly on Ī©: given
3637:, the probability distribution is approximated by a separable function:
34:
An illustration of Monte Carlo integration. In this example, the domain
3969:
4004:
3979:
3743:{\displaystyle g(x_{1},x_{2},\ldots )=g_{1}(x_{1})g_{2}(x_{2})\ldots }
114:
The problem Monte Carlo integration addresses is the computation of a
2278:(*Sample from truncated normal distribution to speed up convergence*)
2868:{\displaystyle E(f)={\tfrac {1}{2}}\left(E_{a}(f)+E_{b}(f)\right)}
2441:
2440:
1483:{\displaystyle Q_{N}=4{\frac {1}{N}}\sum _{i=1}^{N}H(x_{i},y_{i})}
1156:
29:
1579:
Keep in mind that a true random number generator should be used.
3856:
3813:
3801:
3146:
There are a variety of importance sampling algorithms, such as
987:
is bounded, this variance decreases asymptotically to zero as 1/
4084:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007).
2105:
The code below describes a process of integrating the function
3963:
Weinzierl, S. (2000). "Introduction to Monte Carlo methods".
3273:
Formally, given a set of samples chosen from a distribution
257:{\displaystyle V=\int _{\Omega }d{\overline {\mathbf {x} }}}
2556:
1291:
4189:
Monte Carlo applet applied in statistical physics problems
3615:, thus providing an efficient way of computing integrals.
3259:
can be chosen to decrease the variance of the measurement
2595:
The popular MISER routine implements a similar algorithm.
2185:{\displaystyle f(x)={\frac {1}{1+\sinh(2x)\log(x)^{2}}}}
1381:{\displaystyle I_{\pi }=\int _{\Omega }H(x,y)dxdy=\pi .}
106:. Monte Carlo integration, on the other hand, employs a
4088:(3rd ed.). New York: Cambridge University Press.
2803:
1548:
1500:
1169:
1098:
4116:
Information Theory, Inference and
Learning Algorithms
3924:(1998). "Monte Carlo and quasi-Monte Carlo methods".
3643:
3583:
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3381:
3279:
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123:
4086:Numerical Recipes: The Art of Scientific Computing
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1978:# If the point is inside circle, increase variable
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456:
335:
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208:
179:
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478:
3828:
3826:
3824:
3822:
3548:is one of the most used algorithms to generate
1492:In the figure on the right, the relative error
3901:
3899:
98:In numerical integration, methods such as the
3844:
3832:
3608:{\displaystyle p({\overline {\mathbf {x} }})}
3534:{\displaystyle p({\overline {\mathbf {x} }})}
3252:{\displaystyle p({\overline {\mathbf {x} }})}
3214:{\displaystyle p({\overline {\mathbf {x} }})}
8:
4053:Hammersley, J. M.; Handscomb, D. C. (1964).
3750:so that the number of bins required is only
682:
676:
515:{\displaystyle \lim _{N\to \infty }Q_{N}=I.}
448:
442:
4108:"chapter 4.4 Typicality & chapter 29.1"
3879:
2622:with Monte Carlo estimates of the integral
1909:# Choose random X and Y centered around 0,0
1529:{\displaystyle {\tfrac {Q_{N}-\pi }{\pi }}}
4144:Monte Carlo Methods in Statistical Physics
3570:{\displaystyle {\overline {\mathbf {x} }}}
3176:{\displaystyle {\overline {\mathbf {x} }}}
2603:The MISER algorithm is based on recursive
4003:
3968:
3774:Monte Carlo method in statistical physics
3728:
3715:
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167:
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161:
148:
146:
134:
122:
4178:CafƩ math : Monte Carlo Integration
3794:
2579:is a generalization of one-dimensional
1564:{\displaystyle {\tfrac {1}{\sqrt {N}}}}
1185:{\displaystyle {\tfrac {1}{\sqrt {N}}}}
1114:{\displaystyle {\tfrac {1}{\sqrt {N}}}}
86:(also known as a particle filter), and
3905:
3867:
7:
3978:Press, W. H.; Farrar, G. R. (1990).
4075:; Taimre, T.; Botev, Z. I. (2011).
2889:
2886:
2883:
2769:{\displaystyle \sigma _{b}^{2}(f)}
2728:{\displaystyle \sigma _{a}^{2}(f)}
1334:
1030:
1027:
1024:
834:
831:
828:
798:
795:
792:
723:
720:
717:
566:
563:
560:
488:
327:
234:
135:
116:multidimensional definite integral
25:
4142:Newman, MEJ; Barkema, GT (1999).
991:. The estimation of the error of
548:unbiased estimate of the variance
4016:Journal of Computational Physics
3593:
3558:
3519:
3469:
3439:
3337:
3309:
3289:
3237:
3199:
3164:
2224:using the Monte-Carlo method in
1394:random numbers on Ī© and compute
655:
418:
309:
281:
245:
209:{\displaystyle \mathbb {R} ^{m}}
168:
149:
4153:Monte Carlo Statistical Methods
4151:Robert, CP; Casella, G (2004).
