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

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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, . .. 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 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...
View Full Document

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.

Page1 / 12

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

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online