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Shanks Transformation
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.Weniger (2003). Formulation For a sequence \left\_ the series :A = \sum_^\infty a_m\, is to be determined. First, the partial sum A_n is defined as: :A_n = \sum_^n a_m\, and forms a new sequence \left\_. Provided the series converges, A_n will also approach the limit A as n\to\infty. The Shanks transformation S(A_n) of the sequence A_n is the new sequence defined byBender & Orszag (1999), pp. 368–375.Van Dyke (1975), pp. 202–205. :S(A_n) = \frac = A_ - \frac where this sequence S(A_n) often converges more rapidly than the sequence A_n. Further speed-up may be obtained by repeated use of the Shanks transformation, by computing S^2(A_n)=S(S(A_n)), S^3(A_n)=S(S(S(A_n))), etc. No ... [...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] |
Transient (oscillation)
In electrical engineering and mechanical engineering, a transient response is the response of a system to a change from an equilibrium or a steady state. The transient response is not necessarily tied to abrupt events but to any event that affects the equilibrium of the system. The impulse response and step response are transient responses to a specific input (an impulse and a step, respectively). In electrical engineering specifically, the transient response is the circuit’s temporary response that will die out with time. It is followed by the steady state response, which is the behavior of the circuit a long time after an external excitation is applied. Damping The response can be classified as one of three types of damping that describes the output in relation to the steady-state response. ;Underdamped :An underdamped response is one that oscillates within a decaying envelope. The more underdamped the system, the more oscillations and longer it takes to reach steady-stat ... [...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] |
Sequence Transformation
In mathematics, a sequence transformation is an operator acting on a given space of sequences (a sequence space). Sequence transformations include linear mappings such as convolution with another sequence, and resummation of a sequence and, more generally, are commonly used for series acceleration, that is, for improving the rate of convergence of a slowly convergent sequence or series. Sequence transformations are also commonly used to compute the antilimit of a divergent series numerically, and are used in conjunction with extrapolation methods. Overview Classical examples for sequence transformations include the binomial transform, Möbius transform, Stirling transform and others. Definitions For a given sequence :S=\_,\, the transformed sequence is :\mathbf(S)=S'=\_,\, where the members of the transformed sequence are usually computed from some finite number of members of the original sequence, i.e. :s_n' = T(s_n,s_,\dots,s_) for some k which often depends on n ( ... [...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] |
Rate Of Convergence
In numerical analysis, the order of convergence and the rate of convergence of a convergent sequence are quantities that represent how quickly the sequence approaches its limit. A sequence (x_n) that converges to x^* is said to have ''order of convergence'' q \geq 1 and ''rate of convergence'' \mu if : \lim _ \frac=\mu. The rate of convergence \mu is also called the ''asymptotic error constant''. Note that this terminology is not standardized and some authors will use ''rate'' where this article uses ''order'' (e.g., ). In practice, the rate and order of convergence provide useful insights when using iterative methods for calculating numerical approximations. If the order of convergence is higher, then typically fewer iterations are necessary to yield a useful approximation. Strictly speaking, however, the asymptotic behavior of a sequence does not give conclusive information about any finite part of the sequence. Similar concepts are used for discretization methods. The solutio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anderson Acceleration
In mathematics, Anderson acceleration, also called Anderson mixing, is a method for the acceleration of the convergence rate of fixed-point iterations. Introduced by Donald G. Anderson, this technique can be used to find the solution to fixed point equations f(x) = x often arising in the field of computational science. Definition Given a function f:\mathbb^n \to \mathbb^n, consider the problem of finding a fixed point of f, which is a solution to the equation f(x) = x. A classical approach to the problem is to employ a fixed-point iteration scheme; that is, given an initial guess x_0 for the solution, to compute the sequence x_ = f(x_i) until some convergence criterion is met. However, the convergence of such a scheme is not guaranteed in general; moreover, the rate of convergence is usually linear, which can become too slow if the evaluation of the function f is computationally expensive. Anderson acceleration is a method to accelerate the convergence of the fixed-point sequen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padé Table
In complex analysis, a Padé table is an array, possibly of infinite extent, of the rational Padé approximants :''R''''m'', ''n'' to a given complex formal power series. Certain sequences of approximants lying within a Padé table can often be shown to correspond with successive convergents of a continued fraction representation of a holomorphic or meromorphic function. History Although earlier mathematicians had obtained sporadic results involving sequences of rational approximations to transcendental functions, Frobenius (in 1881) was apparently the first to organize the approximants in the form of a table. Henri Padé further expanded this notion in his doctoral thesis ''Sur la representation approchee d'une fonction par des fractions rationelles'', in 1892. Over the ensuing 16 years Padé published 28 additional papers exploring the properties of his table, and relating the table to analytic continued fractions. Modern interest in Padé tables was revived by H. S. Wall a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods— in some sense inspired by the Padé theory— typically replace them. Since Padé approximant is a rational function, an artificial singul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant
In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one). The determinant of a matrix is denoted , , or . The determinant of a matrix is :\begin a & b\\c & d \end=ad-bc, and the determinant of a matrix is : \begin a & b & c \\ d & e & f \\ g & h & i \end= aei + bfg + cdh - ceg - bdi - afh. The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is n!, the factorial of (t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shanks Transformation
In numerical analysis, the Shanks transformation is a non-linear series acceleration method to increase the rate of convergence of a sequence. This method is named after Daniel Shanks, who rediscovered this sequence transformation in 1955. It was first derived and published by R. Schmidt in 1941.Weniger (2003). Formulation For a sequence \left\_ the series :A = \sum_^\infty a_m\, is to be determined. First, the partial sum A_n is defined as: :A_n = \sum_^n a_m\, and forms a new sequence \left\_. Provided the series converges, A_n will also approach the limit A as n\to\infty. The Shanks transformation S(A_n) of the sequence A_n is the new sequence defined byBender & Orszag (1999), pp. 368–375.Van Dyke (1975), pp. 202–205. :S(A_n) = \frac = A_ - \frac where this sequence S(A_n) often converges more rapidly than the sequence A_n. Further speed-up may be obtained by repeated use of the Shanks transformation, by computing S^2(A_n)=S(S(A_n)), S^3(A_n)=S(S(S(A_n))), etc. No ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-linear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists because most systems are inherently nonlinear in nature. Nonlinear dynamical systems, describing changes in variables over time, may appear chaotic, unpredictable, or counterintuitive, contrasting with much simpler linear systems. Typically, the behavior of a nonlinear system is described in mathematics by a nonlinear system of equations, which is a set of simultaneous equations in which the unknowns (or the unknown functions in the case of differential equations) appear as variables of a polynomial of degree higher than one or in the argument of a function which is not a polynomial of degree one. In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