4077:Handbook of Monte Carlo Methods
3892:Kroese, Taimre & Botev 2011
3304:
4119:. Cambridge University Press.
3734:
3721:
3708:
3695:
3679:
3647:
3602:
3587:
3528:
3513:
3484:
3462:
3454:
3432:
3298:
3283:
3246:
3231:
3208:
3193:
2977:
2971:
2929:
2923:
2899:
2893:
2857:
2851:
2835:
2829:
2796:
2790:
2763:
2757:
2722:
2716:
2681:
2675:
2645:
2639:
2467:
2455:
2170:
2163:
2154:
2145:
2121:
2115:
1477:
1451:
1354:
1342:
1047:
1034:
844:
838:
808:
802:
740:
727:
670:
648:
576:
570:
485:
433:
411:
158:
143:
1:
3546:MetropolisāHastings algorithm
3150:Importance sampling algorithm
2577:Recursive stratified sampling
2431:Recursive stratified sampling
2217:{\displaystyle 0.8<x<3}
1536:is measured as a function of
4046:Cornell Preprint CLNS 80-447
4037:10.1016/0021-9991(78)90004-9
3597:
3562:
3523:
3473:
3443:
3341:
3313:
3293:
3241:
3203:
3168:
2425:(*Compare with real answer*)
659:
422:
313:
285:
249:
172:
153:
63:that numerically computes a
3769:Auxiliary field Monte Carlo
2780:) of the combined estimate
2101:Wolfram Mathematica example
88:mean-field particle methods
4220:
4155:(2nd ed.). Springer.
3622:
3139:
2434:
1123:standard error of the mean
3948:10.1017/S0962492900002804
3845:Newman & Barkema 1999
3833:Newman & Barkema 1999
1305:and the set Ī© = Ć with
542:can be estimated by the
4079:. John Wiley & Sons.
3764:Quasi-Monte Carlo method
2687:{\displaystyle E_{b}(f)}
2651:{\displaystyle E_{a}(f)}
2230:
1843:
1581:
903:As long as the sequence
524:Given the estimation of
3880:Press & Farrar 1990
348:can be approximated by
49:Monte Carlo integration
18:Monte-Carlo integration
3744:
3609:
3571:
3535:
3494:
3425:
3365:
3253:
3215:
3177:
3102:
3002:
2869:
2770:
2729:
2688:
2652:
2573:
2564:
2218:
2186:
1565:
1530:
1484:
1447:
1382:
1299:
1192:
1186:
1139:
1115:
1084:
981:
895:
790:
703:
638:
516:
458:
407:
337:
258:
210:
181:
104:deterministic approach
84:sequential Monte Carlo
40:
3745:
3610:
3572:
3536:
3495:
3405:
3366:
3254:
3216:
3178:
3103:
3003:
2870:
2771:
2730:
2689:
2653:
2565:
2444:
2219:
2187:
1566:
1531:
1485:
1427:
1383:
1300:
1187:
1160:
1140:
1116:
1085:
982:
896:
770:
704:
618:
517:
459:
387:
338:
259:
211:
187:where Ī©, a subset of
182:
59:. It is a particular
53:numerical integration
33:
3984:Computers in Physics
3641:
3581:
3552:
3507:
3379:
3277:
3225:
3187:
3158:
3014:
2879:
2784:
2739:
2698:
2662:
2626:
2581:adaptive quadratures
2449:
2196:
2109:
1544:
1496:
1398:
1313:
1199:
1165:
1129:
1094:
1002:
907:
713:
556:
535:, the error bars of
474:
468:law of large numbers
466:This is because the
352:
274:
220:
191:
121:
4204:Monte Carlo methods
4055:Monte Carlo Methods
4029:1978JCoPh..27..192L
3996:1990ComPh...4..190P
3940:1998AcNum...7....1C
3221:. The idea is that
3142:Importance sampling
3136:Importance sampling
2970:
2922:
2776:, the variance Var(
2756:
2715:
2605:stratified sampling
2437:Stratified sampling
1090:which decreases as
965:
947:
929:
882:
596:
80:importance sampling
76:stratified sampling
51:is a technique for
27:Numerical technique
4146:. Clarendon Press.
3882:, pp. 190ā195
3870:, pp. 284ā292
3784:Variance reduction
3779:Monte Carlo method
3740:
3605:
3567:
3531:
3490:
3371:the estimator for
3361:
3249:
3211:
3173:
3098:
2998:
2956:
2908:
2865:
2812:
2766:
2742:
2725:
2701:
2684:
2648:
2574:
2560:
2555:
2214:
2182:
1561:
1559:
1526:
1524:
1480:
1378:
1295:
1290:
1193:
1182:
1180:
1135:
1111:
1109:
1080:
977:
951:
933:
915:
891:
868:
699:
582:
512:
492:
454:
333:
254:
206:
177:
61:Monte Carlo method
41:
4162:978-1-4419-1939-7
4126:978-0-521-64298-9
4095:978-0-521-88068-8
4064:978-0-416-52340-9
4005:10.1063/1.4822899
3857:Press et al. 2007
3814:Press et al. 2007
3802:Press et al. 2007
3619:VEGAS Monte Carlo
3600:
3565:
3526:
3488:
3476:
3446:
3403:
3344:
3316:
3296:
3244:
3206:
3171:
3096:
3054:
2996:
2948:
2811:
2599:MISER Monte Carlo
2180:
1558:
1557:
1540:, confirming the
1523:
1425:
1309:= 4. Notice that
1286:
1243:
1179:
1178:
1138:{\displaystyle V}
1108:
1107:
1075:
1074:
1050:
886:
851:
768:
662:
616:
477:
425:
385:
316:
288:
270:uniform samples,
252:
175:
156:
108:non-deterministic
65:definite integral
16:(Redirected from
4211:
4166:
4147:
4138:
4112:
4099:
4080:
4068:
4049:
4040:
4009:
4007:
3974:
3972:
3959:
3908:
3903:
3894:
3889:
3883:
3877:
3871:
3865:
3859:
3854:
3848:
3842:
3836:
3830:
3817:
3811:
3805:
3799:
3749:
3747:
3746:
3741:
3733:
3732:
3720:
3719:
3707:
3706:
3694:
3693:
3672:
3671:
3659:
3658:
3614:
3612:
3611:
3606:
3601:
3596:
3591:
3576:
3574:
3573:
3568:
3566:
3561:
3556:
3540:
3538:
3537:
3532:
3527:
3522:
3517:
3499:
3497:
3496:
3491:
3489:
3487:
3483:
3482:
3477:
3472:
3467:
3457:
3453:
3452:
3447:
3442:
3437:
3427:
3424:
3419:
3404:
3396:
3391:
3390:
3370:
3368:
3367:
3362:
3351:
3350:
3345:
3340:
3335:
3323:
3322:
3317:
3312:
3307:
3297:
3292:
3287:
3258:
3256:
3255:
3250:
3245:
3240:
3235:
3220:
3218:
3217:
3212:
3207:
3202:
3197:
3182:
3180:
3179:
3174:
3172:
3167:
3162:
3107:
3105:
3104:
3099:
3097:
3095:
3094:
3093:
3081:
3080:
3070:
3069:
3060:
3055:
3053:
3052:
3051:
3039:
3038:
3028:
3027:
3018:
3007:
3005:
3004:
2999:
2997:
2995:
2994:
2993:
2980:
2969:
2964:
2954:
2949:
2947:
2946:
2945:
2932:
2921:
2916:
2906:
2892:
2874:
2872:
2871:
2866:
2864:
2860:
2850:
2849:
2828:
2827:
2813:
2804:
2775:
2773:
2772:
2767:
2755:
2750:
2734:
2732:
2731:
2726:
2714:
2709:
2693:
2691:
2690:
2685:
2674:
2673:
2657:
2655:
2654:
2649:
2638:
2637:
2569:
2567:
2566:
2561:
2559:
2558:
2546:
2545:
2533:
2532:
2508:
2507:
2495:
2494:
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:
2223:
2221:
2220:
2215:
2191:
2189:
2188:
2183:
2181:
2179:
2178:
2177:
2128:
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:
1949:
1946:
1943:
1940:
1937:
1934:
1931:
1928:
1925:
1922:
1919:
1916:
1913:
1910:
1907:
1904:
1901:
1898:
1895:
1892:
1889:
1886:
1883:
1880:
1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1789:
1786:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1759:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1672:
1669:
1666:
1663:
1660:
1657:
1654:
1651:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1627:
1624:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1570:
1568:
1567:
1562:
1560:
1553:
1549:
1535:
1533:
1532:
1527:
1525:
1519:
1512:
1511:
1501:
1489:
1487:
1486:
1481:
1476:
1475:
1463:
1462:
1446:
1441:
1426:
1418:
1410:
1409:
1387:
1385:
1384:
1379:
1338:
1337:
1325:
1324:
1304:
1302:
1301:
1296:
1294:
1293:
1287:
1284:
1267:
1266:
1254:
1253:
1244:
1241:
1223:
1219:
1191:
1189:
1188:
1183:
1181:
1174:
1170:
1144:
1142:
1141:
1136:
1125:multiplied with
1120:
1118:
1117:
1112:
1110:
1103:
1099:
1089:
1087:
1086:
1081:
1076:
1070:
1069:
1068:
1059:
1051:
1046:
1045:
1033:
1022:
1017:
1016:
986:
984:
983:
978:
976:
972:
964:
959:
946:
941:
928:
923:
900:
898:
897:
892:
887:
881:
876:
867:
865:
864:
852:
847:
837:
825:
823:
822:
801:
789:
784:
769:
767:
766:
757:
756:
747:
739:
738:
726:
708:
706:
705:
700:
695:
694:
689:
685:
669:
668:
663:
658:
653:
637:
632:
617:
615:
601:
595:
590:
569:
521:
519:
518:
513:
502:
501:
491:
463:
461:
460:
455:
432:
431:
426:
421:
416:
406:
401:
386:
378:
370:
369:
342:
340:
339:
334:
323:
322:
317:
312:
307:
295:
294:
289:
284:
279:
263:
261:
260:
255:
253:
248:
243:
238:
237:
215:
213:
212:
207:
205:
204:
199:
186:
184:
183:
178:
176:
171:
166:
157:
152:
147:
139:
138:
100:trapezoidal rule
72:uniform sampling
39:4*0.8 = 3.2 ā Ļ.
