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Adaptive Stepsize
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Periodic 3-body RKF Integration
Periodicity or periodic may refer to: Mathematics * Bott periodicity theorem, addresses Bott periodicity: a modulo-8 recurrence relation in the homotopy groups of classical groups * Periodic function, a function whose output contains values that repeat periodically * Periodic mapping Physical sciences * Periodic table of chemical elements * Periodic trends, relative characteristics of chemical elements observed * Redshift periodicity, astronomical term for redshift quantization Other uses * Fokker periodicity blocks, which mathematically relate musical intervals * Periodic acid, a compound of iodine * Principle of periodicity, a concept in generally accepted accounting principles * Quasiperiodicity, property of a system that displays irregular periodicity See also * Aperiodic (other) * Cycle (other) * Frequency (other) * Period (other) * Periodical * Seasonality In time series data, seasonality refers to the trends that occur at specif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Runge–Kutta–Fehlberg Method
In mathematics, the Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the large class of Runge–Kutta methods. The novelty of Fehlberg's method is that it is an embedded method from the Runge–Kutta family, meaning that it reuses the same intermediate calculations to produce two estimates of different accuracy, allowing for automatic error estimation. The method presented in Fehlberg's 1969 paper has been dubbed the RKF45 method, and is a method of order O(''h''4) with an error estimator of order O(''h''5). By performing one extra calculation, the error in the solution can be estimated and controlled by using the higher-order embedded method that allows for an adaptive stepsize to be determined automatically. Butcher tableau for Fehlberg's 4(5) method Any Runge–Kutta method is uniquely identifi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Adaptive Numerical Differentiation
In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or 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 and . Choosing a small number , represents a small change in , 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 . As approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true derivative of at is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 tol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dormand–Prince Method
In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations (ODE). The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Runge–Kutta–Fehlberg method, Fehlberg (RKF) and Cash–Karp (RKCK). The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the "First Same As Last" (FSAL) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince chose the coefficients of their method to minimize the error of the fifth-order solution. This is the main difference wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cash–Karp Method
In numerical analysis, the Cash–Karp method is a method for solving ordinary differential equations (ODEs). It was proposed by Professor Jeff R. Cash from Imperial College London and Alan H. Karp from IBM Scientific Center. The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Dormand–Prince (RKDP). The Butcher tableau is: The first row of ''b'' coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order solution. See also * Adaptive Runge–Kutta methods * List of Runge–Kutta methods Runge–Kutta methods are methods for the numerical solution of the ordinary differential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Richardson Extrapolation
In numerical analysis, Richardson extrapolation is a Series acceleration, 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 Christiaan_Huygens#De_Circuli_Magnitudine_Inventa, his calculation of \pi. In the words of Garrett Birkhoff, Birkhoff and Gian-Carlo Rota, 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. General formula Notation Let A_0(h) be an approximation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Taylor's Theorem
In calculus, Taylor's theorem gives an approximation of a k-times differentiable function around a given point by a polynomial of degree k, called the k-th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order ''k'' of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation. There are several versions of Taylor's theorem, some giving explicit estimates of the approximation error of the function by its Taylor polynomial. Taylor's theorem is named after the mathematician Brook Taylor, who stated a version of it in 1715, although an earlier version of the result was already mentioned in 1671 in science, 1671 by James Gregory (astronomer and mathematician), James Gregory. Taylor's theorem is taught in introductory-level calculus courses and is one of the central elementary tools in mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Romberg's Method
In numerical analysis, Romberg's method is used to estimate the Integral, 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 formulas, 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, who published the method in 1955. Method Using h_n = \frac, the method can be inductively defined by \begin R(0,0) &= h_0 (f(a) + f(b)) \\ R(n,0) &= \tfrac R(n-1,0) + 2h_n \sum_^ f(a + (2k-1)h_) \\ R(n,m) &= R(n,m-1) + \tfrac (R(n,m-1) - R(n-1,m-1)) \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
