20 Function Approximation

V.D. Pathak

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1.   Introduction:

 

An evaluation of an arbitrary function f on a digital computer involves approximating this function by a polynomial function P. This is because computational capabilities of digital computers are restricted to arithmetic operations and evaluation of a polynomial function involves only basic arithmetic operations.

 

Interpolation and curve fitting techniques can be used for function approximation. However, both these techniques use finite number of data points obtained either by evaluating a function to be approximated at finite number of arguments, or obtained experimentally. Such approximations are completely dependent on these finite data values and do not use any other properties of the function to be approximated. Also, the interpolating polynomial is constrained to pass through the given data points where as least square polynomial minimizes the total square error. And hence, the approximating polynomial obtained by Interpolation or least square curve fitting techniques may not have much control over the error |f(x) – P(x)| at some points in the domain of approximation. Thus, though these techniques are efficient for estimating a functional relationship from the available finite data points, they are not adequate for function approximation.

 

The  main  focus  in  function  approximation  is  to  find  a  polynomial  P  that  minimizes {Maximum |f(x) – P(x)|: x in the domain of approximation}. For this purpose, two techniques namely, Taylor’s / Maclaurin’s series approximation and approximation by Chebyshev polynomials, are discussed in this module.

 

2.   Taylor’s Polynomial Approximation:

 

This approximation is applicable to the functions which are sufficiently smooth and based on well-known Taylor’s theorem as stated below.

 

Taylor’s theorem: Suppose that function f has (n + 1) continuous derivatives on the interval [a, b] and suppose that x0 ϵ [a, b]. Then for every x ϵ [a, b] there exists a number c(x) is in between x0 and x, such that f(x) = Pn(x) + Rn(x) where

 

Additional Reading Material

 

The following are some of the references on Interpolation. Detailed discussion on Newton’s divided differences and Newton’s divided differences interpolation is given in reference 3 and 4. Computational aspects are discussed in reference 1. And MATLAB programs can be found in reference 5.

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References:

  1. Steven C Chapra & Raymond Canale: N. Methods for Engineers, 5th Ed. TMH, 2009
  2. Conte Samuel D. & Carl de Boor: Elementary N A – An Algorithmic approach, 3rd Ed., TMH, 2005.
  3. Alfio Quarteroni, Recardo Sacco, Fausto Saleri: N. Mathematics (Text in Applied Math.) 2nd Ed. SV, 2007
  4. Richard L. Burden & J. Douglas Faires: N A –Theory & Applications, Cengage Learning, 2005.
  5. Won Y. Yang, Wenwu Cao, Tae-Sang Chung, & John Morris: Applied NM Using Matlab, WSE, 2005
  6. Kendall Atkinson & Weimin Han: Elimentary Numerical Analysis, Third edition, Wiley Dreamtech India(P) Ltd., 2004.
  7. Matheus Grasselli & Dimitry Pelinovsky: Numerical Mathematics, Narosa Publishing House, 2009.
  8. s. s. ssatry: Introductory Methods of Numerical Analysis, PHI, 1989.
  9. Polynomial interpolation, From Wikipedia, the free encyclopaedia