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Journal of Applied Sciences and Environmental Management
World Bank assisted National Agricultural Research Project (NARP) - University of Port Harcourt
ISSN: 1119-8362
Vol. 9, Num. 1, 2005, pp. 87-91

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.
2University of Agriculture Department of Mathematical Sciences Abeokuta, Ogun State, Nigeria.
3Ladoke AkintolaUniversity of Technology Department of Computer Science & Engineering, Ogbomoso, OyoState.

 Code Number: ja05016

ABSTRACT: In this paper, two methods are described for obtaining estimates of the error of rational functions the Pade’s and Meahly’s methods of approximation were used and it was discovered that Maehly’s proved more accurate than the Pade’s 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 Meahly’s 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 Maehly’s method leads to better accuracy than Pade’s method.

Method 1: (Pade’s 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: Meahly’s 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 coefficient’s thereafter, we obtain system of linear equations.

Where aj = 0 for j>m

                bj = 0 for j>n

                j = 1, 2, a, a. g.

Numerical Example 1

Using Pade and Meahly’s 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 Meahly’s 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.


Use both Pade and Maehly’s method to approximate aresinx using m=5, and n=4.

Solution Following illustration from 1 to 10 we have the following graphs for the result.


Table 5
: Comparison of Maehly’s  Approximation with Chebyshev  Series for Arctan x

Table 6: Comparison of Result of Table 5.

Table 7: Comparison of Pade Approximation to Maclaurin series for arc sin x

Table 8: Comparison of Result of Table 7.

Illustration from 1-10

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.

Maehly’s 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.

Maehly’s 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

  • Fox, L (1962) Chebyshev methods for Ordinary differential Equations. Compt. J. 4:318-331.
  • Lanczos (1938) Trigonometry Interpretation of empirical and analytical functions. J. Math. Phys. 17:123-177.
  • Onumayi, P Ortizi E. L. (1982) Numerical Solution of higher order boundary value problems for ordinary differential equations with an estimation of the error Int. J. Numer. Meth. Engrg. 18:775-781.
  • Ortizi, E. L. (1969) The Tau method. SIAM J. Numerical. Anal 6:480-492

Copyright 2005 - Journal of Applied Sciences & Environmental Management


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