1138:
31:
1574:
1353:
1799:
4016:
1062:
3739:
2078:
2457:
type is generally a function of the number of evaluation points. The result is usually more accurate as the number of evaluation points increases, or, equivalently, as the width of the step size between the points decreases. It is natural to ask what the result would be if the step size were allowed
4095:
were originally developed by
Smolyak for the quadrature of high-dimensional functions. The method is always based on a one-dimensional quadrature rule, but performs a more sophisticated combination of univariate results. However, whereas the tensor product rule guarantees that the weights of all of
1325:
An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations. A method that yields a small error for a small number of evaluations is usually considered superior. Reducing the number of
2443:
using static quadrature rules on adaptively refined subintervals of the region of integration. Generally, adaptive algorithms are just as efficient and effective as traditional algorithms for "well behaved" integrands, but are also effective for "badly behaved" integrands for which traditional
3734:
3259:
972:
It may be possible to find an antiderivative symbolically, but it may be easier to compute a numerical approximation than to compute the antiderivative. That may be the case if the antiderivative is given as an infinite series or product, or if its evaluation requires a
3049:
1873:
4011:{\displaystyle {\begin{aligned}\int _{a}^{\infty }f(x)\,dx&=\int _{0}^{1}f\left(a+{\frac {t}{1-t}}\right){\frac {dt}{(1-t)^{2}}},\\\int _{-\infty }^{a}f(x)\,dx&=\int _{0}^{1}f\left(a-{\frac {1-t}{t}}\right){\frac {dt}{t^{2}}},\end{aligned}}}
4026:
The quadrature rules discussed so far are all designed to compute one-dimensional integrals. To compute integrals in multiple dimensions, one approach is to phrase the multiple integral as repeated one-dimensional integrals by applying
3560:
1133:
as the diameter, then the height BH (from a point of their connection to crossing with a circle) equals their geometric mean. The similar geometrical construction solves a problem of a quadrature for a parallelogram and a triangle.
1794:
1549:
3477:
1439:
4311:, can be applied to the restated problem and thus be used to evaluate the integral. For instance, the standard fourth-order RungeâKutta method applied to the differential equation yields Simpson's rule from above.
1316:
Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral. The integrand is evaluated at a finite set of points called
4299:
2886:
3744:
3087:
2753:
1322:
and a weighted sum of these values is used to approximate the integral. The integration points and weights depend on the specific method used and the accuracy required from the approximation.
877:
is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision.
4190:
4044:
A great many additional techniques for forming multidimensional cubature integration rules for a variety of weighting functions are given in the monograph by Stroud. Integration on the
802:) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as
4069:
are easy to apply to multi-dimensional integrals. They may yield greater accuracy for the same number of function evaluations than repeated integrations using one-dimensional methods.
5012:
4808:
2310:
861:
4304:
1673:
1628:
3334:
2798:
2248:
4361:
2204:
1111:
2596:
1683:
4643:
Kerstin Hesse, Ian H. Sloan, and Robert S. Womersley: Numerical
Integration on the Sphere. In W. Freeden et al. (eds.), Handbook of Geomathematics, Springer: Berlin 2015,
2664:
5407:
3381:
2163:
2073:{\displaystyle \int _{a}^{b}f(x)\,dx\approx {\frac {b-a}{n}}\left({f(a) \over 2}+\sum _{k=1}^{n-1}\left(f\left(a+k{\frac {b-a}{n}}\right)\right)+{f(b) \over 2}\right),}
5412:
1453:
765:
3076:
4096:
the cubature points will be positive if the weights of the quadrature points were positive, Smolyak's rule does not guarantee that the weights will all be positive.
3512:
5275:
4845:
506:
2437:
81:
5402:
3557:
for integrals on the positive reals. Monte Carlo methods can also be used, or a change of variables to a finite interval; e.g., for the whole line one could use
2623:
4401:
4381:
3396:
2881:
2861:
2841:
2818:
2539:
2504:
2369:
formulas. A Gaussian quadrature rule is typically more accurate than a NewtonâCotes rule that uses the same number of function evaluations, if the integrand is
2268:
1856:
52:
2345:
1836:
5524:
5005:
5062:
249:
3549:
Several methods exist for approximate integration over unbounded intervals. The standard technique involves specially derived quadrature rules, such as
895:
5175:
4427:
3729:{\displaystyle \int _{-\infty }^{\infty }f(x)\,dx=\int _{-1}^{+1}f\left({\frac {t}{1-t^{2}}}\right){\frac {1+t^{2}}{\left(1-t^{2}\right)^{2}}}\,dt,}
380:
5219:
4998:
4536:
Mathieu
Ossendrijver (Jan 29, 2016). "Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph".
1040:
421:
1367:
5113:
4710:
4599:
311:
4923:
3053:
We can further approximate the integral on the right-hand side by bringing the absolute value into the integrand, and replacing the term in
2474:. Extrapolation methods are described in more detail by Stoer and Bulirsch (Section 3.4) and are implemented in many of the routines in the
2381:
5397:
758:
213:
1003:"Quadrature" is a historical mathematical term that means calculating area. Quadrature problems have served as one of the main sources of
5322:
5103:
4838:
1285:
came a universal method for area calculation. In response, the term "quadrature" has become traditional, and instead the modern phrase "
632:
286:
4403:). This simplifies the theory and algorithms considerably. The problem of evaluating integrals is thus best studied in its own right.
4077:
5108:
5072:
4777:
4759:
4751:
4662:
4628:
1151:
with compass and straightedge had been proved in the 19th century to be impossible. Nevertheless, for some figures (for example the
307:
259:
2362:. The corresponding rule with each interval subdivided includes all the current points, so those integrand values can be re-used.
2669:
5170:
5082:
375:
294:
269:
5138:
4204:
751:
5453:
5209:
4964:
4938:
4432:
2374:
686:
411:
5052:
4831:
4411:
4200:
1156:
1036:
329:
239:
5224:
4933:
4928:
4221:
3254:{\displaystyle \left|\int _{a}^{b}f(x)\,dx-(b-a)f(a)\right|\leq {(b-a)^{2} \over 2}\sup _{a\leq x\leq b}\left|f'(x)\right|,}
426:
880:
Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure (
5067:
5057:
3479:
for the error term of that particular approximation. (Note that this is precisely the error we calculated for the example
2365:
If we allow the intervals between interpolation points to vary, we find another group of quadrature formulas, such as the
1248:
254:
244:
4913:
5077:
5047:
4885:
4407:
1210:
701:
552:
455:
342:
264:
4918:
5519:
5312:
5307:
5182:
5143:
906:
There are several reasons for carrying out numerical integration, as opposed to analytical integration by finding the
387:
302:
936:
A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative that is an
592:
1858:
of subintervals, computing an approximation for each subinterval, then adding up all the results. This is called a
1326:
evaluations of the integrand reduces the number of arithmetic operations involved, and therefore reduces the total
1222:
997:
460:
5352:
4954:
4862:
4818:
2454:
2348:
563:
5488:
5458:
5433:
3554:
706:
541:
4129:
5347:
4812:
4437:
4073:
4066:
3550:
2463:
1148:
1028:
987:
891:
557:
465:
5357:
4308:
2458:
to approach zero. This can be answered by extrapolating the result from two or more nonzero step sizes, using
619:
3044:{\displaystyle \left|\int _{a}^{b}f(x)\,dx-(b-a)f(a)\right|=\left|\int _{a}^{b}(x-a)f'(\xi _{x})\,dx\right|.}
1338:
A "brute force" kind of numerical integration can be done, if the integrand is reasonably well-behaved (i.e.
5327:
5317:
5214:
4057:
4037:
2273:
637:
627:
575:
416:
815:
4802:
4466:
4108:
2440:
2408:
1306:
1271:
1194:
966:
450:
1633:
1588:
5473:
5468:
5362:
5265:
5133:
5128:
5123:
5118:
5042:
5021:
4196:
1364:(a piecewise constant function, or a segmented polynomial of degree zero) that passes through the point
1004:
926:
882:
809:
The basic problem in numerical integration is to compute an approximate solution to a definite integral
696:
681:
570:
495:
334:
90:
4107:
is a statistical approach to the numerical problem of computing integrals and falls under the field of
3288:
2758:
1567:
3514:.) Using more derivatives, and by tweaking the quadrature, we can do a similar error analysis using a
2373:(i.e., if it is sufficiently differentiable). Other quadrature methods with varying intervals include
2209:
5367:
5291:
5260:
4979:
4545:
4461:
4111:. It can provide a full handling of the uncertainty over the solution of the integral expressed as a
1806:
For either one of these rules, we can make a more accurate approximation by breaking up the interval
1183:
582:
518:
491:
2168:
1125:). For this purpose it is possible to use the following fact: if we draw the circle with the sum of
1083:
5377:
5270:
5164:
4959:
4905:
4895:
4735:
4456:
4104:
4028:
3530:
2544:
2459:
2400:
2394:
2366:
1342:
1264:
1152:
1137:
1032:
937:
614:
599:
500:
364:
203:
170:
161:
30:
5498:
5337:
5332:
5255:
5087:
4974:
4672:
4569:
4062:
4032:
2628:
2599:
1260:
1202:
779:
609:
604:
487:
4406:
Conversely, the term "quadrature" may also be used for the solution of differential equations: "
4317:
4790:
3342:
2085:
4880:
4875:
4773:
4755:
4747:
4706:
4624:
4595:
4587:
4561:
4487:
2471:
2352:
1346:
1282:
1275:
1240:
1187:
1020:
1008:
721:
666:
445:
180:
5483:
5463:
4969:
4870:
4690:
4644:
4617:
4553:
4451:
4446:
4112:
3482:
1677:
1447:
1252:
1051:
1047:
974:
930:
731:
716:
5443:
5372:
4722:
4658:
4035:
as the number of dimensions increases. Three methods are known to overcome this so-called
2413:
2370:
1582:
1327:
1198:
1193:
In medieval Europe the quadrature meant calculation of area by any method. More often the
960:
671:
587:
114:
57:
5448:
4890:
1178:
cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment.