21:
4219:
4218:
4214:
4213:
4212:
4210:
4209:
4208:
4194:
4193:
4174:
4169:
4163:
4150:
4141:
4127:
4110:
4102:
4096:
4083:
4071:
4065:
4052:
4043:
4012:
3977:
3962:
3922:Caflisch, R. E.
3920:
3916:
3911:
3904:
3897:
3890:
3886:
3878:
3874:
3866:
3862:
3855:
3851:
3843:
3839:
3831:
3820:
3812:
3808:
3800:
3796:
3792:
3760:
3724:
3711:
3698:
3685:
3663:
3650:
3639:
3638:
3627:
3625:VEGAS algorithm
3621:
3579:
3578:
3550:
3549:
3505:
3504:
3465:
3458:
3435:
3428:
3382:
3377:
3376:
3333:
3305:
3275:
3274:
3264:
3223:
3222:
3185:
3184:
3156:
3155:
3152:
3144:
3138:
3130:
3123:
3085:
3072:
3071:
3061:
3043:
3030:
3029:
3019:
3012:
3011:
2985:
2981:
2955:
2937:
2933:
2907:
2877:
2876:
2841:
2819:
2818:
2814:
2782:
2781:
2737:
2736:
2696:
2695:
2665:
2660:
2659:
2629:
2624:
2623:
2601:
2570:
2554:
2553:
2537:
2524:
2522:
2516:
2515:
2499:
2486:
2484:
2474:
2447:
2446:
2439:
2433:
2428:
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:
2194:
2193:
2169:
2132:
2107:
2106:
2103:
2098:
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:
1947:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
1872:
1869:
1866:
1863:
1860:
1857:
1854:
1851:
1848:
1845:
1835:
1830:
1829:
1826:
1823:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1697:
1694:
1691:
1688:
1685:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1658:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1589:
1586:
1583:
1577:
1542:
1541:
1503:
1502:
1494:
1493:
1467:
1454:
1401:
1396:
1395:
1329:
1316:
1311:
1310:
1289:
1288:
1281:
1275:
1274:
1258:
1245:
1238:
1228:
1209:
1205:
1197:
1196:
1163:
1162:
1155:
1127:
1126:
1092:
1091:
1060:
1037:
1008:
1000:
999:
996:
914:
910:
905:
904:
856:
826:
814:
758:
748:
730:
711:
710:
709:which leads to
651:
644:
640:
639:
605:
554:
553:
544:sample variance
540:
533:
493:
472:
471:
414:
361:
350:
349:
305:
277:
272:
271:
229:
218:
217:
194:
189:
188:
130:
119:
118:
96:
28:
23:
22:
15:
12:
11:
5:
4217:
4215:
4207:
4206:
4196:
4195:
4192:
4191:
4186:
4181:
4173:
4172:External links
4170:
4168:
4167:
4161:
4148:
4139:
4125:
4100:
4094:
4081:
4069:
4063:
4050:
4041:
4023:(2): 192ā203.
4010:
3975:
3970:hep-ph/0006269
3960:
3917:
3915:
3912:
3910:
3909:
3895:
3884:
3872:
3860:
3849:
3837:
3818:
3806:
3793:
3791:
3788:
3787:
3786:
3781:
3776:
3771:
3766:
3759:
3756:
3739:
3736:
3731:
3727:
3723:
3718:
3714:
3710:
3705:
3701:
3697:
3692:
3688:
3684:
3681:
3678:
3675:
3670:
3666:
3662:
3657:
3653:
3649:
3646:
3623:Main article:
3620:
3617:
3604:
3599:
3595:
3589:
3586:
3564:
3560:
3530:
3525:
3521:
3515:
3512:
3486:
3481:
3475:
3471:
3464:
3461:
3456:
3451:
3445:
3441:
3434:
3431:
3423:
3418:
3415:
3412:
3408:
3402:
3399:
3394:
3389:
3385:
3360:
3357:
3354:
3349:
3343:
3339:
3332:
3329:
3326:
3321:
3315:
3311:
3303:
3300:
3295:
3291:
3285:
3282:
3262:
3248:
3243:
3239:
3233:
3230:
3210:
3205:
3201:
3195:
3192:
3170:
3166:
3151:
3148:
3140:Main article:
3137:
3134:
3128:
3121:
3092:
3088:
3084:
3079:
3075:
3068:
3064:
3058:
3050:
3046:
3042:
3037:
3033:
3026:
3022:
2992:
2988:
2984:
2979:
2976:
2973:
2968:
2963:
2959:
2952:
2944:
2940:
2936:
2931:
2928:
2925:
2920:
2915:
2911:
2904:
2901:
2898:
2895:
2891:
2888:
2885:
2863:
2859:
2856:
2853:
2848:
2844:
2840:
2837:
2834:
2831:
2826:
2822:
2817:
2810:
2807:
2801:
2798:
2795:
2792:
2789:
2765:
2762:
2759:
2754:
2749:
2745:
2724:
2721:
2718:
2713:
2708:
2704:
2694:and variances
2683:
2680:
2677:
2672:
2668:
2647:
2644:
2641:
2636:
2632:
2600:
2597:
2557:
2552:
2549:
2544:
2540:
2536:
2531:
2527:
2523:
2521:
2518:
2517:
2514:
2511:
2506:
2502:
2498:
2493:
2489:
2485:
2483:
2480:
2479:
2477:
2472:
2469:
2466:
2463:
2460:
2457:
2454:
2432:
2429:
2231:
2213:
2210:
2207:
2204:
2201:
2176:
2172:
2168:
2165:
2162:
2159:
2156:
2153:
2150:
2147:
2144:
2141:
2138:
2135:
2131:
2126:
2123:
2120:
2117:
2114:
2102:
2099:
1844:
1834:
1833:Python example
1831:
1582:
1576:
1573:
1556:
1552:
1522:
1518:
1515:
1510:
1506:
1479:
1474:
1470:
1466:
1461:
1457:
1453:
1450:
1445:
1440:
1437:
1434:
1430:
1424:
1421:
1416:
1413:
1408:
1404:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1336:
1332:
1328:
1323:
1319:
1292:
1282:
1280:
1277:
1276:
1273:
1270:
1265:
1261:
1257:
1252:
1248:
1239:
1237:
1234:
1233:
1231:
1226:
1222:
1218:
1215:
1212:
1208:
1204:
1177:
1173:
1154:
1151:
1134:
1106:
1102:
1079:
1073:
1067:
1063:
1057:
1054:
1049:
1044:
1040:
1036:
1032:
1029:
1026:
1020:
1015:
1011:
1007:
994:
975:
971:
968:
963:
958:
954:
950:
945:
940:
936:
932:
927:
922:
918:
913:
890:
885:
880:
875:
871:
863:
859:
855:
850:
846:
843:
840:
836:
833:
830:
821:
817:
813:
810:
807:
804:
800:
797:
794:
788:
783:
780:
777:
773:
765:
761:
755:
751:
745:
742:
737:
733:
729:
725:
722:
719:
698:
693:
688:
684:
681:
678:
675:
672:
667:
661:
657:
650:
647:
643:
636:
631:
628:
625:
621:
614:
611:
608:
604:
599:
594:
589:
585:
581:
578:
575:
572:
568:
565:
562:
538:
531:
511:
508:
505:
500:
496:
490:
487:
484:
480:
453:
450:
447:
444:
441:
438:
435:
430:
424:
420:
413:
410:
405:
400:
397:
394:
390:
384:
381:
376:
373:
368:
364:
360:
357:
332:
329:
326:
321:
315:
311:
304:
301:
298:
293:
287:
283:
251:
247:
241:
236:
232:
228:
225:
203:
198:
174:
170:
164:
160:
155:
151:
145:
142:
137:
133:
129:
126:
95:
92:
57:random numbers
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4216:
4205:
4202:
4201:
4199:
4190:
4187:
4185:
4182:
4179:
4176:
4175:
4171:
4164:
4158:
4154:
4149:
4145:
4140:
4136:
4132:
4128:
4122:
4118:
4117:
4109:
4105:
4104:MacKay, David
4101:
4097:
4091:
4087:
4082:
4078:
4074:
4073:Kroese, D. P.
4070:
4066:
4060:
4056:
4051:
4047:
4042:
4038:
4034:
4030:
4026:
4022:
4018:
4017:
4011:
4006:
4001:
3997:
3993:
3989:
3985:
3981:
3976:
3971:
3966:
3961:
3957:
3953:
3949:
3945:
3941:
3937:
3933:
3929:
3928:
3927:Acta Numerica
3923:
3919:
3918:
3913:
3907:
3902:
3900:
3896:
3893:
3888:
3885:
3881:
3876:
3873:
3869:
3864:
3861:
3858:
3853:
3850:
3846:
3841:
3838:
3834:
3829:
3827:
3825:
3823:
3819:
3815:
3810:
3807:
3803:
3798:
3795:
3789:
3785:
3782:
3780:
3777:
3775:
3772:
3770:
3767:
3765:
3762:
3761:
3757:
3755:
3753:
3737:
3729:
3725:
3716:
3712:
3703:
3699:
3690:
3686:
3682:
3676:
3673:
3668:
3664:
3660:
3655:
3651:
3644:
3636:
3632:
3626:
3618:
3616:
3584:
3547:
3542:
3541:is constant.
