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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 27 October 24, 2011 Prof. Kamesh Madduri Class Overview: Interpolation Chebyshev polynomials Chebyshev points Issues with higher order polynomial interpolation Runges function 2 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Monomial, Lagrange, and Newton Interpolation Orthogonal Polynomials Accuracy and Convergence Chebyshev Polynomials k th Chebyshev polynomial of first kind, defined on interval [ 1 , 1] by T k ( t ) = cos( k arccos( t )) are orthogonal with respect to weight function (1 t 2 ) 1 / 2 First few Chebyshev polynomials are given by 1 , t, 2 t 2 1 , 4 t 3 3 t, 8 t 4 8 t 2 + 1 , 16 t 5 20 t 3 + 5 t, . .. Equioscillation property : successive extrema of T k are equal in magnitude and alternate in sign, which distributes error uniformly when approximating arbitrary continuous function Michael T. Heath Scientific Computing 30 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Monomial, Lagrange, and Newton Interpolation Orthogonal Polynomials Accuracy and Convergence Chebyshev Basis Functions < interactive example > Michael T. Heath Scientific Computing 31 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Monomial, Lagrange, and Newton Interpolation Orthogonal Polynomials Accuracy and Convergence Chebyshev Points Chebyshev points are zeros of T k , given by t i = cos...
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This note was uploaded on 01/19/2012 for the course CMPSC 451 taught by Professor Staff during the Spring '08 term at Pennsylvania State University, University Park.
 Spring '08
 staff

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