|
Adaptive Quadrature
Adaptive quadrature is a numerical integration method in which the integral of a function f(x) is approximated 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 algorithms may fail. General scheme Adaptive quadrature follows the general scheme 1. procedure integrate ( f, a, b, τ ) 2. Q \approx \int_a^b f(x)\,\mathrmx 3. \varepsilon \approx \left, Q - \int_a^b f(x)\,\mathrmx\ 4. if ''ε'' > ''τ'' then 5. m = (a + b) / 2 6. Q = integrate(f, a, m, τ/2) + integrate(f, m, b, τ/2) 7. endif 8. return Q An approximation Q to the integral of f(x) over the interval ,b/math> is computed (line 2), as well as an error estimate \varepsilon (line 3). If the estimated error is larger than the required toler ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. This article focuses on calculation of definite integrals. The term numerical quadrature (often abbreviated to ''quadrature'') 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 cubature; others take ''quadrature'' to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If 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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richardson Extrapolation
In numerical analysis, Richardson extrapolation is a sequence acceleration method used to improve the rate of convergence of a sequence of estimates of some value A^\ast = \lim_ A(h). In essence, given the value of A(h) for several values of h, we can estimate A^\ast by extrapolating the estimates to h=0. It is named after Lewis Fry Richardson, who introduced the technique in the early 20th century, though the idea was already known to Christiaan Huygens in his calculation of π. In the words of Birkhoff and Rota, "its usefulness for practical computations can hardly be overestimated."Page 126 of Practical applications of Richardson extrapolation include Romberg integration, which applies Richardson extrapolation to the trapezoid rule, and the Bulirsch–Stoer algorithm for solving ordinary differential equations. Example of Richardson extrapolation Suppose that we wish to approximate A^*, and we have a method A(h) that depends on a small parameter h in such a way that A( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New York City
New York, often called New York City or NYC, is the List of United States cities by population, most populous city in the United States. With a 2020 population of 8,804,190 distributed over , New York City is also the List of United States cities by population density, most densely populated major city in the United States, and is more than twice as populous as second-place Los Angeles. New York City lies at the southern tip of New York (state), New York State, and constitutes the geographical and demographic center of both the Northeast megalopolis and the New York metropolitan area, the largest metropolitan area in the world by urban area, urban landmass. With over 20.1 million people in its metropolitan statistical area and 23.5 million in its combined statistical area as of 2020, New York is one of the world's most populous Megacity, megacities, and over 58 million people live within of the city. New York City is a global city, global Culture of New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications Of The ACM
''Communications of the ACM'' is the monthly journal of the Association for Computing Machinery (ACM). It was established in 1958, with Saul Rosen as its first managing editor. It is sent to all ACM members. Articles are intended for readers with backgrounds in all areas of computer science and information systems. The focus is on the practical implications of advances in information technology and associated management issues; ACM also publishes a variety of more theoretical journals. The magazine straddles the boundary of a science magazine, trade magazine, and a scientific journal. While the content is subject to peer review, the articles published are often summaries of research that may also be published elsewhere. Material published must be accessible and relevant to a broad readership. From 1960 onward, ''CACM'' also published algorithms, expressed in ALGOL. The collection of algorithms later became known as the Collected Algorithms of the ACM. See also * ''Journal of the A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QUADPACK
QUADPACK is a FORTRAN 77 library for numerical integration of one-dimensional functions. It was included in the SLATEC Common Mathematical Library and is therefore in the public domain. The individual subprograms are also available on netlib. The GNU Scientific Library reimplemented the QUADPACK routines in C. SciPy provides a Python interface to part of QUADPACK. Routines The main focus of QUADPACK is on ''automatic'' integration routines in which the user inputs the problem and an absolute or relative error tolerance and the routine attempts to perform the integration with an error no larger than that requested. There are nine such automatic routines in QUADPACK, in addition to a number of non-automatic routines. All but one of the automatic routines use adaptive quadrature. Each of the adaptive routines also have versions suffixed by E that have an extended parameter list that provides more information and allows more control. Double precision versions of all routines wer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adaptive Simpson's Method