3510:
3500:
3479:
3459:
3449:
3429:
3421:
3416:
3413:
3410:
3406:
3400:
3397:
3392:
3387:
3383:
3374:
3358:
3355:
3352:
3347:
3330:
3327:
3324:
3319:
3301:
3280:
3271:
3267:
3265:
3228:
3190:
3149:
3147:
3143:
3135:
3133:
3131:
3124:
3116:
3111:
3108:
3090:
3086:
3082:
3077:
3073:
3066:
3062:
3056:
3048:
3044:
3040:
3035:
3031:
3024:
3020:
3008:
2990:
2986:
2982:
2974:
2966:
2961:
2957:
2950:
2942:
2938:
2934:
2926:
2918:
2913:
2909:
2902:
2896:
2875:is given by,
2861:
2854:
2846:
2842:
2838:
2832:
2824:
2820:
2815:
2808:
2805:
2799:
2793:
2787:
2779:
2760:
2752:
2747:
2743:
2719:
2711:
2706:
2702:
2678:
2670:
2666:
2642:
2634:
2630:
2621:
2617:
2613:
2608:
2606:
2598:
2596:
2593:
2589:
2585:
2582:
2578:
2550:
2547:
2542:
2538:
2534:
2529:
2525:
2519:
2512:
2509:
2504:
2500:
2496:
2491:
2487:
2481:
2475:
2470:
2464:
2461:
2458:
2452:
2443:
2438:
2430:
2329:RandomVariate
2229:
2227:
2211:
2208:
2205:
2202:
2199:
2174:
2166:
2160:
2157:
2151:
2148:
2142:
2139:
2136:
2133:
2129:
2124:
2118:
2112:
2100:
2074:inside_circle
2020:inside_circle
1867:inside_circle
1842:
1840:
1832:
1580:
1574:
1572:
1554:
1550:
1539:
1520:
1516:
1513:
1508:
1504:
1490:
1472:
1468:
1464:
1459:
1455:
1448:
1443:
1438:
1435:
1432:
1428:
1422:
1419:
1414:
1411:
1406:
1402:
1393:
1388:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1351:
1348:
1345:
1339:
1330:
1326:
1321:
1317:
1308:
1278:
1271:
1268:
1263:
1259:
1255:
1250:
1246:
1235:
1229:
1224:
1220:
1216:
1213:
1210:
1206:
1202:
1175:
1171:
1159:
1152:
1150:
1146:
1132:
1124:
1104:
1100:
1077:
1071:
1065:
1061:
1055:
1052:
1042:
1038:
1018:
1013:
1009:
1005:
997:
990:
973:
969:
966:
961:
956:
952:
948:
943:
938:
934:
930:
925:
920:
916:
911:
901:
888:
883:
878:
873:
869:
861:
857:
853:
848:
841:
819:
815:
811:
805:
786:
781:
778:
775:
771:
763:
759:
753:
749:
743:
735:
731:
696:
691:
686:
679:
673:
665:
645:
641:
634:
629:
626:
623:
619:
612:
609:
606:
602:
597:
592:
587:
583:
579:
573:
551:
549:
545:
541:
534:
527:
522:
509:
506:
503:
498:
494:
482:
470:ensures that
469:
464:
451:
445:
439:
436:
428:
408:
403:
398:
395:
392:
388:
382:
379:
374:
371:
366:
362:
358:
355:
347:
343:
330:
324:
319:
302:
299:
296:
291:
269:
264:
239:
230:
226:
223:
216:, has volume
201:
162:
140:
131:
127:
124:
117:
112:
109:
105:
101:
93:
91:
89:
85:
81:
77:
73:
68:
66:
62:
58:
54:
50:
46:
37:
32:
19:
4152:
4143:
4115:
4085:
4076:
4054:
4045:
4020:
4014:
3987:
3983:
3931:
3925:
3887:
3875:
3863:
3852:
3840:
3809:
3797:
3751:
3634:
3630:
3628:
3543:
3501:
3375:is given by
3372:
3272:
3268:
3260:
3153:
3145:
3126:
3119:
3114:
3112:
3109:
3009:
2777:
2619:
2615:
2609:
2602:
2594:
2590:
2586:
2576:
2575:
2104:
1836:
1818:insideCircle
1797:insideCircle
1605:insideCircle
1578:
1537:
1491:
1391:
1389:
1306:
1194:
1147:
992:
988:
902:
552:
536:
529:
525:
523:
465:
345:
344:
267:
265:
113:
97:
69:
48:
42:
35:
4057:. Methuen.
3906:Lepage 1978
3868:MacKay 2003
2226:Mathematica
45:mathematics
3990:(2): 190.
