The computer method of variable piecewise polynomial approximation of functions and solutions of ordinary differential e
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THE COMPUTER METHOD OF VARIABLE PIECEWISE POLYNOMIAL APPROXIMATION OF FUNCTIONS AND SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS Ya. E. Romma† and G. A. Dzhanuntsa‡
UDC 681.3.06:681.323 (519.6)
Abstract. The authors present a computer method of piecewise polynomial approximation of functions and solution of Cauchy problem for systems of ordinary differential equations based on the Newton polynomial. The approximating polynomial on a subinterval is converted to the form with numerical coefficients, the degree of the polynomial and the number of subintervals varies. The method is shown to uniformly converge at the rate of geometric progression under conditions of double continuous differentiability of the function and of the right-hand side of the system. The approximate solution of the system is continuous, continuously differentiable, and is characterized by low error rate, in particular, when solving stiff problems. Keywords: piecewise polynomial approximation of functions, solution of ordinary differential equations, error minimization, stiff systems. INTRODUCTION We formulate the problem of computing a real function of one real variable with a priori defined error bound and simultaneous minimization of the time complexity. The solution is based on piecewise polynomial approximation; an interpolation Newton polynomial is selected on a subinterval and is transformed to a polynomial with numerical coefficients. As a result, the composition of standard functions is calculated in a time O(1) accurate to 10-18 . This computer approximation of subintegral functions increses the accuracy of integral computation. The method is transferred to the solution of ordinary differential equations (ODEs). The well-known solution methods for the Cauchy problem for ODEs allow attaining high accuracy in the case of nonstiff problems, and high order of smoothness of the right-hand side of the equations is required [1]. However, direct application of these methods does not retain the solution accuracy of stiff problems. Below, we will show that it is possible to reduce the error of the computer solution of the Cauchy problem for systems of ODEs based on piecewise interpolation of difference approximations for comparatively general constraints. The solution of nonstiff problems is approximated with the accuracy 10-19 , stiff problems are solved with a lower error rate as compared with specialized methods. COMPUTER PIECEWISE POLYNOMIAL APPROXIMATION OF FUNCTIONS A real function u = u( x ) of one real variable is approximated on an arbitrary interval [ a, b] as follows [2, 3]. We select a system of subintervals of equal length whose union covers the interval [ a, b] : [ a, b ] =
P -1
U [ xi , xi + 1 ] ,
(1)
i=0 a
A. P. Chekhov Taganrog State Pedagogical Institute, Taganrog, Russia, †[email protected]; ‡[email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 3, May–June, 2013, pp. 95–112. Original article submitted February 15, 2012. 1060-0396/13/4903-0409
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