Adaptive Simpson's method, also called adaptive Simpson's rule, is a method of numerical integration proposed by G.F. Kuncir in 1962. It is probably the first recursive adaptive algorithm for numerical integration to appear in print,For an earlier, non-recursive adaptive integrator more reminiscent of ODE solvers, see although more modern adaptive methods based on Gauss–Kronrod quadrature and Clenshaw–Curtis quadrature are now generally preferred. Adaptive Simpson's method uses an estimate of the error we get from calculating a definite integral using Simpson's rule. If the error exceeds a user-specified tolerance, the algorithm calls for subdividing the interval of integration in two and applying adaptive Simpson's method to each subinterval in a recursive manner. The technique is usually much more efficient than composite Simpson's rule since it uses fewer function evaluations in places where the function is well-approximated by a cubic function. Simpson's rule is an inte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adaptive Step Size
In mathematics and numerical analysis, an adaptive step size is used in some methods for the numerical solution of ordinary differential equations (including the special case of numerical integration) in order to control the errors of the method and to ensure stability properties such as A-stability. Using an adaptive stepsize is of particular importance when there is a large variation in the size of the derivative. For example, when modeling the motion of a satellite about the earth as a standard Kepler orbit, a fixed time-stepping method such as the Euler method may be sufficient. However things are more difficult if one wishes to model the motion of a spacecraft taking into account both the Earth and the Moon as in the Three-body problem. There, scenarios emerge where one can take large time steps when the spacecraft is far from the Earth and Moon, but if the spacecraft gets close to colliding with one of the planetary bodies, then small time steps are needed. Romberg's m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adaptive Numerical Differentiation
In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the function. Finite differences The simplest method is to use finite difference approximations. A simple two-point estimation is to compute the slope of a nearby secant line through the points (''x'', ''f''(''x'')) and (''x'' + ''h'', ''f''(''x'' + ''h'')). Choosing a small number ''h'', ''h'' represents a small change in ''x'', and it can be either positive or negative. The slope of this line is : \frac. This expression is Newton's difference quotient (also known as a first-order divided difference). The slope of this secant line differs from the slope of the tangent line by an amount that is approximately proportional to ''h''. As ''h'' approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true derivative of ''f'' at ''x'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Romberg's Method
In numerical analysis, Romberg's method is used to estimate the definite integral \int_a^b f(x) \, dx by applying Richardson extrapolation repeatedly on the trapezium rule or the rectangle rule (midpoint rule). The estimates generate a triangular array. Romberg's method is a Newton–Cotes formula – it evaluates the integrand at equally spaced points. The integrand must have continuous derivatives, though fairly good results may be obtained if only a few derivatives exist. If it is possible to evaluate the integrand at unequally spaced points, then other methods such as Gaussian quadrature and Clenshaw–Curtis quadrature are generally more accurate. The method is named after Werner Romberg (1909–2003), who published the method in 1955. Method Using h_n = \frac, the method can be inductively defined by \begin R(0,0) &= h_1 (f(a) + f(b)) \\ R(n,0) &= \tfrac R(n-1,0) + h_n \sum_^ f(a + (2k-1)h_n) \\ R(n,m) &= R(n,m-1) + \tfrac (R(n,m-1) - R(n-1,m-1)) \\ &= \frac ( 4^m R( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Kronrod Quadrature Formula
The Gauss–Kronrod quadrature formula is an adaptive method for numerical integration. It is a variant of Gaussian quadrature, in which the evaluation points are chosen so that an accurate approximation can be computed by re-using the information produced by the computation of a less accurate approximation. It is an example of what is called a nested quadrature rule: for the same set of function evaluation points, it has two quadrature rules, one higher order and one lower order (the latter called an ''embedded'' rule). The difference between these two approximations is used to estimate the calculational error of the integration. These formulas are named after Alexander Kronrod, who invented them in the 1960s, and Carl Friedrich Gauss. Description The problem in numerical integration is to approximate definite integrals of the form :\int_a^b f(x)\,dx. Such integrals can be approximated, for example, by ''n''-point Gaussian quadrature :\int_a^b f(x)\,dx \approx \sum_^n w_i f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., an integral assigns numbers to functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be int ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Quadrature
In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for more on quadrature rules.) An -point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree or less by a suitable choice of the nodes and weights for . The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826. The most common domain of integration for such a rule is taken as , so the rule is stated as :\int_^1 f(x)\,dx \approx \sum_^n w_i f(x_i), which is exact for polynomials of degree or less. This exact rule is known as the Gauss-Legendre quadrature rule. The quadrature rule will only be an accurate approximation to the integral above if is well-approximated by a polynomial of degree or less on . The Gaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