726:
4549:
3056:
2605:
5438:
5245:
4694:
4676:
4415:
4386:
4366:
4081:
3534:
2866:
2846:
2826:
2803:
2509:
2489:
2253:
1841:
1114:
1066:
1012:
956:
907:
691:
676:
482:
470:
189:
37:
4796:
2318:
1809:
1789:{\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)\left({\frac {f(a)+f(b)}{2}}\right).}
1573:
1352:
1330:. Also, each evaluation takes time, and the integrand may be arbitrarily complicated.
5513:
4573:
4363:
has a special form: the right-hand side contains only the independent variable (here
3522:. This error analysis gives a strict upper bound on the error, if the derivatives of
3515:
1559:
1361:
17:
5493:
5428:
5342:
4743:
4698:
1798:
1310:
1244:
1168:
711:
661:
547:
175:
34:
Numerical integration is used to calculate a numerical approximation for the value
1562:
functions that are easy to integrate. Typically these interpolating functions are
4648:
2358:
Quadrature rules with equally spaced points have the very convenient property of
1544:{\displaystyle \int _{a}^{b}f(x)\,dx\approx (b-a)f\left({\frac {a+b}{2}}\right).}
994:
A Course in
Interpolation and Numeric Integration for the Mathematical Laboratory
5250:
4765:
4718:
4442:
4092:
3390:
1232:
119:
1225:, de Saint-Vincent's pupil and commentator, noted the relation of this area to
933:
and other computer applications may need numerical integration for this reason.
5478:
5148:
4490:
4031:(the tensor product rule). This approach requires the function evaluations to
2467:
1563:
1270:
The quadrature of the hyperbola by Saint-Vincent and de Sarasa provided a new
1160:
736:
2355:, which is based on a polynomial of order 2, is also a NewtonâCotes formula.
5240:
4557:
4495:
1339:
1226:
1214:
1197:
was used; it was less rigorous, but more simple and powerful. With its help
1147:
Problems of quadrature for curvilinear figures are much more difficult. The
1061:
787:
477:
198:
141:
131:
4565:
1570:, only polynomials of low degree are used, typically linear and quadratic.
1155:) a quadrature can be performed. The quadratures of a sphere surface and a
4207:. By differentiating both sides of the above with respect to the argument
3472:{\displaystyle {\frac {n^{-1}}{2}}\sup _{0\leq x\leq 1}\left|f'(x)\right|}
992:
The term "numerical integration" first appears in 1915 in the publication
5280:
4990:
3279:
2475:
2404:
1434:{\textstyle \left({\frac {a+b}{2}},f\left({\frac {a+b}{2}}\right)\right)}
1313:
of function values at specified points within the domain of integration.
1302:
1175:
1055:
791:
4823:
1167:
The area of the surface of a sphere is equal to quadruple the area of a
1206:
152:
147:
136:
806:; others take "quadrature" to include higher-dimensional integration.
4045:
2351:, of which the rectangle rule and the trapezoidal rule are examples.
2315:
Interpolation with polynomials evaluated at equally spaced points in
1256:
4511:
1797:
1572:
1351:
1060:
1027:). That is why the process was named "quadrature". For example, a
29:
1558:
A large class of quadrature rules can be derived by constructing
1016:
4994:
4827:
1870:. For example, the composite trapezoidal rule can be stated as
4072:
A large class of useful Monte Carlo methods are the so-called
1566:. In practice, since polynomials of very high degree tend to
4512:"Earliest Known Uses of Some of the Words of Mathematics (Q)"
1585:, i.e. a polynomial of degree 1) passing through the points
1349:), by evaluating the integrand with very small increments.
3337:
925:
may be known only at certain points, such as obtained by
4809:
Implementations of many quadrature and cubature formulae
1163:
became the highest achievement of the antique analysis.
2380:
Gaussian quadrature rules do not nest, but the related
2377:(also called Fejér quadrature) methods, which do nest.
1039:. This construction must be performed only by means of
4705:(3rd ed.), New York: Cambridge University Press,
4294:{\displaystyle {\frac {dF(x)}{dx}}=f(x),\quad F(a)=0.}
3291:
2883:
on both sides and take the absolute values, we obtain
2171:
1581:
The interpolating function may be a straight line (an
1370:
4389:
4369:
4320:
4305:
Numerical methods for ordinary differential equations
4224:
4132:
3742:
3563:
3485:
3399:
3389:. We can convert this into an error analysis for the
3345:
3090:
3059:
2889:
2869:
2849:
2829:
2806:
2761:
2672:
2631:
2608:
2547:
2512:
2492:
2416:
2321:
2276:
2256:
2212:
2088:
1876:
1844:
1812:
1686:
1636:
1591:
1456:
1086:
818:
60:
40:
3383:
our error is no greater than the right hand side of
1360:
This simplest method approximates the function by a
1080:
it is necessary to construct a square with the side
5421:
5390:
5300:
5233:
5202:
5195:
5157:
5096:
5035:
5028:
4947:
4904:
4861:
2270:but one could also use intervals of varying length
4703:Numerical Recipes: The Art of Scientific Computing
4616:
4395:
4375:
4355:
4293:
4184:
4010:
3728:
3506:
3471:
3375:
3328:
3253:
3070:
3043:
2875:
2855:
2835:
2812:
2792:
2747:
2658:
2617:
2590:
2533:
2498:
2431:
2339:
2304:
2262:
2242:
2198:
2157:
2072:
1850:
1830:
1788:
1667:
1622:
1543:
1433:
1105:
894:. The term is also sometimes used to describe the
855:
790:for calculating the numerical value of a definite
75:
46:
5408:List of nonlinear ordinary differential equations
1554:Quadrature rules based on interpolating functions
1182:For the proof of the results Archimedes used the
5413:List of nonlinear partial differential equations
4123:The problem of evaluating the definite integral
3421:
3200:
2403:is a numerical integration method in which the
1072:For a quadrature of a rectangle with the sides
1019:as the process of constructing geometrically a
5403:List of linear ordinary differential equations
4681:Computer Methods for Mathematical Computations
3736:and for semi-infinite intervals one could use
3518:(using a partial sum with remainder term) for
5006:
4839:
4770:An Introduction to the History of Mathematics
4683:. Englewood Cliffs, NJ: Prentice-Hall, 1977.
4619:Approximate Calculation of Multiple Integrals
4592:Numerical Analysis and Scientific Computation
3529:This integration method can be combined with
2250:Here we used subintervals of the same length
1287:computation of a univariate definite integral
759:
8:
4414:" means expressing its solution in terms of
2748:{\displaystyle (x-a)f'(\xi _{x})=f(x)-f(a),}
1263:successfully performed a quadrature of some
1251:made further progress: quadratures for some
896:numerical solution of differential equations
4697:; Vetterling, W.T.; Flannery, B.P. (2007),
5199:
5032:
5013:
4999:
4991:
4846:
4832:
4824:
4791:Integration: Background, Simulations, etc.
4185:{\displaystyle F(x)=\int _{a}^{x}f(u)\,du}
4048:has been reviewed by Hesse et al. (2015).
766:
752:
232:
107:
86:
4388:
4368:
4319:
4225:
4223:
4175:
4157:
4152:
4131:
3993:
3979:
3956:
3936:
3931:
3913:
3895:
3887:
3867:
3840:
3817:
3797:
3792:
3774:
3756:
3751:
3743:
3741:
3716:
3708:
3697:
3674:
3661:
3648:
3632:
3616:
3608:
3594:
3576:
3568:
3562:
3553:for integrals on the whole real line and
3484:
3424:
3406:
3400:
3398:
3344:
3319:
3301:
3296:
3290:
3203:
3187:
3168:
3123:
3105:
3100:
3089:
3058:
3026:
3017:
2981:
2976:
2922:
2904:
2899:
2888:
2868:
2848:
2828:
2805:
2766:
2760:
2703:
2671:
2630:
2607:
2558:
2546:
2511:
2491:
2453:The accuracy of a quadrature rule of the
2415:
2320:
2296:
2286:
2275:
2255:
2211:
2178:
2170:
2087:
2041:
2010:
1976:
1965:
1937:
1914:
1904:
1886:
1881:
1875:
1843:
1811:
1743:
1714:
1696:
1691:
1685:
1635:
1590:
1516:
1484:
1466:
1461:
1455:
1404:
1376:
1369:
1093:
1085:
959:, times a constant) cannot be written in
846:
828:
823:
817:
59:
39:
27:Methods of calculating definite integrals
4428:Truncation error (numerical integration)
2482:Conservative (a priori) error estimation
1334:Quadrature rules based on step functions
1235:algebrised this method: he wrote in his
1136:
4793:at Holistic Numerical Methods Institute
4478:
4383:) and not the dependent variable (here
647:
398:
320:
278:
221:
188:
160:
110:
96:
89:
4119:Connection with differential equations
3285:Hence, if we approximate the integral
2466:. The extrapolation function may be a
2305:{\displaystyle \left(h_{k}\right)_{k}}
54:, the area under the curve defined by
4699:"Chapter 4. Integration of Functions"
2506:have a bounded first derivative over
2082:where the subintervals have the form
1577:Illustration of the trapezoidal rule.
