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lecture27

# lecture27 - CMPSC/MATH 451 Numerical Computations Lecture...

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CMPSC/MATH 451 Numerical Computations Lecture 27 October 24, 2011 Prof. Kamesh Madduri

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Class Overview: Interpolation Chebyshev polynomials Chebyshev points Issues with higher order polynomial interpolation Runge’s 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, . . . Equi-oscillation 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

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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 (2 i - 1) π 2 k , i = 1 , . . . , k or extrema of T k , given by t i

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lecture27 - CMPSC/MATH 451 Numerical Computations Lecture...

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