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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] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equations
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial differential equation which may be with respect to ''more than'' one independent variable. Differential equations A linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form :a_0(x)y +a_1(x)y' + a_2(x)y'' +\cdots +a_n(x)y^+b(x)=0, where , ..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of the unknown function of the variable . Among ordinary differential equations, linear differential equations play a prominent role for several reasons. Most elementary and special functions that are encountered in physics and applied mathematics are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imperial College London
Imperial College London (legally Imperial College of Science, Technology and Medicine) is a public research university in London, United Kingdom. Its history began with Prince Albert, consort of Queen Victoria, who developed his vision for a cultural area that included the Royal Albert Hall, Victoria & Albert Museum, Natural History Museum and royal colleges. In 1907, Imperial College was established by a royal charter, which unified the Royal College of Science, Royal School of Mines, and City and Guilds of London Institute. In 1988, the Imperial College School of Medicine was formed by merging with St Mary's Hospital Medical School. In 2004, Queen Elizabeth II opened the Imperial College Business School. Imperial focuses exclusively on science, technology, medicine, and business. The main campus is located in South Kensington, and there is an innovation campus in White City. Facilities also include teaching hospitals throughout London, and with Imperial College Healthcare ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dormand–Prince Method
In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations . 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 Cash–Karp (RKCK). The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the FSAL (First Same As Last) 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 with the Fehlberg method, which was con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butcher Tableau
A butcher is a person who may slaughter animals, dress their flesh, sell their meat, or participate within any combination of these three tasks. They may prepare standard cuts of meat and poultry for sale in retail or wholesale food establishments. A butcher may be employed by supermarkets, grocery stores, butcher shops and fish markets, slaughter houses, or may be self-employed. Butchery is an ancient trade, whose duties may date back to the domestication of livestock; its practitioners formed guilds in England as far back as 1272. Since the 20th century, many countries and local jurisdictions offer trade certifications for butchers in order to ensure quality, safety, and health standards but not all butchers have formal certification or training. Trade qualification in English-speaking countries is often earned through an apprenticeship although some training organisations also certify their students. In Canada, once a butcher is trade qualified, they can learn to become a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Runge–Kutta Methods
In numerical analysis, the Runge–Kutta methods ( ) are a family of implicit and explicit iterative methods, which include the Euler method, used in temporal discretization for the approximate solutions of simultaneous nonlinear equations. These methods were developed around 1900 by the German mathematicians Carl Runge and Wilhelm Kutta. The Runge–Kutta method The most widely known member of the Runge–Kutta family is generally referred to as "RK4", the "classic Runge–Kutta method" or simply as "the Runge–Kutta method". Let an initial value problem be specified as follows: : \frac = f(t, y), \quad y(t_0) = y_0. Here y is an unknown function (scalar or vector) of time t, which we would like to approximate; we are told that \frac, the rate at which y changes, is a function of t and of y itself. At the initial time t_0 the corresponding y value is y_0. The function f and the initial conditions t_0, y_0 are given. Now we pick a step-size ''h'' > 0 and define: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Runge–Kutta Methods
Runge–Kutta methods are methods for the numerical solution of the ordinary differential equation :\frac = f(t, y). Explicit Runge–Kutta methods take the form :\begin y_ &= y_n + h \sum_^s b_i k_i \\ k_1 &= f(t_n, y_n), \\ k_2 &= f(t_n+c_2h, y_n+h(a_k_1)), \\ k_3 &= f(t_n+c_3h, y_n+h(a_k_1+a_k_2)), \\ &\;\;\vdots \\ k_i &= f\left(t_n + c_i h, y_n + h \sum_^ a_ k_j\right). \end Stages for implicit methods of s stages take the more general form, with the solution to be found over all s :k_i = f\left(t_n + c_i h, y_n + h \sum_^ a_ k_j\right). Each method listed on this page is defined by its Butcher tableau, which puts the coefficients of the method in a table as follows: : \begin c_1 & a_ & a_& \dots & a_\\ c_2 & a_ & a_& \dots & a_\\ \vdots & \vdots & \vdots& \ddots& \vdots\\ c_s & a_ & a_& \dots & a_ \\ \hline & b_1 & b_2 & \dots & b_s\\ \end For adaptive and implicit methods, the Butcher tableau is extended to give values of b^*_i, and the estim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Differential Equations
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Numerical may refer to: * Number * Numerical digit * Numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