1293:Methods for one-dimensional integrals
940:. An example of such an integrand is
856:{\displaystyle \int _{a}^{b}f(x)\,dx}
7:
5398:List of named differential equations
3080:
1015:doctrine, understood calculation of
214:List of named differential equations
5323:Method of undetermined coefficients
5104:Dependent and independent variables
4729:. New York: Springer-Verlag, 1980.
1668:{\displaystyle \left(b,f(b)\right)}
1623:{\displaystyle \left(a,f(a)\right)}
1356:Illustration of the rectangle rule.
1239:(1656) series that we now call the
1142:The area of a segment of a parabola
955:, the antiderivative of which (the
287:Dependent and independent variables
5525:Numerical integration (quadrature)
4727:Introduction to Numerical Analysis
4203:by applying the first part of the
3891:
3757:
3577:
3572:
3329:{\textstyle \int _{a}^{b}f(x)\,dx}
1243:, and he calculated their values.
866:to a given degree of accuracy. If
25:
4623:. Cliffs, NJ: Prentice-Hall Inc.
3545:Integrals over infinite intervals
2793:{\displaystyle \xi _{x}\in (a,x]}
2382:GaussâKronrod quadrature formulas
1046:The ancient Babylonians used the
5220:Carathéodory's existence theorem
4924:GaussâKronrod quadrature formula
4805:from Encyclopedia of Mathematics
4667:Methods of Numerical Integration
2393:This section is an excerpt from
2243:{\displaystyle k=0,\ldots ,n-1.}
1009:Mathematicians of Ancient Greece
422:Carathéodory's existence theorem
4272:
4211:, it is seen that the function
4205:fundamental theorem of calculus
2199:{\textstyle h={\frac {b-a}{n}}}
1802:Illustration of Simpson's rule.
4350:
4344:
4335:
4329:
4282:
4276:
4266:
4260:
4240:
4234:
4201:ordinary differential equation
4172:
4166:
4142:
4136:
4076:algorithms, which include the
3910:
3904:
3864:
3851:
3771:
3765:
3591:
3585:
3495:
3489:
3461:
3455:
3370:
3364:
3358:
3346:
3316:
3310:
3240:
3234:
3184:
3171:
3157:
3151:
3145:
3133:
3120:
3114:
3023:
3010:
2999:
2987:
2956:
2950:
2944:
2932:
2919:
2913:
2787:
2775:
2739:
2733:
2724:
2718:
2709:
2696:
2685:
2673:
2650:
2638:
2582:
2579:
2567:
2564:
2525:
2513:
2426:
2420:
2334:
2322:
2149:
2137:
2131:
2125:
2113:
2089:
2053:
2047:
1949:
1943:
1901:
1895:
1825:
1813:
1770:
1764:
1755:
1749:
1736:
1724:
1711:
1705:
1657:
1651:
1612:
1606:
1506:
1494:
1481:
1475:
1213:investigated the area under a
1106:{\displaystyle x={\sqrt {ab}}}
1037:The Quadrature of the Parabola
843:
837:
509: / Integral solutions
70:
64:
1:
4078:MetropolisâHastings algorithm
4018:as possible transformations.
2591:{\displaystyle f\in C^{1}().}
1174:The area of a segment of the
5048:Notation for differentiation
4649:10.1007/978-3-642-54551-1_40
786:comprises a broad family of
553:Exponential response formula
299:Coupled / Decoupled
5144:Exact differential equation
3393:, giving an upper bound of
3385:
2659:{\displaystyle x\in [a,b),}
1301:is an approximation of the
1065:Antique method to find the
1050:to integrate the motion of
5541:
4965:ClenshawâCurtis quadrature
4939:ChebyshevâGauss quadrature
4819:SageMath Online Integrator
4803:Lobatto quadrature formula
4675:, Michael A. Malcolm, and
4433:ClenshawâCurtis quadrature
4356:{\displaystyle F'(x)=f(x)}
4314:The differential equation
4055:
4022:Multidimensional integrals
2392:
2375:ClenshawâCurtis quadrature
1278:, of critical importance.
1223:Alphonse Antonio de Sarasa
985:
964:
5454:JĂłzef Maria Hoene-WroĆski
5434:Gottfried Wilhelm Leibniz
5225:CauchyâKowalevski theorem
4934:GaussâLegendre quadrature
4929:GaussâLaguerre quadrature
4886:Adaptive Simpson's method
4813:Tracker Component Library
4067:quasi-Monte Carlo methods
3555:Gauss-Laguerre quadrature
3376:{\displaystyle (b-a)f(a)}
3282:was used to approximate.
2158:{\displaystyle \subset ,}
1211:Grégoire de Saint-Vincent
687:JĂłzef Maria Hoene-WroĆski
633:Undetermined coefficients
542:Method of characteristics
427:CauchyâKowalevski theorem
5348:Finite difference method
4914:GaussâHermite quadrature
4746:, New York: Wiley, 1989
4740:A History of Mathematics
4438:Gauss-Kronrod quadrature
4074:Markov chain Monte Carlo
3551:Gauss-Hermite quadrature
2464:Richardson extrapolation
1149:quadrature of the circle
1041:compass and straightedge
1029:quadrature of the circle
988:Quadrature (mathematics)
892:quadrature of the circle
412:PicardâLindelöf theorem
406:Existence and uniqueness
5328:Variation of parameters
5318:Separation of variables
5215:Peano existence theorem
5210:PicardâLindelöf theorem
5097:Attributes of variables
4919:GaussâJacobi quadrature
4558:10.1126/science.aad8085
4412:reduction to quadrature
4058:Monte Carlo integration
4038:curse of dimensionality
1237:Arithmetica Infinitorum
638:Variation of parameters
628:Separation of variables
417:Peano existence theorem
5489:Carl David Tolmé Runge
5063:Differential-algebraic
5022:Differential equations
4799:from Wolfram Mathworld
4615:Stroud, A. H. (1971).
4467:Nonelementary Integral
4397:
4377:
4357:
4295:
4186:
4109:probabilistic numerics
4012:
3730:
3508:
3507:{\displaystyle f(x)=x}
3473:
3377:
3330:
3255:
3072:
3045:
2877:
2857:
2837:
2814:
2794:
2749:
2660:
2619:
2592:
2535:
2500:
2433:
2341:
2306:
2264:
2244:
2200:
2159:
2074:
1987:
1852:
1832:
1803:
1790:
1669:
1624:
1578:
1545:
1435:
1357:
1309:, usually stated as a
1281:With the invention of
1195:Method of indivisibles
1144:
1107:
1069:
1023:having the same area (
977:that is not available.
967:nonelementary integral
857:
798:(often abbreviated to
707:Carl David Tolmé Runge
250:Differential-algebraic
91:Differential equations
84:
77:
48:
5474:Augustin-Louis Cauchy
5469:Joseph-Louis Lagrange
5363:Finite element method
5353:CrankâNicolson method
5287:Numerical integration
5266:Exponential stability
5158:Relation to processes
5043:Differential operator
4955:BarnesâHut simulation
4863:NewtonâCotes formulas
4855:Numerical integration
4408:solving by quadrature
4398:
4378:
4358:
4296:
4197:initial value problem
4195:can be reduced to an
4187:
4013:
3731:
3509:
3474:
3378:
3331:
3256:
3073:
3046:
2878:
2858:
2838:
2815:
2795:
2750:
2661:
2620:
2593:
2536:
2501:
2449:Extrapolation methods
2434:
2349:NewtonâCotes formulas
2342:
2307:
2265:
2245:
2201:
2160:
2075:
1961:
1853:
1833:
1801:
1791:
1675:. This is called the
1670:
1625:
1576:
1546:
1441:. This is called the
1436:
1355:
1140:
1108:
1064:
1005:mathematical analysis
858:
784:numerical integration
697:Augustin-Louis Cauchy
682:Joseph-Louis Lagrange
514:Numerical integration
496:Exponential stability
359:Relation to processes
78:
49:
33:
5368:Finite volume method
5292:Dirac delta function
5261:Asymptotic stability
5203:Existence/uniqueness
5068:Integro-differential
4980:Tanh-sinh quadrature
4462:Tanh-sinh quadrature
4387:
4367:
4318:
4222:
4130:
4115:posterior variance.
3740:
3561:
3483:
3397:
3343:
3289:
3088:
3057:
2887:
2867:
2847:
2827:
2804:
2759:
2670:
2629:
2606:
2545:
2510:
2490:
2444:algorithms may fail.
