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General Linear Methods
General linear methods (GLMs) are a large class of numerical methods used to obtain numerical solutions to ordinary differential equations. They include multistage Runge–Kutta methods that use intermediate collocation points, as well as linear multistep methods that save a finite time history of the solution. John C. Butcher originally coined this term for these methods, and has written a series of review papers a book chapter and a textbook on the topic. His collaborator, Zdzislaw Jackiewicz also has an extensive textbook on the topic. The original class of methods were originally proposed by Butcher (1965), Gear (1965) and Gragg and Stetter (1964). Some definitions Numerical methods for first-order ordinary differential equations approximate solutions to initial value problems of the form : y' = f(t,y), \quad y(t_0) = y_0. The result is approximations for the value of y(t) at discrete times t_i : : y_i \approx y(t_i) \quad\text\quad t_i = t_0 + i h, where ''h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Linear Model
The general linear model or general multivariate regression model is a compact way of simultaneously writing several multiple linear regression models. In that sense it is not a separate statistical linear model. The various multiple linear regression models may be compactly written as : \mathbf = \mathbf\mathbf + \mathbf, where Y is a matrix with series of multivariate measurements (each column being a set of measurements on one of the dependent variables), X is a matrix of observations on independent variables that might be a design matrix (each column being a set of observations on one of the independent variables), B is a matrix containing parameters that are usually to be estimated and U is a matrix containing errors (noise). The errors are usually assumed to be uncorrelated across measurements, and follow a multivariate normal distribution. If the errors do not follow a multivariate normal distribution, generalized linear models may be used to relax assumptions about Y and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Linear Model
In statistics, a generalized linear model (GLM) is a flexible generalization of ordinary linear regression. The GLM generalizes linear regression by allowing the linear model to be related to the response variable via a ''link function'' and by allowing the magnitude of the variance of each measurement to be a function of its predicted value. Generalized linear models were formulated by John Nelder and Robert Wedderburn as a way of unifying various other statistical models, including linear regression, logistic regression and Poisson regression. They proposed an iteratively reweighted least squares method for maximum likelihood estimation (MLE) of the model parameters. MLE remains popular and is the default method on many statistical computing packages. Other approaches, including Bayesian regression and least squares fitting to variance stabilized responses, have been developed. Intuition Ordinary linear regression predicts the expected value of a given unknown quantity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Methods For Ordinary Differential Equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved. The problem A first-order differentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Ordinary Differential Equations
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation. An alternative method is to use techniques from calculus to obtain a series expansion of the solution. Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved. The problem A first-order differentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordinary Differential Equation
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] |
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] |
Collocation Method
In mathematics, a collocation method is a method for the numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate solutions (usually polynomials up to a certain degree) and a number of points in the domain (called ''collocation points''), and to select that solution which satisfies the given equation at the collocation points. Ordinary differential equations Suppose that the ordinary differential equation : y'(t) = f(t,y(t)), \quad y(t_0)=y_0, is to be solved over the interval _0,t_0+c_k h/math>. Choose c_k from 0 ≤ ''c''1< ''c''2< … < ''c''''n'' ≤ 1. The corresponding (polynomial) collocation method approximates the solution ''y'' by the polynomial ''p'' of degree ''n'' which satisfies the initial condition , and the differential equation at all ''collocation points' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Multistep Method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of ''linear'' multistep methods, a linear combination of the previous points and derivative values is used. Definitions Numerical methods ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John C
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died c. AD 30), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (lived c. AD 30), one of the twelve apostles of Jesus * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * Pope Jo ... [...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] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted v \otimes w. An element of the form v \otimes w is called the tensor product of and . An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for and , a basis of V \otimes W is formed by all tensor products of a basis element of and a basis element of . The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space factors uniquely through a linear map V\otimes W\to Z (see Universal property). Tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Multistep Method
Linear multistep methods are used for the numerical solution of ordinary differential equations. Conceptually, a numerical method starts from an initial point and then takes a short step forward in time to find the next solution point. The process continues with subsequent steps to map out the solution. Single-step methods (such as Euler's method) refer to only one previous point and its derivative to determine the current value. Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it. Consequently, multistep methods refer to several previous points and derivative values. In the case of ''linear'' multistep methods, a linear combination of the previous points and derivative values is used. Definitions Numerical methods ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
