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Truncation Error (numerical Integration)
Truncation errors in numerical integration are of two kinds: * ''local truncation errors'' – the error caused by one iteration, and * ''global truncation errors'' – the cumulative error caused by many iterations. Definitions Suppose we have a continuous differential equation : y' = f(t,y), \qquad y(t_0) = y_0, \qquad t \geq t_0 and we wish to compute an approximation y_n of the true solution y(t_n) at discrete time steps t_1,t_2,\ldots,t_N . For simplicity, assume the time steps are equally spaced: : h = t_n - t_, \qquad n=1,2,\ldots,N. Suppose we compute the sequence y_n with a one-step method of the form : y_n = y_ + h A(t_, y_, h, f). The function A is called the ''increment function'', and can be interpreted as an estimate of the slope \frac . Local truncation error The local truncation error \tau_n is the error that our increment function, A , causes during a single iteration, assuming perfect knowledge of the true solution at the previous ite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Little-o Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Lipschitz Continuity
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the ''Lipschitz constant'' of the function (or '' modulus of uniform continuity''). For instance, every function that has bounded first derivatives is Lipschitz continuous. In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. We have the following chain of str ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Picard–Lindelöf Theorem
In mathematics – specifically, in differential equations – the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named after Émile Picard, Ernst Lindelöf, Rudolf Lipschitz and Augustin-Louis Cauchy. Theorem Let D \subseteq \R \times \R^nbe a closed rectangle with (t_0, y_0) \in D. Let f: D \to \R^n be a function that is continuous in t and Lipschitz continuous in y. Then, there exists some such that the initial value problem y'(t)=f(t,y(t)),\qquad y(t_0)=y_0. has a unique solution y(t) on the interval _0-\varepsilon, t_0+\varepsilon/math>. Note that D is often instead required to be open but even under such an assumption, the proof only uses a closed rectangle within D. Proof sketch The proof relies on transforming the differential equation, and ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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 method ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Zero-stability
Linear multistep methods are used for the numerical ordinary differential equations, 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 methods, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Order Of Accuracy
In numerical analysis, order of accuracy quantifies the rate of convergence of a numerical approximation of a differential equation to the exact solution. Consider u, the exact solution to a differential equation in an appropriate normed space (V,, , \ , , ). Consider a numerical approximation u_h, where h is a parameter characterizing the approximation, such as the step size in a finite difference scheme or the diameter of the cells in a finite element method. The numerical solution u_h is said to be nth-order accurate if the error E(h):= , , u-u_h, , is proportional to the step-size h to the nth power: : E(h) = , , u-u_h, , \leq Ch^n where the constant C is independent of h and usually depends on the solution u. Using the big O notation an nth-order accurate numerical method is notated as : , , u-u_h, , = O(h^n) This definition is strictly dependent on the norm used in the space; the choice of such norm is fundamental to estimate the rate of convergence and, in general, al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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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 differenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Truncation Error
In numerical analysis and scientific computing, truncation error is an error caused by approximating a mathematical process. Examples Infinite series A summation series for e^x is given by an infinite series such as e^x=1+ x+ \frac + \frac+ \frac+ \cdots In reality, we can only use a finite number of these terms as it would take an infinite amount of computational time to make use of all of them. So let's suppose we use only three terms of the series, then e^x\approx 1+x+ \frac In this case, the truncation error is \frac+\frac+ \cdots Example A: Given the following infinite series, find the truncation error for if only the first three terms of the series are used. S = 1 + x + x^2 + x^3 + \cdots, \qquad \left, x\<1. Solution Using only first three terms of the series gives The sum of an infinite geometrical series [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also publishes Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Spo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |