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Journal of Applied Sciences & Environmental Management, Vol. 9, No. 1, 2005, pp. 87-91 Error Estimation in the Numerical Solution of Rational Functions *1 AKINPELU F. O.2 I. A. ADETUNDE3 OMIDIORA E. O. 1Ladoke
AkintolaUniversity of Technology Department of Pure and Applied
Mathematics Ogbomoso, Oyo StateNigeria. Code Number: ja05016 ABSTRACT: In this paper, two methods are described for obtaining estimates of the error of rational functions the Pades and Meahlys methods of approximation were used and it was discovered that Maehlys proved more accurate than the Pades method. @JASERM Accurate polynomial approximations solution of rational functions with polynomial coefficients can be obtained by the Tau method introduced by Lanczos (B) in 1938. Techniques based on this method have been reported in the literature with application to more general equation (C, D) while technique based on direct Chebyshev series replacements have been discussed by Fox (A) in this work, a polynomial is constructed, based on rational function, in doing so, the number of undetermined constants are kept to a minimum upon which evaluations at points of the intervals are considered. By approximating rational functions, we try to find an accurate approximation, which will give a small overall error on the considered (a, b). Two methods were considered in this paper Pade and Meahlys method of approximation by rational functions. Pade formulated a general method with the aid of which power series can be used for rational approximations to function represented by this power series. Meahly developed the fundamental idea with the aid of which any Chebyshev expansion can generate a rational approximation. We stress the point that a rational approximation is useful only in the equivalent form of a finite continued fraction the accuracy of a rational approximation depends on the range, and in general decreasing the range increasing the accuracy. Numerical examples are given which shows that Maehlys method leads to better accuracy than Pades method. Method 1: (Pades method of approximation) Pade suggested that a function F (x) be equated to the rational function RG(x) Where Pm (x) and Qn (x) gare polynomials of degree m and n respectively, such that (1) can be written as The polynomials in 2 are formed so that value of F(x) and RG(x) agree at x=0 and their derivatives up to n + m agree at x=0. When n = 0 in (2), then the denominator implies Q(x) = 1, then (2) will just be Maclaurin expansion for F(x). Furthermore, assuming F(x) to be analytical and has Maclaurin series expansion. Then equation (3) Becomes i.e. 4 can be expressed as
Method 2: Meahlys Method of Approximation Maehly Method Starting With The Chebyshev Expansion, of our Subsequent Use, We Recall Some Well-known Properties of the Chebyshev Polynomials Defined the Interval [-1, 1]
Other number can be generated with the use of a three terms recursion formula
To find expression for the numerator we resolve the product of Chebyshev polynomial before equating it to zero.
solving for coefficients thereafter, we obtain system of linear equations.
Where aj = 0 for j>m bj = 0 for j>n j
= 1, 2, a, a. g. Using Pade and Meahlys method to approximate arctanx when m = 5 and n =4, and we compare the results. Hence we obtained error estimate for the two method considered. Following illustration from 1 to 10 we have the following graphs for the results Table 1: Comparison of Meahlys Approximation with Chebyshev series for Arctanx Table 2: Comparison of Result of Table 1. Table 3: Comparison of Pades Approximation with Maclaurin Series for Arctanx Table 4: Comparison of Result of Table 3.
Solution Following illustration from 1 to 10 we have the following graphs for the result.
Table 6: Comparison of Result of Table 5. Table 7: Comparison of Pade Approximation to Maclaurin series for arc sin x Conclusions
The Pade approximation method is more effective when the polynomial degree of the numerator is greater than or equal to the polynomial degree of the denominator. Pade approximation used to approximate arctan x and arcsin x in the interval [-1,1] assumes accuracy at origin but error bumps up at the extreme much rapidly. Maehlys method used to apprximate arctanx and arcsin x in the interval (-1,1) assumes accuracy at the extreme of the interval and also the errors at this extreme almost reduced equal values. Maehlys method is a better approximation method of rational function than the Pade method though longer and more time consuming but gives a better accuracy. Finally, the accuracy of a rational function approximation depends on the interval and in general the internal increases and improves the accuracy. Acknowledgement: The authors wish to thank Adejumo Oluwaseyi a of Ladoke Akintola University of Technology, Ogbomoso for performing the numerical experiments and the drawing of the graphs in this paper. REFERENCES
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