lecture25 - CMPSC/MATH 451 Numerical Computations Lecture...

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Unformatted text preview: CMPSC/MATH 451 Numerical Computations Lecture 25 October 19, 2011 Prof. Kamesh Madduri Class Overview: Interpolation Lagrange basis Newton basis 2 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Monomial, Lagrange, and Newton Interpolation Orthogonal Polynomials Accuracy and Convergence Lagrange Interpolation For given set of data points ( t i ,y i ) , i = 1 ,. .. ,n , Lagrange basis functions are defined by ` j ( t ) = n Y k =1 ,k 6 = j ( t- t k ) / n Y k =1 ,k 6 = j ( t j- t k ) , j = 1 ,. .. ,n For Lagrange basis, ` j ( t i ) = 1 if i = j if i 6 = j , i,j = 1 ,. .. ,n so matrix of linear system Ax = y is identity matrix Thus, Lagrange polynomial interpolating data points ( t i ,y i ) is given by p n- 1 ( t ) = y 1 ` 1 ( t ) + y 2 ` 2 ( t ) + + y n ` n ( t ) Michael T. Heath Scientific Computing 18 / 56 Interpolation Polynomial Interpolation Piecewise Polynomial Interpolation Monomial, Lagrange, and Newton Interpolation Orthogonal Polynomials Accuracy and Convergence Lagrange Basis Functions < interactive example > Lagrange interpolant is easy to determine but more expensive to evaluate for given argument, compared with monomial basis representation Lagrangian form is also more difficult to differentiate, integrate, etc....
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lecture25 - CMPSC/MATH 451 Numerical Computations Lecture...

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