2432:{\displaystyle f(x)}
2414:
2319:
2274:
2254:
2210:
2169:
2086:
1874:
1842:
1810:
1684:
1634:
1589:
1454:
1368:
1265:Solids of revolution
1205:found the area of a
1184:Method of exhaustion
1084:
816:
796:numerical quadrature
519:Dirac delta function
255:Integro-differential
76:{\displaystyle f(x)}
58:
38:
18:Numerical quadrature
5378:Perturbation theory
5358:RungeâKutta methods
5338:Integral transforms
5271:Rate of convergence
5167:(discrete analogue)
4960:Bayesian quadrature
4906:Gaussian quadrature
4550:2016Sci...351..482O
4309:RungeâKutta methods
4162:
4105:Bayesian quadrature
4100:Bayesian quadrature
4063:Monte Carlo methods
3941:
3900:
3802:
3761:
3624:
3581:
3531:interval arithmetic
3306:
3110:
2986:
2909:
2823:If we integrate in
2460:series acceleration
2401:Adaptive quadrature
2395:Adaptive quadrature
2388:Adaptive algorithms
2367:Gaussian quadrature
1891:
1701:
1471:
1153:Lune of Hippocrates
1033:Lune of Hippocrates
1011:, according to the
938:elementary function
902:Motivation and need
833:
615:Perturbation theory
610:Integral transforms
501:Rate of convergence
367:(discrete analogue)
204:Population dynamics
171:Continuum mechanics
162:Applied mathematics
5520:Numerical analysis
5499:Sofya Kovalevskaya
5333:Integrating factor
5256:Lyapunov stability
5176:Stochastic partial
4975:Lebedev quadrature
4881:Simpson's 3/8 rule
4797:Lobatto Quadrature
4772:, Saunders, 1990,
4742:, 2nd ed. rev. by
4673:George E. Forsythe
4594:. Addison Wesley.
4588:Leader, Jeffery J.
4516:jeff560.tripod.com
4488:Weisstein, Eric W.
4393:
4373:
4353:
4291:
4182:
4148:
4033:grow exponentially
4008:
4006:
3927:
3883:
3788:
3747:
3726:
3604:
3564:
3504:
3469:
3441:
3373:
3326:
3292:
3251:
3220:
3096:
3078:by an upper bound
3071:{\displaystyle f'}
3068:
3041:
2972:
2895:
2873:
2853:
2833:
2810:
2790:
2745:
2656:
2618:{\displaystyle f,}
2615:
2600:mean value theorem
2588:
2531:
2496:
2429:
2337:
2302:
2260:
2240:
2196:
2155:
2070:
1877:
1848:
1828:
1804:
1786:
1687:
1665:
1620:
1579:
1541:
1457:
1431:
1358:
1319:integration points
1289:" is more common.
1261:Christiaan Huygens
1203:Gilles de Roberval
1145:
1103:
1070:
853:
819:
605:Integrating factor
446:Initial conditions
381:Stochastic partial
85:
73:
44:
5507:
5506:
5386:
5385:
5191:
5190:
4988:
4987:
4712:978-0-521-88068-8
4663:Philip Rabinowitz
4601:978-0-201-73499-7
4544:(6272): 482â484.
4396:{\displaystyle F}
4376:{\displaystyle x}
4252:
3999:
3972:
3874:
3833:
3714:
3655:
3420:
3418:
3276:
3275:
3199:
3197:
2876:{\displaystyle b}
2856:{\displaystyle a}
2836:{\displaystyle x}
2813:{\displaystyle x}
2534:{\displaystyle ,}
2499:{\displaystyle f}
2472:rational function
2263:{\displaystyle h}
2194:
2060:
2026:
1956:
1930:
1851:{\displaystyle n}
1838:into some number
1777:
1532:
1420:
1392:
1347:bounded variation
1303:definite integral
1283:integral calculus
1276:natural logarithm
1241:definite integral
1101:
776:
775:
667:Gottfried Leibniz
558:Finite difference
350:
349:
211:
210:
181:Dynamical systems
47:{\displaystyle S}
16:(Redirected from
5532:
5484:Phyllis Nicolson
5464:Rudolf Lipschitz
5301:Solution methods
5276:Series solutions
5200:
5033:
5015:
5008:
5001:
4992:
4970:Filon quadrature
4896:Romberg's method
4871:Trapezoidal rule
4848:
4841:
4834:
4825:
4811:within the free
4731:(See Chapter 3.)
4715:
4685:(See Chapter 5.)
4651:
4641:
4635:
4634:
4622:
4612:
4606:
4605:
4584:
4578:
4577:
4533:
4527:
4526:
4524:
4522:
4508:
4502:
4501:
4500:
4483:
4457:Romberg's method
4452:Trapezoidal rule
4447:Riemann Integral
4402:
4400:
4399:
4394:
4382:
4380:
4379:
4374:
4362:
4360:
4359:
4354:
4328:
4300:
4298:
4297:
4292:
4253:
4251:
4243:
4226:
4191:
4189:
4188:
4183:
4161:
4156:
4113:Gaussian process
4029:Fubini's theorem
4017:
4015:
4014:
4009:
4007:
4000:
3998:
3997:
3988:
3980:
3978:
3974:
3973:
3968:
3957:
3940:
3935:
3899:
3894:
3875:
3873:
3872:
3871:
3849:
3841:
3839:
3835:
3834:
3832:
3818:
3801:
3796:
3760:
3755:
3735:
3733:
3732:
3727:
3715:
3713:
3712:
3707:
3703:
3702:
3701:
3680:
3679:
3678:
3662:
3660:
3656:
3654:
3653:
3652:
3633:
3623:
3615:
3580:
3575:
3513:
3511:
3510:
3505:
3478:
3476:
3475:
3470:
3468:
3464:
3454:
3440:
3419:
3414:
3413:
3401:
3382:
3380:
3379:
3374:
3335:
3333:
3332:
3327:
3305:
3300:
3270:
3260:
3258:
3257:
3252:
3247:
3243:
3233:
3219:
3198:
3193:
3192:
3191:
3169:
3164:
3160:
3109:
3104:
3081:
3077:
3075:
3074:
3069:
3067:
3050:
3048:
3047:
3042:
3037:
3033:
3022:
3021:
3009:
2985:
2980:
2963:
2959:
2908:
2903:
2882:
2880:
2879:
2874:
2862:
2860:
2859:
2854:
2842:
2840:
2839:
2834:
2819:
2817:
2816:
2811:
2799:
2797:
2796:
2791:
2771:
2770:
2754:
2752:
2751:
2746:
2708:
2707:
2695:
2665:
2663:
2662:
2657:
2624:
2622:
2621:
2616:
2597:
2595:
2594:
2589:
2563:
2562:
2540:
2538:
2537:
2532:
2505:
2503:
2502:
2497:
2462:methods such as
2438:
2436:
2435:
2430:
2346:
2344:
2343:
2340:{\displaystyle }
2338:
2311:
2309:
2308:
2303:
2301:
2300:
2295:
2291:
2290:
2269:
2267:
2266:
2261:
2249:
2247:
2246:
2241:
2205:
2203:
2202:
2197:
2195:
2190:
2179:
2164:
2162:
2161:
2156:
2079:
2077:
2076:
2071:
2066:
2062:
2061:
2056:
2042:
2037:
2033:
2032:
2028:
2027:
2022:
2011:
1986:
1975:
1957:
1952:
1938:
1931:
1926:
1915:
1890:
1885:
1857:
1855:
1854:
1849:
1837:
1835:
1834:
1831:{\displaystyle }
1829:
1795:
1793:
1792:
1787:
1782:
1778:
1773:
1744:
1700:
1695:
1678:trapezoidal rule
1674:
1672:
1671:
1666:
1664:
1660:
1629:
1627:
1626:
1621:
1619:
1615:
1568:oscillate wildly
1550:
1548:
1547:
1542:
1537:
1533:
1528:
1517:
1470:
1465:
1440:
1438:
1437:
1432:
1430:
1426:
1425:
1421:
1416:
1405:
1393:
1388:
1377:
1253:algebraic curves
1219:Opus Geometricum
1157:parabola segment
1112:
1110:
1109:
1104:
1102:
1094:
1048:trapezoidal rule
975:special function
954:
931:embedded systems
924:
876:
862:
860:
859:
854:
832:
827:
768:
761:
754:
732:Phyllis Nicolson
717:Rudolf Lipschitz
600:Green's function
576:Infinite element
567:
532:Solution methods
510:
368:
279:By variable type
233:
115:Natural sciences
108:
87:
82:
80:
79:
74:
53:
51:
50:
45:
21:
5540:
5539:
5535:
5534:
5533:
5531:
5530:
5529:
5510:
5509:
5508:
5503:
5444:Jacob Bernoulli
5417:
5382:
5373:Galerkin method
5296:
5234:Solution topics
5229:
5187:
5153:
5092:
5024:
5019:
4989:
4984:
4943:
4900:
4857:
4852:
4787:
4744:Uta C. Merzbach
4723:Roland Bulirsch
4713:
4695:Teukolsky, S.A.
