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A textbook of Computer Based Numerical and Statiscal Techniques part 30

A textbook of Computer Based Numerical and Statiscal Techniques part 30. By joining statistical analysis with computer-based numerical methods, this book bridges the gap between theory and practice with software-based examples, flow charts, and applications. Designed for engineering students as well as practicing engineers and scientists, the book has numerous examples with in-text solutions. | 276 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES P X I ta.T. x n i 1 1 is very near solution to the problem. n max f x -V Bixi -1 x 1M à i - min -1 x 1 n f x -X Bixi i 0 i.e. the partial sum 1 is closely the best approximation to f x . Chebyshev polynomial approximation FIG. 5.2. Chebbyshev polynomiats T0 x through T6 x . Note that T. has j roots in the interval -1 1 and that all the polynomials are bounded between 1. iii Economization of power series To describe the process of economization which is essential due to Lanczos we first express the given function as a power series in x. Let power series expansion of x is. f x A0 A1x A2x2 . Anxn -1 x 1 . 1 Now convert each term in the power series in terms of Chebyshev polynomials. Thus we obtain the Chebyshev series expansion of the given continuous function f x on the interval -1 1 . i.e. n Pn x X BiT x . 2 i 0 or Pn x Bo B1T1 x B2T2 x . BnT x INTERPOLATION WITH UNEQUAL INTERVAL 277 Now if the truncated Chebyshew expansion is taken by 2 then max f x - Pn x B 1 Bn d . e and Hence Pn x is a good uniform approximation to f x in which the number of terms retained depends on the given tolerance of e . However for a large number of functions an expansion as in 2 converges more rapidly than the initial power series for the given function. This process is known as economization of the power series which is essentially due to Lanczos. Replacing each Chebyshev polynomial T x by its polynomial form and rearranging the terms we get the required economized polynomial approximation. We have thus economized the initial power series in the sense of using fewer terms to achieve almost the same accuracy. Example 5. Economize the power series. 2 5 7 sin x x - x- X- . x . 6 120 5040 to 3 significant digit accuracy. 2 5 7 x x x Sol. Here we have sin x x------ ------ ------ . 6 120 5040 Now it is required to compute sin x correct to 3 significant digits. So truncating after 3 terms as the truncation error after 3 terms of the given