Chapter 8. Chebyshev spectral methods

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8.2. CHEBYSHEV DIFFERENTIATION MATRICES TREFETHEN 1994 271 A note of caution: D N is rarely used in exactly the form described in Theorem 8.4, for boundary conditions will modify it slightly, and these depend on the problem. EXERCISES . 8.2.1. Prove that for any N, DN is nilpotent: Dn N = 0 for a su ciently high integer n.
8.3. CHEBYSHEV DIFFERENTIATION BY THE FFT TREFETHEN 1994 272 8.3. Chebyshev di erentiation by the FFT Polynomial interpolation in Chebyshev points is equivalent to trigonometric interpolation in equally spaced points, and hence can be carried out by the FFT. The algorithm described below has the optimal order O(N logN),* but we donotworry about achieving the optimal constant factor. For more practical discussions, see Appendix B of the book by Canuto, et al., and also P. N.Swarztrauber, \Symmetric FFTs," Math. Comp. 47 (1986), 323{346. Valuable additional references are the book The Chebyshev Polynomials by Rivlin and Chapter 13 of P. Henrici, Applied and Computational Complex Analysis, 1986. Consider three independent variables 2 R, x 2 [;11], and z 2 S, where S is the complex unit circle fz : jzj =1g. They are related as follows: which implies z = e i x =Rez = 1 2 (z +z;1 ) = cos (8:3:1) dx d = ;sin = ;p 1;x 2 : (8:3:2) See Figure 8.3.1. Note that there are two conjugate values z 2 S for each x 2 (;11), and an in nite number of possible choices of . ;1 0 x Figure 8.3.1. z, x, and . *optimal, that is, so far as anyone knows as of 1994. o* o* o* z z 1
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Chapter 8. Chebyshev spectral methods

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