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# Linear difference equation

About: Linear difference equation is a research topic. Over the lifetime, 392 publications have been published within this topic receiving 4572 citations.

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TL;DR: In this article, a simple method is proposed for investigating the computational stability of finite-difference equations with variable coefficients, based on an examination of certain kinds of truncation errors, which is illustrated by applying it to a simple linear difference equation.

414 citations

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TL;DR: In this article, a linear difference equation is used as a model to generate a stationary discrete sequence of random variables from the experimental data, and the parameters are estimated by first computing the first 100 correlation coefficients and then applying one of two available algorithms.

Abstract: A linear difference equation is used as a model to generate a stationary discrete sequence of random variables. From the experimental data the parameters are estimated by first computing the first 100 correlation coefficients and then applying one of two available algorithms. The first algorithm gives a maximum likelihood estimate by replacing the original equation with an autoregressive scheme and a moving average scheme. The parameters of the latter are estimated by numerical optimization, while the others are obtained in closed form. For a system of order p it is sufficient to optimize in q ⩽ p variables. The second algorithm is based on a linear recursive relation for the correlation coefficients. It is considerably faster than the first one but gives about the same accuracy expressed in terms of maximum or mean square error for the reconstructed correlation coefficients. The estimation procedures are tried on a few recorded EEG signals and it is found that typically a fifth-order system suffices to give an accuracy of about 5% for the correlation coefficients. The sampling interval is found to be of considerable importance. It is argued that our model may be used with advantage to define suitable parameters to described the stationary parts of EEG signals; two sets of parameters are actually discussed.

304 citations

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TL;DR: In this paper, two nonlinear transformations, the D-transformation to accelerate the convergence of infinite integrals and the d -transformation for infinite series, are presented, and the computational aspects of these transformations are described in detail.

146 citations

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Kyoto University

^{1}TL;DR: This paper derives a closed form expression for the performance of a class of dynamic quantizers in the form of a linear difference equation such that the system composed of a given linear plant and the quantizer is an optimal approximation of the givenlinear plant in the sense of the input-output relation.

144 citations

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TL;DR: This work considers consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension and provides the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes.

Abstract: We consider consistent, conservative-form, monotone difference schemes for nonlinear convection-diffusion equations in one space dimension. Since we allow the diffusion term to be strongly degenerate, solutions can be discontinuous and, in general, are not uniquely determined by their data. Here we choose to work with weak solutions that belong to the BV (in space and time) class and, in addition, satisfy an entropy condition. A recent result of Wu and Yin [ Northeastern Math J., 5 (1989), pp. 395--422] states that these so-called BV entropy weak solutions are unique. The class of equations under consideration is very large and contains, to mention only a few, the heat equation, the porous medium equation, the two phase flow equation, and hyperbolic conservation laws. The difference schemes are shown to converge to the unique BV entropy weak solution of the problem. In view of the classical theory for monotone difference approximations of conservation laws, the main difficulty in obtaining a similar convergence theory in the present context is to show that the (strongly degenerate) discrete diffusion term is sufficiently smooth. We provide the necessary regularity estimates by deriving and carefully analyzing a linear difference equation satisfied by the numerical flux of the difference schemes. Finally, we make some concluding remarks about monotone difference schemes for multidimensional equations.

132 citations