4689:
4659:Philip J. Davis
4655:
4654:
4642:
4638:
4631:
4614:
4613:
4609:
4602:
4586:
4585:
4581:
4535:
4534:
4530:
4520:
4518:
4510:
4509:
4505:
4486:
4485:
4484:
4480:
4475:
4424:
4385:
4384:
4365:
4364:
4321:
4316:
4315:
4244:
4227:
4220:
4219:
4128:
4127:
4121:
4102:
4090:
4060:
4054:
4024:
4005:
4004:
3989:
3981:
3958:
3949:
3945:
3920:
3880:
3879:
3863:
3850:
3842:
3822:
3810:
3806:
3781:
3738:
3737:
3693:
3686:
3682:
3681:
3670:
3663:
3644:
3637:
3628:
3559:
3558:
3547:
3535:computer proofs
3526:are available.
3481:
3480:
3447:
3446:
3442:
3402:
3395:
3394:
3341:
3340:
3338:quadrature rule
3287:
3286:
3268:
3226:
3225:
3221:
3183:
3170:
3095:
3091:
3086:
3085:
3060:
3055:
3054:
3013:
3002:
2971:
2967:
2894:
2890:
2885:
2884:
2865:
2864:
2845:
2844:
2825:
2824:
2802:
2801:
2762:
2757:
2756:
2699:
2688:
2668:
2667:
2627:
2626:
2604:
2603:
2554:
2543:
2542:
2508:
2507:
2488:
2487:
2484:
2451:
2446:
2445:
2412:
2411:
2398:
2390:
2317:
2316:
2282:
2278:
2277:
2272:
2271:
2252:
2251:
2208:
2207:
2180:
2167:
2166:
2084:
2083:
2043:
2012:
2000:
1996:
1992:
1988:
1939:
1936:
1932:
1916:
1872:
1871:
1840:
1839:
1808:
1807:
1745:
1739:
1682:
1681:
1641:
1637:
1632:
1631:
1596:
1592:
1587:
1586:
1583:affine function
1556:
1518:
1512:
1452:
1451:
1406:
1400:
1378:
1375:
1371:
1366:
1365:
1336:
1328:round-off error
1299:quadrature rule
1295:
1199:Galileo Galilei
1171:of this sphere.
1143:
1082:
1081:
990:
984:
969:
961:elementary form
941:
915:
904:
867:
814:
813:
772:
743:
742:
741:
672:Jacob Bernoulli
656:
643:
642:
624:
593:PetrovâGalerkin
561:
546:
533:
525:
524:
523:
505:
451:Boundary values
440:
432:
431:
407:
394:
393:
392:
366:
360:
352:
351:
339:
316:
274:
230:
217:
216:
212:
190:Social sciences
146:
124:
105:
56:
55:
36:
35:
28:
23:
22:
15:
12:
11:
5:
5538:
5536:
5528:
5527:
5522:
5512:
5511:
5505:
5504:
5502:
5501:
5496:
5491:
5486:
5481:
5476:
5471:
5466:
5461:
5459:Ernst Lindelöf
5456:
5451:
5446:
5441:
5439:Leonhard Euler
5436:
5431:
5425:
5423:
5422:Mathematicians
5419:
5418:
5416:
5415:
5410:
5405:
5400:
5394:
5392:
5388:
5387:
5384:
5383:
5381:
5380:
5375:
5370:
5365:
5360:
5355:
5350:
5345:
5340:
5335:
5330:
5325:
5320:
5315:
5310:
5304:
5302:
5298:
5297:
5295:
5294:
5289:
5284:
5278:
5273:
5268:
5263:
5258:
5253:
5248:
5246:Phase portrait
5243:
5237:
5235:
5231:
5230:
5228:
5227:
5222:
5217:
5212:
5206:
5204:
5197:
5193:
5192:
5189:
5188:
5186:
5185:
5180:
5179:
5178:
5168:
5161:
5159:
5155:
5154:
5152:
5151:
5149:On jet bundles
5146:
5141:
5136:
5131:
5126:
5121:
5116:
5114:Nonhomogeneous
5111:
5106:
5100:
5098:
5094:
5093:
5091:
5090:
5085:
5080:
5075:
5070:
5065:
5060:
5055:
5050:
5045:
5039:
5037:
5030:
5029:Classification
5026:
5025:
5020:
5018:
5017:
5010:
5003:
4995:
4986:
4985:
4983:
4982:
4977:
4972:
4967:
4962:
4957:
4951:
4949:
4945:
4944:
4942:
4941:
4936:
4931:
4926:
4921:
4916:
4910:
4908:
4902:
4901:
4899:
4898:
4893:
4888:
4883:
4878:
4876:Simpson's rule
4873:
4867:
4865:
4859:
4858:
4853:
4851:
4850:
4843:
4836:
4828:
4822:
4821:
4816:
4806:
4800:
4794:
4786:
4785:External links
4783:
4782:
4781:
4763:
4754:(1991 pbk ed.
4733:
4716:
4711:
4687:
4677:Cleve B. Moler
4670:
4653:
4652:
4636:
4629:
4607:
4600:
4579:
4528:
4503:
4477:
4476:
4474:
4471:
4470:
4469:
4464:
4459:
4454:
4449:
4440:
4435:
4430:
4423:
4420:
4392:
4372:
4352:
4349:
4346:
4343:
4340:
4337:
4334:
4331:
4327:
4324:
4302:
4301:
4290:
4287:
4284:
4281:
4278:
4275:
4271:
4268:
4265:
4262:
4259:
4256:
4250:
4247:
4242:
4239:
4236:
4233:
4230:
4193:
4192:
4181:
4178:
4174:
4171:
4168:
4165:
4160:
4155:
4151:
4147:
4144:
4141:
4138:
4135:
4120:
4117:
4101:
4098:
4089:
4086:
4082:Gibbs sampling
4056:Main article:
4053:
4050:
4023:
4020:
4003:
3996:
3992:
3987:
3984:
3977:
3971:
3967:
3964:
3961:
3955:
3952:
3948:
3944:
3939:
3934:
3930:
3926:
3923:
3921:
3919:
3916:
3912:
3909:
3906:
3903:
3898:
3893:
3890:
3886:
3882:
3881:
3878:
3870:
3866:
3862:
3859:
3856:
3853:
3848:
3845:
3838:
3831:
3828:
3825:
3821:
3816:
3813:
3809:
3805:
3800:
3795:
3791:
3787:
3784:
3782:
3780:
3777:
3773:
3770:
3767:
3764:
3759:
3754:
3750:
3746:
3745:
3725:
3722:
3719:
3711:
3706:
3700:
3696:
3692:
3689:
3685:
3677:
3673:
3669:
3666:
3659:
3651:
3647:
3643:
3640:
3636:
3631:
3627:
3622:
3619:
3614:
3611:
3607:
3603:
3600:
3597:
3593:
3590:
3587:
3584:
3579:
3574:
3571:
3567:
3546:
3543:
3541:calculations.
3503:
3500:
3497:
3494:
3491:
3488:
3467:
3463:
3460:
3457:
3453:
3450:
3445:
3439:
3436:
3433:
3430:
3427:
3423:
3417:
3412:
3409:
3405:
3372:
3369:
3366:
3363:
3360:
3357:
3354:
3351:
3348:
3325:
3322:
3318:
3315:
3312:
3309:
3304:
3299:
3295:
3274:
3273:
3264:
3262:
3250:
3246:
3242:
3239:
3236:
3232:
3229:
3224:
3218:
3215:
3212:
3209:
3206:
3202:
3196:
3190:
3186:
3182:
3179:
3176:
3173:
3167:
3163:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3122:
3119:
3116:
3113:
3108:
3103:
3099:
3094:
3066:
3063:
3040:
3036:
3032:
3029:
3025:
3020:
3016:
3012:
3008:
3005:
3001:
2998:
2995:
2992:
2989:
2984:
2979:
2975:
2970:
2966:
2962:
2958:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2928:
2925:
2921:
2918:
2915:
2912:
2907:
2902:
2898:
2893:
2872:
2852:
2832:
2809:
2789:
2786:
2783:
2780:
2777:
2774:
2769:
2765:
2744:
2741:
2738:
2735:
2732:
2729:
2726:
2723:
2720:
2717:
2714:
2711:
2706:
2702:
2698:
2694:
2691:
2687:
2684:
2681:
2678:
2675:
2655:
2652:
2649:
2646:
2643:
2640:
2637:
2634:
2614:
2611:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2561:
2557:
2553:
2550:
2530:
2527:
2524:
2521:
2518:
2515:
2495:
2483:
2480:
2450:
2447:
2428:
2425:
2422:
2419:
2399:
2391:
2389:
2386:
2353:Simpson's rule
2336:
2333:
2330:
2327:
2324:
2299:
2294:
2289:
2285:
2281:
2259:
2239:
2236:
2233:
2230:
2227:
2224:
2221:
2218:
2215:
2193:
2189:
2186:
2183:
2177:
2174:
2154:
2151:
2148:
2145:
2142:
2139:
2136:
2133:
2130:
2127:
2124:
2121:
2118:
2115:
2112:
2109:
2106:
2103:
2100:
2097:
2094:
2091:
2069:
2065:
2059:
2055:
2052:
2049:
2046:
2040:
2036:
2031:
2025:
2021:
2018:
2015:
2009:
2006:
2003:
1999:
1995:
1991:
1985:
1982:
1979:
1974:
1971:
1968:
1964:
1960:
1955:
1951:
1948:
1945:
1942:
1935:
1929:
1925:
1922:
1919:
1913:
1910:
1907:
1903:
1900:
1897:
1894:
1889:
1884:
1880:
1860:composite rule
1847:
1827:
1824:
1821:
1818:
1815:
1785:
1781:
1776:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1742:
1738:
1735:
1732:
1729:
1726:
1723:
1720:
1717:
1713:
1710:
1707:
1704:
1699:
1694:
1690:
1663:
1659:
1656:
1653:
1650:
1647:
1644:
1640:
1618:
1614:
1611:
1608:
1605:
1602:
1599:
1595:
1555:
1552:
1540:
1536:
1531:
1527:
1524:
1521:
1515:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1487:
1483:
1480:
1477:
1474:
1469:
1464:
1460:
1448:rectangle rule
1429:
1424:
1419:
1415:
1412:
1409:
1403:
1399:
1396:
1391:
1387:
1384:
1381:
1374:
1335:
1332:
1294:
1291:
1180:
1179:
1172:
1141:
1115:Geometric mean
1100:
1097:
1092:
1089:
1067:Geometric mean
986:Main article:
983:
980:
979:
978:
970:
957:error function
934:
914:The integrand
908:antiderivative
903:
900:
864:
863:
852:
849:
845:
842:
839:
836:
831:
826:
822:
774:
773:
771:
770:
763:
756:
748:
745:
744:
740:
739:
734:
729:
724:
722:Ernst Lindelöf
719:
714:
709:
704:
699:
694:
692:Joseph Fourier
689:
684:
679:
677:Leonhard Euler
674:
669:
664:
658:
657:
654:
653:
650:
649:
645:
644:
641:
640:
635:
630:
623:
622:
617:
612:
607:
602:
597:
596:
595:
585:
580:
579:
578:
571:Finite element
568:
564:CrankâNicolson
555:
550:
544:
539:
535:
534:
531:
530:
527:
526:
522:
521:
516:
511:
503:
498:
485:
483:Phase portrait
480:
475:
474:
473:
471:Cauchy problem
468:
463:
458:
448:
442:
441:
439:General topics
438:
437:
434:
433:
430:
429:
424:
419:
414:
408:
405:
404:
401:
400:
396:
395:
391:
390:
385:
384:
383:
372:
371:
370:
361:
358:
357:
354:
353:
348:
347:
346:
345:
338:
337:
332:
326:
323:
322:
318:
317:
315:
314:
312:Nonhomogeneous
305:
300:
297:
291:
290:
289:
281:
280:
276:
275:
273:
272:
267:
262:
257:
252:
247:
242:
236:
231:
228:
227:
224:
223:
222:Classification
219:
218:
209:
208:
207:
206:
201:
193:
192:
186:
185:
184:
183:
178:
173:
165:
164:
158:
157:
156:
155:
150:
144:
139:
134:
126:
125:
123:
122:
117:
111:
106:
103:
102:
99:
98:
94:
93:
72:
69:
66:
63:
43:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5537:
5526:
5523:
5521:
5518:
5517:
5515:
5500:
5497:
5495:
5492:
5490:
5487:
5485:
5482:
5480:
5477:
5475:
5472:
5470:
5467:
5465:
5462:
5460:
5457:
5455:
5452:
5450:
5447:
5445:
5442:
5440:
5437:
5435:
5432:
5430:
5427:
5426:
5424:
5420:
5414:
5411:
5409:
5406:
5404:
5401:
5399:
5396:
5395:
5393:
5389:
5379:
5376:
5374:
5371:
5369:
5366:
5364:
5361:
5359:
5356:
5354:
5351:
5349:
5346:
5344:
5341:
5339:
5336:
5334:
5331:
5329:
5326:
5324:
5321:
5319:
5316:
5314:
5311:
5309:
5306:
5305:
5303:
5299:
5293:
5290:
5288:
5285:
5282:
5279:
5277:
5274:
5272:
5269:
5267:
5264:
5262:
5259:
5257:
5254:
5252:
5249:
5247:
5244:
5242:
5239:
5238:
5236:
5232:
5226:
5223:
5221:
5218:
5216:
5213:
5211:
5208:
5207:
5205:
5201:
5198:
5194:
5184:
5181:
5177:
5174:
5173:
5172:
5169:
5166:
5163:
5162:
5160:
5156:
5150:
5147:
5145:
5142:
5140:
5137:
5135:
5132:
5130:
5127:
5125:
5122:
5120:
5117:
5115:
5112:
5110:
5107:
5105:
5102:
5101:
5099:
5095:
5089:
5086:
5084:
5081:
5079:
5076:
5074:
5071:
5069:
5066:
5064:
5061:
5059:
5056:
5054:
5051:
5049:
5046:
5044:
5041:
5040:
5038:
5034:
5031:
5027:
5023:
5016:
5011:
5009:
5004:
5002:
4997:
4996:
4993:
4981:
4978:
4976:
4973:
4971:
4968:
4966:
4963:
4961:
4958:
4956:
4953:
4952:
4950:
4946:
4940:
4937:
4935:
4932:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4911:
4909:
4907:
4903:
4897:
4894:
4892:
4889:
4887:
4884:
4882:
4879:
4877:
4874:
4872:
4869:
4868:
4866:
4864:
4860:
4856:
4849:
4844:
4842:
4837:
4835:
4830:
4829:
4826:
4820:
4817:
4814:
4810:
4807:
4804:
4801:
4798:
4795:
4792:
4789:
4788:
4784:
4779:
4778:0-03-029558-0
4775:
4771:
4767:
4764:
4761:
4760:0-471-54397-7
4757:
4753:
4752:0-471-09763-2
4749:
4745:
4741:
4737:
4734:
4732:
4728:
4724:
4720:
4717:
4714:
4708:
4704:
4700:
4696:
4692:
4688:
4686:
4682:
4678:
4674:
4671:
4668:
4664:
4660:
4657:
4656:
4650:
4646:
4640:
4637:
4632:
4630:9780130438935
4626:
4621:
4620:
4611:
4608:
4603:
4597:
4593:
4589:
4583:
4580:
4575:
4571:
4567:
4563:
4559:
4555:
4551:
4547:
4543:
4539:
4532:
4529:
4517:
4513:
4507:
4504:
4498:
4497:
4492:
4489:
4482:
4479:
4472:
4468:
4465:
4463:
4460:
4458:
4455:
4453:
4450:
4448:
4444:
4441:
4439:
4436:
4434:
4431:
4429:
4426:
4425:
4421:
4419:
4417:
4413:
4409:
4404:
4390:
4370:
4347:
4341:
4338:
4332:
4325:
4322:
4312:
4310:
4306:
4288:
4285:
4279:
4273:
4269:
4263:
4257:
4254:
4248:
4245:
4237:
4231:
4228:
4218:
4217:
4216:
4214:
4210:
4206:
4202:
4198:
4179:
4176:
4169:
4163:
4158:
4153:
4149:
4145:
4139:
4133:
4126:
4125:
4124:
4118:
4116:
4114:
4110:
4106:
4099:
4097:
4094:
4087:
4085:
4083:
4079:
4075:
4070:
4068:
4064:
4059:
4051:
4049:
4047:
4042:
4040:
4039:
4034:
4030:
4021:
4019:
4001:
3994:
3990:
3985:
3982:
3975:
3969:
3965:
3962:
3959:
3953:
3950:
3946:
3942:
3937:
3932:
3928:
3924:
3922:
3917:
3914:
3907:
3901:
3896:
3888:
3884:
3876:
3868:
3860:
3857:
3854:
3846:
3843:
3836:
3829:
3826:
3823:
3819:
3814:
3811:
3807:
3803:
3798:
3793:
3789:
3785:
3783:
3778:
3775:
3768:
3762:
3752:
3748:
3723:
3720:
3717:
3709:
3704:
3698:
3694:
3690:
3687:
3683:
3675:
3671:
3667:
3664:
3657:
3649:
3645:
3641:
3638:
3634:
3629:
3625:
3620:
3617:
3612:
3609:
3605:
3601:
3598:
3595:
3588:
3582:
3569:
3565:
3556:
3552:
3544:
3542:
3540:
3536:
3532:
3527:
3525:
3521:
3517:
3516:Taylor series
3501:
3498:
3492:
3486:
3465:
3458:
3451:
3448:
3443:
3437:
3434:
3431:
3428:
3425:
3415:
3410:
3407:
3403:
3392:
3388:
3387:
3367:
3361:
3355:
3352:
3349:
3339:
3323:
3320:
3313:
3307:
3302:
3297:
3293:
3283:
3281:
3272:
3265:
3263:
3261:
3248:
3244:
3237:
3230:
3227:
3222:
3216:
3213:
3210:
3207:
3204:
3194:
3188:
3180:
3177:
3174:
3165:
3161:
3154:
3148:
3142:
3139:
3136:
3130:
3127:
3124:
3117:
3111:
3106:
3101:
3097:
3092:
3083:
3082:
3079:
3064:
3061:
3051:
3038:
3034:
3030:
3027:
3018:
3014:
3006:
3003:
2996:
2993:
2990:
2982:
2977:
2973:
2968:
2964:
2960:
2953:
2947:
2941:
2938:
2935:
2929:
2926:
2923:
2916:
2910:
2905:
2900:
2896:
2891:
2870:
2850:
2830:
2821:
2807:
2800:depending on
2784:
2781:
2778:
2772:
2767:
2763:
2742:
2736:
2730:
2727:
2721:
2715:
2712:
2704:
2700:
2692:
2689:
2682:
2679:
2676:
2653:
2647:
2644:
2641:
2635:
2632:
2612:
2609:
2601:
2585:
2576:
2573:
2570:
2559:
2555:
2551:
2548:
2528:
2522:
2519:
2516:
2493:
2481:
2479:
2477:
2473:
2469:
2465:
2461:
2456:
2448:
2442:
2423:
2417:
2410:
2406:
2402:
2396:
2387:
2385:
2383:
2378:
2376:
2372:
2368:
2363:
2361:
2356:
2354:
2350:
2331:
2328:
2325:
2313:
2297:
2292:
2287:
2283:
2279:
2257:
2237:
2234:
2231:
2228:
2225:
2222:
2219:
2216:
2213:
2191:
2187:
2184:
2181:
2175:
2172:
2152:
2146:
2143:
2140:
2134:
2128:
2122:
2119:
2116:
2110:
2107:
2104:
2101:
2098:
2095:
2092:
2080:
2067:
2063:
2057:
2050:
2044:
2038:
2034:
2029:
2023:
2019:
2016:
2013:
2007:
2004:
2001:
1997:
1993:
1989:
1983:
1980:
1977:
1972:
1969:
1966:
1962:
1958:
1953:
1946:
1940:
1933:
1927:
1923:
1920:
1917:
1911:
1908:
1905:
1898:
1892:
1887:
1882:
1878:
1869:
1868:iterated rule
1865:
1864:extended rule
1861:
1845:
1822:
1819:
1816:
1800:
1796:
1783:
1779:
1774:
1767:
1761:
1758:
1752:
1746:
1740:
1733:
1730:
1727:
1721:
1718:
1715:
1708:
1702:
1697:
1692:
1688:
1680:
1679:
1661:
1654:
1648:
1645:
1642:
1638:
1616:
1609:
1603:
1600:
1597:
1593:
1584:
1575:
1571:
1569:
1565:
1561:
1560:interpolating
1553:
1551:
1538:
1534:
1529:
1525:
1522:
1519:
1513:
1509:
1503:
1500:
1497:
1491:
1488:
1485:
1478:
1472:
1467:
1462:
1458:
1450:
1449:
1444:
1443:midpoint rule
1427:
1422:
1417:
1413:
1410:
1407:
1401:
1397:
1394:
1389:
1385:
1382:
1379:
1372:
1363:
1362:step function
1354:
1350:
1348:
1344:
1341:
1333:
1331:
1329:
1323:
1321:
1320:
1314:
1312:
1308:
1304:
1300:
1292:
1290:
1288:
1284:
1279:
1277:
1273:
1268:
1266:
1262:
1258:
1254:
1250:
1249:James Gregory
1246:
1242:
1238:
1234:
1230:
1228:
1224:
1221:, 1647), and
1220:
1216:
1212:
1208:
1204:
1200:
1196:
1191:
1189:
1185:
1177:
1173:
1170:
1166:
1165:
1164:
1162:
1158:
1154:
1150:
1139:
1135:
1132:
1128:
1124:
1120:
1116:
1098:
1095:
1090:
1087:
1079:
1075:
1068:
1063:
1059:
1057:
1053:
1049:
1044:
1042:
1038:
1034:
1030:
1026:
1022:
1018:
1014:
1010:
1006:
1001:
999:
995:
989:
981:
976:
971:
968:
962:
958:
952:
948:
944:
939:
935:
932:
928:
922:
918:
913:
912:
911:
909:
901:
899:
897:
893:
890:), as in the
889:
885:
884:
878:
874:
870:
850:
847:
840:
834:
829:
824:
820:
812:
811:
810:
807:
805:
801:
797:
793:
789:
785:
781:
769:
764:
762:
757:
755:
750:
749:
747:
746:
738:
735:
733:
730:
728:
725:
723:
720:
718:
715:
713:
710:
708:
705:
703:
700:
698:
695:
693:
690:
688:
685:
683:
680:
678:
675:
673:
670:
668:
665:
663:
660:
659:
652:
651:
646:
639:
636:
634:
631:
629:
626:
625:
621:
618:
616:
613:
611:
608:
606:
603:
601:
598:
594:
591:
590:
589:
586:
584:
583:Finite volume
581:
577:
574:
573:
572:
569:
565:
559:
556:
554:
551:
549:
545:
543:
540:
537:
536:
529:
528:
520:
517:
515:
512:
508:
504:
502:
499:
497:
493:
489:
486:
484:
481:
479:
476:
472:
469:
467:
464:
462:
459:
457:
454:
453:
452:
449:
447:
444:
443:
436:
435:
428:
425:
423:
420:
418:
415:
413:
410:
409:
403:
402:
397:
389:
386:
382:
379:
378:
377:
374:
373:
369:
363:
362:
356:
355:
344:
341:
340:
336:
333:
331:
328:
327:
325:
324:
319:
313:
309:
306:
304:
301:
298:
296:
293:
292:
288:
285:
284:
283:
282:
277:
271:
268:
266:
263:
261:
258:
256:
253:
251:
248:
246:
243:
241:
238:
237:
235:
234:
226:
225:
220:
215:
205:
202:
200:
197:
196:
195:
194:
191:
187:
182:
179:
177:
174:
172:
169:
168:
167:
166:
163:
159:
154:
151:
149:
145:
143:
140:
138:
135:
133:
130:
129:
128:
127:
121:
118:
116:
113:
112:
109:
101:
100:
95:
92:
88:
67:
61:
41:
32:
19:
5494:Martin Kutta
5449:Ămile Picard
5429:Isaac Newton
5343:Euler method
5313:Substitution
5286:
4891:Boole's rule
4854:
4769:
4766:Eves, Howard
4739:
4736:Boyer, C. B.
4730:
4726:
4702:
4684:
4680:
4666:
4639:
4618:
4610:
4591:
4582:
4541:
4537:
4531:
4519:. Retrieved
4515:
4506:
4494:
4481:
4405:
4313:
4303:
4212:
4208:
4194:
4122:
4103:
4093:Sparse grids
4091:
4088:Sparse grids
4071:
4061:
4043:
4036:
4025:
3548:
3538:
3528:
3523:
3519:
3384:
3284:
3277:
3266:
3084:
3052:
2822:
2485:
2455:NewtonâCotes
2452:
2441:approximated
2379:
2364:
2359:
2357:
2314:
2081:
1867:
1863:
1859:
1805:
1676:
1580:
1557:
1446:
1442:
1359:
1337:
1324:
1318:
1317:
1315:
1311:weighted sum
1298:
1296:
1286:
1280:
1269:
1245:Isaac Barrow
1236:
1231:
1218:
1192:
1181:
1169:great circle
1146:
1130:
1126:
1122:
1118:
1077:
1073:
1071:
1045:
1024:
1002:
993:
991:
950:
946:
942:
920:
916:
905:
887:
881:
879:
872:
868:
865:
808:
803:
799:
795:
783:
777:
727:Ămile Picard
712:Martin Kutta
702:George Green
662:Isaac Newton
513:
494: /
490: /
310: /
176:Chaos theory
5251:Phase space
5109:Homogeneous
4719:Josef Stoer
4691:Press, W.H.
4443:Riemann Sum
4052:Monte Carlo
3533:to produce
3391:Riemann sum
2347:yields the
1564:polynomials
1233:John Wallis
1013:Pythagorean
794:. The term
620:RungeâKutta
365:Difference
308:Homogeneous
120:Engineering
5514:Categories
5479:John Crank
5308:Inspection
5171:Stochastic
5165:Difference
5139:Autonomous
5083:Non-linear
5073:Fractional
5036:Operations
4491:"Cubature"
4473:References
4307:, such as
4215:satisfies
3278:where the
2468:polynomial
1343:continuous
1227:logarithms
1161:Archimedes
1054:along the
998:David Gibb
965:See also:
883:quadrature
800:quadrature
788:algorithms
737:John Crank
538:Inspection
492:Asymptotic
376:Stochastic
295:Autonomous
270:Non-linear
260:Fractional
5283:solutions
5241:Wronskian
5196:Solutions
5124:Decoupled
5088:Holonomic
4574:206644971
4496:MathWorld
4416:integrals
4150:∫
3963:−
3954:−
3929:∫
3892:∞
3889:−
3885:∫
3858:−
3827:−
3790:∫
3758:∞
3749:∫
3691:−
3642:−
3610:−
3606:∫
3578:∞
3573:∞
3570:−
3566:∫
3435:≤
3429:≤
3408:−
3353:−
3294:∫
3214:≤
3208:≤
3178:−
3166:≤
3140:−
3131:−
3098:∫
3015:ξ
2994:−
2974:∫
2939:−
2930:−
2897:∫
2773:∈
2764:ξ
2755:for some
2728:−
2701:ξ
2680:−
2636:∈
2552:∈
2478:library.
