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MATLAB : a ubiquitous tool for the practical engineer by Ionescu, Clara-Mihaela

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By Ionescu, Clara-Mihaela

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X N , see (Trefethen, 2000) for more details on the coefficients ak versus ck . The next theorem of (Boyd, 2000) estimates the error: T HEOREM 2. Let y have at least N derivatives on dom. Then y ( x ) − p N −1 ( x ) = N 1 (N) y (ξ ) ∏ ( x − xk ) N! k =1 for some ξ on the interval spanned by x and the interpolation points. The point ξ depends on the function y, upon N, upon x and upon the location of the interpolation points. , N. 2N (7) For these points x, the polynomials { p N −1 } are generally nearly as good approximations to y as the polynomials {y N −1 } and if y is analytic on dom, then both y − y N −1 and y − p N −1 decrease geometrically as N → ∞.

To approximate the solution y by a finite sum of the Chebyshev series y( x ) = 1 c T ( x ) + c1 T1 ( x ) + c2 T2 ( x ).... 2 0 0 (3) 38 2 MATLAB – A Ubiquitous Tool for the Practical Engineer Will-be-set-by-IN-TECH Here Tk ( x ) = cos(k cos−1 ( x )), k = 0, 1, ... are unknown. A spectral method is characterized by a specific way to determine these coefficients. The Chebyshev spectral methods could be implemented as - Galerkin and tau methods, where we work in the spectral space of the coefficients c = c0 , c1 , c2 , ...

Let us calculate now the differentiation matrix D such that if f is the column of the Chebyshev coefficients of a function f , then Df is the column of the Chebyshev coefficients of the df derivative function dx . , Ti = 4 2( i + 1) 2( i − 1) T0 = T1 , T1 = from where T0 = 0, Ti = 2i ( Ti−1 + Ti−3 + ... + T1 ) , i even 2 Ti = 2i ( Ti−1 + Ti−3 + ... 5T0 ) , i odd. Consequently, D is a sparse upper triangular matrix with Dii = 0, Dij = 0 for ( j − i ) even and Dij = 2j otherwise. e. the coefficients of f ( p) are D p f.

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