3914:References
2435:See also:
2398:NIntegrate
1121:. This is
546:using the
3847:, Chap. 1
3835:, Chap. 2
3816:, Chap. 7
3804:, Chap. 4
3738:…
3677:…
3598:¯
3563:¯
3524:¯
3474:¯
3444:¯
3407:∑
3393:≡
3353:∈
3342:¯
3328:⋯
3314:¯
3294:¯
3242:¯
3204:¯
3169:¯
3087:σ
3074:σ
3063:σ
2958:σ
2910:σ
2744:σ
2703:σ
2548:≥
2371:Integrate
2161:
2143:
1575:C example
1521:π
1517:π
1514:−
1429:∑
1373:π
1335:Ω
1331:∫
1322:π
1269:≤
1062:σ
1019:≈
1006:δ
970:…
953:σ
935:σ
917:σ
870:σ
772:∑
683:⟩
677:⟨
674:−
660:¯
620:∑
610:−
584:σ
580:≡
489:∞
486:→
449:⟩
443:⟨
423:¯
389:∑
372:≡
359:≈
328:Ω
325:∈
314:¯
300:⋯
286:¯
250:¯
235:Ω
231:∫
173:¯
154:¯
136:Ω
132:∫
4198:Category
4106:(2003).
3934:: 1ā49.
3758:See also
2614:regions
2612:disjoint
2572:largest.
1837:Made in
1752:RAND_MAX
1722:RAND_MAX
1242:if
998:is thus
94:Overview
4135:2012999
4025:Bibcode
3992:Bibcode
3956:5708790
3936:Bibcode
2362:Distrib
2281:Distrib
1957:uniform
1924:uniform
1153:Example
4159:
4133:
4123:
4092:
4061:
3954:
2083:throws
2056:radius
2008:radius
1972:radius
1966:radius
1951:random
1939:radius
1933:radius
1918:random
1903:throws
1885:radius
1858:throws
1855:random
1852:import
1839:Python
1824:throws
1746:double
1716:double
1680:throws
1617:double
1593:throws
102:use a
55:using
4111:(PDF)
3965:arXiv
3952:S2CID
3790:Notes
3577:from
2356:Total
2192:from
2086:print
2005:<=
1894:while
1849:numpy
1782:randY
1776:randY
1770:randX
1764:randX
1728:randY
1698:randX
1638:srand
1626:randY
1620:randX
1599:99999
528:from
4157:ISBN
4121:ISBN
4090:ISBN
4059:ISBN
3544:The
3125:and
2735:and
2658:and
2618:and
2510:<
2254:Sinh
2233:func
2209:<
2203:<
2140:sinh
2092:area
2041:area
1900:<
1864:2000
1846:from
1785:<
1734:rand
1704:rand
1677:<
1650:NULL
1644:time
1285:else
4033:doi
4000:doi
3944:doi
2413:0.8
2386:0.8
2341:Int
2305:0.1
2293:1.1
2287:PDF
2263:Log
2200:0.8
2158:log
2047:(((
1841:.
1812:4.0
1656:for
1653:));
1584:int
479:lim
43:In
4200::
4131:MR
4129:.
4113:.
4031:.
4021:27
4019:.
3998:.
3986:.
3982:.
3950:.
3942:.
3930:.
3898:^
3821:^
3752:Kd
3266:.
2422:}]
2395:}]
2338:];
2332:],
2323:RV
2317:10
2308:];
2284::=
2275:);
2236::=
2228::
2062:**
2032:+=
2023:+=
2011:**
1999:**
1987:**
1981:if
1806:pi
1794:++
1758:if
1737:()
1707:()
1686:++
1632:pi
1571:.
550:.
90:.
82:,
78:,
74:,
47:,
4165:.
4137:.
4098:.
4067:.
4048:.
4039:.
4035::
4027::
4008:.
4002::
3994::
3988:4
3973:.
3967::
3958:.
