L7 - Approximation by Polynomials - Frances Kuo MATH2089...

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MATH2089 Numerical Methods Frances Kuo § 7 Approximation by polynomials Motivation. § 6 Data fitting by least squares Lots of data with possible measurement errors Take line/curve of ‘best fit’ -2 0 2 4 6 8 10 12 1 2 3 4 5 6 7 8 9 Linear least squares -3 -2 -1 0 1 2 3 4 -100 -80 -60 -40 -20 0 20 Polynomial interpolation § 7 Approximating a complicated function by polynomial interpolation Assume function values are known exactly Fit polynomial through data points exactly Degree n ⇐⇒ n + 1 points Benefits: easily evaluated (Horner’s method), differentiated, integrated Example scenario: instead of evaluating a complicated function at 100 points, evaluate it only at 10 points, fit a polynomial through those 10 points, and approximate the remaining 90 points using the polynomial Key concepts. Choice of interpolation points Equally spaced points – large error near end points Chebyshev points minimize the error -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 -1 -0.5 0 0.5 1 x Interpolating 1 / (1 + 25 x 2 ) by polynomial of degree 8 at equally spaced points Interpolation points Interpolating polynomial Function -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 -1 -0.5 0 0.5 1 x Interpolating 1 / (1 + 25 x 2 ) by polynomial of degree 8 at Chebyshev points Interpolation points Interpolating polynomial Function
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