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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 26 October 21, 2011 Prof. Kamesh Madduri Class Overview: Interpolation • Newton method, evaluating coefficients using Divided differences • Orthonormal polynomials • Legendre polynomials 2 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Monomial, Lagrange, and Newton Interpolation Orthogonal Polynomials Accuracy and Convergence Divided Differences Given data points ( t i ,y i ) , i = 1 ,. .. ,n , divided differences , denoted by f [ ] , are defined recursively by f [ t 1 ,t 2 ,. .. ,t k ] = f [ t 2 ,t 3 ,. .. ,t k ] f [ t 1 ,t 2 ,. .. ,t k 1 ] t k t 1 where recursion begins with f [ t k ] = y k , k = 1 ,. .. ,n Coefficient of j th basis function in Newton interpolant is given by x j = f [ t 1 ,t 2 ,. .. ,t j ] Recursion requires O ( n 2 ) arithmetic operations to compute coefficients of Newton interpolant, but is less prone to overflow or underflow than direct formation of triangular Newton basis matrix Michael T. Heath Scientific Computing 26 / 56 Interpolation...
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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 Penn State.
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