tailieunhanh - Burden - Numerical Analysis 5e (PWS, 1993) Episode 3 Part 1

Tham khảo tài liệu 'burden - numerical analysis 5e (pws, 1993) episode 3 part 1', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Fast Fourier Transforms 493 Step 18 For j 1. . . . m 1 set ũj Re ể J j Cj my bj ỈTn e J7rỉ CjỊ m . Step 19 OUTPUT c0 . . . c2m-i Ũ . . . am by . bm y STOP. EXAMPLE 2 Let x X4 3x3 2x2 tanx x 2 . To determine the trigonometric interpolating polynomial of degree four for the data xr yy J 0 where Xj. i 4 and yj f xj requires a transformation of the interval 0 2 to 7T m . The translation is given by Zj 7T x - 1 so the input data to Algorithm is The interpolating polynomial in z is s4 z cos z cos 2z cos 3z - cos 4z sinz sin 2z - sin 3z. The trigonometric polynomial s4 x on 0 2 is obtained by substituting z ir x 1 into s4 z . The graphs of y fix and y s4 x are shown in Figure . Values of f x and s4 x are given in Table on page 494. E Figure 1 2 X 494 CHAPTER 8 Approximation Theory Table X f x s4 x Í G9 - s4 x X IO 2 X 10 3 Ĩ X 10 3 I X IO 3 X 10 3 X 10 3 1 X 10 3 X 10 2 More details on the verification of the validity of the fast Fourier transform procedure can be found in Hamming 69 which presents the method from a mathematical approach or in Bracewell 17 where the presentation is based on methods more likely to be familiar to engineers. Aho Hopcroft and Ullman 1J pages 252-269 is a good reference for a discussion of the computational aspects of the method. Modification of the procedure for the case when m is not a power of two can be found in Winograd 158 . A presentation of the techniques and related material from the point of view of applied abstract algebra is given in Laufer 93 pages 438-465. EXERCISE SET 1. Determine the trigonometric interpolating polynomial s2 of degree two on 77 77 for the following functions and graph f x s2 x a. f x 77 .

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