2235:−
2226:…
2185:−
2135:⊂
2017:−
1981:−
1963:∑
1921:−
1912:≈
1879:∫
1731:−
1722:≈
1689:∫
1501:−
1492:≈
1459:∫
1340:piecewise
1215:hyperbola
949:) = exp(â
821:∫
478:Wronskian
456:Dirichlet
199:Economics
142:Chemistry
132:Astronomy
5391:Examples
5281:Integral
5053:Ordinary
4590:(2004).
4566:26823423
4521:31 March
4422:See also
4326:′
3539:verified
3452:′
3280:supremum
3231:′
3065:′
3007:′
2693:′
2476:QUADPACK
2409:function
2405:integral
1307:function
1272:function
1176:parabola
1159:done by
1056:ecliptic
1025:squaring
927:sampling
888:squaring
804:cubature
792:integral
780:analysis
588:Galerkin
488:Lyapunov
399:Solution
343:Notation
335:Operator
321:Features
240:Ordinary
5119:Coupled
5058:Partial
4546:Bibcode
4538:Science
4199:for an
3336:by the
2360:nesting
1345:and of
1257:spirals
1207:cycloid
1188:Eudoxus
1052:Jupiter
982:History
929:. Some
461:Neumann
245:Partial
153:Geology
148:Biology
137:Physics
5134:Degree
5078:Linear
4776:
4758:
4750:
4709:
4627:
4598:
4572:
4564:
4410:" or "
4046:sphere
2666:gives
2625:where
2371:smooth
1274:, the
1209:arch,
1021:square
648:People
560:
507:Series
265:Linear
104:Fields
5183:Delay
5129:Order
4948:Other
4570:S2CID
2843:from
2541:i.e.
2407:of a
2165:with
1866:, or
1305:of a
1113:(the
548:Euler
466:Robin
388:Delay
330:Order
303:Exact
229:Types
97:Scope
4774:ISBN
4756:ISBN
4748:ISBN
4721:and
4707:ISBN
4661:and
4625:ISBN
4596:ISBN
4562:PMID
4523:2018
4080:and
4065:and
3537:and
2602:for
2598:The
2486:Let
2384:do.
2206:and
1630:and
1255:and
1247:and
1201:and
1129:and
1121:and
1076:and
1017:area
655:List
4645:doi
4554:doi
4542:351
4445:or
3422:sup
3201:sup
2863:to
2470:or
2439:is
1445:or
1186:of
1117:of
996:by
886:or
778:In
5516::
4768:,
4762:).
4738:,
4725:,
4701:,
4693:;
4679:,
4665:,
4568:.
4560:.
4552:.
4540:.
4514:.
4493:.
4418:.
4289:0.
4084:.
4041:.
2820:.
2312:.
2238:1.
1862:,
1297:A
1267:.
1259:.
1229:.
1190:.
1058:.
1043:.
1035:,
1031:,
1007:.
1000:.
963:.
910::
898:.
782:,
5014:e
5007:t
5000:v
4847:e
4840:t
4833:v
4815:.
4780:,
4669:.
4647::
4633:.
4604:.
4576:.
4556::
4548::
4525:.
4499:.
4391:F
4371:x
4351:)
4348:x
4345:(
4342:f
4339:=
4336:)
4333:x
4330:(
4323:F
4286:=
4283:)
4280:a
4277:(
4274:F
4270:,
4267:)
4264:x
4261:(
4258:f
4255:=
4249:x
4246:d
4241:)
4238:x
4235:(
4232:F
4229:d
4213:F
4209:x
4180:u
4177:d
4173:)
4170:u
4167:(
4164:f
4159:x
4154:a
4146:=
4143:)
4140:x
4137:(
4134:F
4002:,
3995:2
3991:t
3986:t
3983:d
3976:)
3970:t
3966:t
3960:1
3951:a
3947:(
3943:f
3938:1
3933:0
3925:=
3918:x
3915:d
3911:)
3908:x
3905:(
3902:f
3897:a
3877:,
3869:2
3865:)
3861:t
3855:1
3852:(
3847:t
3844:d
3837:)
3830:t
3824:1
3820:t
3815:+
3812:a
3808:(
3804:f
3799:1
3794:0
3786:=
3779:x
3776:d
3772:)
3769:x
3766:(
3763:f
3753:a
3724:,
3721:t
3718:d
3710:2
3705:)
3699:2
3695:t
3688:1
3684:(
3676:2
3672:t
3668:+
3665:1
3658:)
3650:2
3646:t
3639:1
3635:t
3630:(
3626:f
3621:1
3618:+
3613:1
3602:=
3599:x
3596:d
3592:)
3589:x
3586:(
3583:f
3524:f
3520:f
3502:x
3499:=
3496:)
3493:x
3490:(
3487:f
3466:|
3462:)
3459:x
3456:(
3449:f
3444:|
3438:1
3432:x
3426:0
3416:2
3411:1
3404:n
3386:1
3371:)
3368:a
3365:(
3362:f
3359:)
3356:a
3350:b
3347:(
3324:x
3321:d
3317:)
3314:x
3311:(
3308:f
3303:b
3298:a
3271:)
3269:1
3267:(
3249:,
3245:|
3241:)
3238:x
3235:(
3228:f
3223:|
3217:b
3211:x
3205:a
3195:2
3189:2
3185:)
3181:a
3175:b
3172:(
3162:|
3158:)
3155:a
3152:(
3149:f
3146:)
3143:a
3137:b
3134:(
3128:x
3125:d
3121:)
3118:x
3115:(
3112:f
3107:b
3102:a
3093:|
3062:f
3039:.
3035:|
3031:x
3028:d
3024:)
3019:x
3011:(
3004:f
3000:)
2997:a
2991:x
2988:(
2983:b
2978:a
2969:|
2965:=
2961:|
2957:)
2954:a
2951:(
2948:f
2945:)
2942:a
2936:b
2933:(
2927:x
2924:d
2920:)
2917:x
2914:(
2911:f
2906:b
2901:a
2892:|
2871:b
2851:a
2831:x
2808:x
2788:]
2785:x
2782:,
2779:a
2776:(
2768:x
2743:,
2740:)
2737:a
2734:(
2731:f
2725:)
2722:x
2719:(
2716:f
2713:=
2710:)
2705:x
2697:(
2690:f
2686:)
2683:a
2677:x
2674:(
2654:,
2651:)
2648:b
2645:,
2642:a
2639:[
2633:x
2613:,
2610:f
2586:.
2583:)
2580:]
2577:b
2574:,
2571:a
2568:[
2565:(
2560:1
2556:C
2549:f
2529:,
2526:]
2523:b
2520:,
2517:a
2514:[
2494:f
2427:)
2424:x
2421:(
2418:f
2397:.
2335:]
2332:b
2329:,
2326:a
2323:[
2298:k
2293:)
2288:k
2284:h
2280:(
2258:h
2232:n
2229:,
2223:,
2220:0
2217:=
2214:k
2192:n
2188:a
2182:b
2176:=
2173:h
2153:,
2150:]
2147:b
2144:,
2141:a
2138:[
2132:]
2129:h
2126:)
2123:1
2120:+
2117:k
2114:(
2111:+
2108:a
2105:,
2102:h
2099:k
2096:+
2093:a
2090:[
2068:,
2064:)
2058:2
2054:)
2051:b
2048:(
2045:f
2039:+
2035:)
2030:)
2024:n
2020:a
2014:b
2008:k
2005:+
2002:a
1998:(
1994:f
1990:(
1984:1
1978:n
1973:1
1970:=
1967:k
1959:+
1954:2
1950:)
1947:a
1944:(
1941:f
1934:(
1928:n
1924:a
1918:b
1909:x
1906:d
1902:)
1899:x
1896:(
1893:f
1888:b
1883:a
1846:n
1826:]
1823:b
1820:,
1817:a
1814:[
1784:.
1780:)
1775:2
1771:)
1768:b
1765:(
1762:f
1759:+
1756:)
1753:a
1750:(
1747:f
1741:(
1737:)
1734:a
1728:b
1725:(
1719:x
1716:d
1712:)
1709:x
1706:(
1703:f
1698:b
1693:a
1662:)
1658:)
1655:b
1652:(
1649:f
1646:,
1643:b
1639:(
1617:)
1613:)
1610:a
1607:(
1604:f
1601:,
1598:a
1594:(
1539:.
1535:)
1530:2
1526:b
1523:+
1520:a
1514:(
1510:f
1507:)
1504:a
1498:b
1495:(
1489:x
1486:d
1482:)
1479:x
1476:(
1473:f
1468:b
1463:a
1428:)
1423:)
1418:2
1414:b
1411:+
1408:a
1402:(
1398:f
1395:,
1390:2
1386:b
1383:+
1380:a
1373:(
1217:(
1131:b
1127:a
1123:b
1119:a
1099:b
1096:a
1091:=
1088:x
1078:b
1074:a
953:)
951:x
947:x
945:(
943:f
923:)
921:x
919:(
917:f
875:)
873:x
871:(
869:f
851:x
848:d
844:)
841:x
838:(
835:f
830:b
825:a
767:e
760:t
753:v
566:)
562:(
83:.
71:)
68:x
65:(
62:f
42:S
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
