L7 - Approximation by Polynomials

# 5 1 075 1 1 075 05 025 0 x 025 05 075 choice

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Unformatted text preview: s Interpolating polynomial Function −0.5 −1 0.75 1 −1 −0.75 −0.5 −0.25 0 x 0.25 0.5 0.75 Choice of representation (basis functions) Standard monomials xi : p(x) = a0 + a1 x + a2 x2 + · · · + an xn Lagrange polynomials Chebyshev polynomials References. Rao 5.4, 5.5, 5.7 MATLAB M-ﬁles. polinterp1.m lagpoly.m 1 finterp.m 1 Polynomial interpolation Polynomial of degree n: p(x) = n i=0 ai xi Basis functions: xi for i = 0, . . . , n n + 1 data points: (xj , yj ) for j = 0, . . . , n Interpolation property : p(xj ) = yj for j = 0, . . . , n Linear system: ⎡ n i=0 1 ⎢1 ⎢ ⎢1 ⎢ ⎢. ⎣. . 1 x0 x1 x2 . . . xn ai xi = yj for j = 0, . . . , n, or in matrix form j ⎤⎡ ⎤ ⎡ ⎤ n−1 2 n y0 x 0 · · · x0 x0 a0 n −1 2 n ⎥⎢ ⎥ ⎢ y1 ⎥ x1 · · · x1 x1 ⎥⎢ a1 ⎥ ⎢ ⎥ n−1 2 x 2 · · · x2 xn ⎥⎢ a2 ⎥ = ⎢ y2 ⎥ 2 ⎥⎢ ⎥ ⎢⎥ . .. . . ⎥⎢ . ⎥ ⎢ . ⎥ . . . ⎦⎣ . ⎦ ⎣ . ⎦ . . . . . . 2 n−1 n yn x n · · · xn xn an a A y A is called the Vandermonde matrix (Matlab vander) Distinct xj =⇒ A nonsingular =⇒ unique solution a for any y Weierstrass Approximation Theorem: every continuous function on a ﬁnite interval can be accurately approximated by a polynomial Example. Linear interpolation : ﬁnd the polynomial (li...
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## This note was uploaded on 05/26/2013 for the course MATHEMATIC 2089 taught by Professor Briancox during the Fall '13 term at National University of Singapore.

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