3946::
3938::
3932:7
3735:)
3730:2
3726:x
3722:(
3717:2
3713:g
3709:)
3704:1
3700:x
3696:(
3691:1
3687:g
3683:=
3680:)
3674:,
3669:2
3665:x
3661:,
3656:1
3652:x
3648:(
3645:g
3635:K
3631:f
3603:)
3594:x
3588:(
3585:p
3559:x
3529:)
3520:x
3514:(
3511:p
3485:)
3480:i
3470:x
3463:(
3460:p
3455:)
3450:i
3440:x
3433:(
3430:f
3422:N
3417:1
3414:=
3411:i
3401:N
3398:1
3388:N
3384:Q
3373:I
3359:,
3356:V
3348:N
3338:x
3331:,
3325:,
3320:1
3310:x
3302::
3299:)
3290:x
3284:(
3281:p
3263:N
3261:Q
3247:)
3238:x
3232:(
3229:p
3209:)
3200:x
3194:(
3191:p
3165:x
3129:b
3127:N
3122:a
3120:N
3115:d
3091:b
3083:+
3078:a
3067:a
3057:=
3049:b
3045:N
3041:+
3036:a
3032:N
3025:a
3021:N
2991:b
2987:N
2983:4
2978:)
2975:f
2972:(
2967:2
2962:b
2951:+
2943:a
2939:N
2935:4
2930:)
2927:f
2924:(
2919:2
2914:a
2903:=
2900:)
2897:f
2894:(
2890:r
2887:a
2884:V
2862:)
2858:)
2855:f
2852:(
2847:b
2843:E
2839:+
2836:)
2833:f
2830:(
2825:a
2821:E
2816:(
2809:2
2806:1
2800:=
2797:)
2794:f
2791:(
2788:E
2778:f
2764:)
2761:f
2758:(
2753:2
2748:b
2723:)
2720:f
2717:(
2712:2
2707:a
2682:)
2679:f
2676:(
2671:b
2667:E
2646:)
2643:f
2640:(
2635:a
2631:E
2620:b
2616:a
2551:1
2543:2
2539:y
2535:+
2530:2
2526:x
2520:0
2513:1
2505:2
2501:y
2497:+
2492:2
2488:x
2482:1
2476:{
2471:=
2468:)
2465:y
2462:,
2459:x
2456:(
2453:f
2419:3
2416:,
2410:,
2407:x
2404:{
2401:,
2392:3
2389:,
2383:,
2380:x
2377:{
2374:,
2368:*
2365:]
2359:/
2353:n
2350:/
2347:1
2344:=
2335:n
2326:=
2320:;
2314:=
2311:n
2302:-
2299:x
2296:*
2290:,
2272:2
2269:^
2266:)
2260:(
2257:*
2251:+
2248:1
2245:(
2242:/
2239:1
2212:3
2206:x
2175:2
2171:)
2167:x
2164:(
2155:)
2152:x
2149:2
2146:(
2137:+
2134:1
2130:1
2125:=
2122:)
2119:x
2116:(
2113:f
2095:)
2089:(
2080:/
2077:)
2071:*
2068:)
2065:2
2059:)
2053:*
2050:2
2044:=
2035:1
2029:i
2026:1
2017::
2014:2
2002:2
1996:y
1993:+
1990:2
1984:x
1975:)
1969:,
1963:-
1960:(
1954:.
1948:=
1945:y
1942:)
1936:,
1930:-
1927:(
1921:.
1915:=
1912:x
1906::
1897:i
1891:1
1888:=
1882:0
1879:=
1876:i
1873:0
1870:=
1861:=
1827:;
1821:/
1815:*
1809:=
1803:}
1800:;
1791:)
1788:1
1779:*
1773:+
1767:*
1761:(
1755:;
1749:)
1743:(
1740:/
1731:=
1725:;
1719:)
1713:(
1710:/
1701:=
1695:{
1692:)
1689:i
1683:;
1674:i
1671:;
1668:0
1665:=
1662:i
1659:(
1647:(
1641:(
1635:;
1629:,
1623:,
1614:;
1611:0
1608:=
1602:,
1596:=
1590:,
1587:i
1555:N
1551:1
1538:N
1509:N
1505:Q
1478:)
1473:i
1469:y
1465:,
1460:i
1456:x
1452:(
1449:H
1444:N
1439:1
1436:=
1433:i
1423:N
1420:1
1415:4
1412:=
1407:N
1403:Q
1392:N
1376:.
1370:=
1367:y
1364:d
1361:x
1358:d
1355:)
1352:y
1349:,
1346:x
1343:(
1340:H
1327:=
1318:I
1307:V
1279:0
1272:1
1264:2
1260:y
1256:+
1251:2
1247:x
1236:1
1230:{
1225:=
1221:)
1217:y
1214:,
1211:x
1207:(
1203:H
1176:N
1172:1
1133:V
1105:N
1101:1
1078:,
1072:N
1066:N
1056:V
1053:=
1048:)
1043:N
1039:Q
1035:(
1031:r
1028:a
1025:V
1014:N
1010:Q
995:N
993:Q
989:N
974:}
967:,
962:2
957:3
949:,
944:2
939:2
931:,
926:2
921:1
912:{
889:.
884:N
879:2
874:N
862:2
858:V
854:=
849:N
845:)
842:f
839:(
835:r
832:a
829:V
820:2
816:V
812:=
809:)
806:f
803:(
799:r
796:a
793:V
787:N
782:1
779:=
776:i
764:2
760:N
754:2
750:V
744:=
741:)
736:N
732:Q
728:(
724:r
721:a
718:V
697:.
692:2
687:)
680:f
671:)
666:i
656:x
649:(
646:f
642:(
635:N
630:1
627:=
624:i
613:1
607:N
603:1
598:=
593:2
588:N
577:)
574:f
571:(
567:r
564:a
561:V
539:N
537:Q
532:N
530:Q
526:I
510:.
507:I
504:=
499:N
495:Q
483:N
452:.
446:f
440:V
437:=
434:)
429:i
419:x
412:(
409:f
404:N
399:1
396:=
393:i
383:N
380:1
375:V
367:N
363:Q
356:I
346:I
331:,
320:N
310:x
303:,
297:,
292:1
282:x
268:N
246:x
240:d
227:=
224:V
202:m
197:R
169:x
163:d
159:)
150:x
144:(
141:f
128:=
125:I
36:D
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
