3,778 research outputs found

    Positive trigonometric polynomials for strong stability of difference equations

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    We follow a polynomial approach to analyse strong stability of linear difference equations with rationally independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5

    The natural algorithmic approach of mixed trigonometric-polynomial problems

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    The aim of this paper is to present a new algorithm for proving mixed trigonometric-polynomial inequalities by reducing to polynomial inequalities. Finally, we show the great applicability of this algorithm and as examples, we use it to analyze some new rational (Pade) approximations of the function cos⁥2(x)\cos^2(x), and to improve a class of inequalities by Z.-H. Yang. The results of our analysis could be implemented by means of an automated proof assistant, so our work is a contribution to the library of automatic support tools for proving various analytic inequalities

    An eighth-order exponentially fitted two-step hybrid method of explicit type for solving orbital and oscillatory problems

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    The construction of an eighth-order exponentially fitted (EF) two-step hybrid method for the numerical integration of oscillatory second-order initial value problems (IVPs) is considered. The EF two-step hybrid methods integrate exactly differential systems whose solutions can be expressed as linear combinations of exponential or trigonometric functions and have variable coefficients depending on the frequency of each problem. Based on the order conditions and the EF conditions for this class of methods, we construct an explicit EF two-step hybrid method with symmetric nodes and algebraic order eight which only uses seven function evaluations per step. This new method has the highest algebraic order we know for the case of explicit EF two-step hybrid methods. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order EF scheme is more efficient than other standard and EF two-step hybrid codes recently proposed in the scientific literature

    Mixed collocation methods for y” = f(x , y)

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    The second-order initial value problem y" = f(x,y), y(x(_0)) = y(_0), y'(x(_0)) = z(_0) which does not contain the first derivative explicitly and where the solution is oscillatory has been of great interest for many years. Our aim is to construct numerical methods which are tuned to act efficiently on strongly oscillating functions. The frequencies involved determine the oscillatory character of the function and as the frequencies approach zero, the classical methods are obtained. The exponential- fitting tool has become increasingly popular as it is specially tailored for oscillating functions. Many classes of methods have been used with exponential-fitting and this will be discussed in more detail in the thesis. Collocation methods are considered for which the basis functions are combinations of polynomial and trigonometric terms. The resulting methods can be regarded as Runge-Kutta-Nyström methods with steplength dependent coefficients. We show how order conditions may be obtained, investigate the stability and other properties of particular methods and present some numerical results
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