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# set1 - \$ Set 1 Polynomial Interpolation Part 1 Kyle A...

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a39 a38 a36 a37 Set 1: Polynomial Interpolation – Part 1 Kyle A. Gallivan Department of Mathematics Florida State University Foundations of Computational Math 2 Spring 2011 1

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a39 a38 a36 a37 Interpolation Topics 1. Interpolation Overview 2. Lagrange Interpolation – Section 8.1 3. Newton Interpolation – Section 8.2 4. Complexity and Barycentric Forms – Sections 8.1, 8.2 and 8.3 5. Conditioning and Error Bounds – Sections 8.1 and 8.2 6. Hermite Interpolation – Section 8.5 7. Piecewise Interpolation and Splines – Section 8.4 and 8.8 8. Multidimensional Interpolation – Section 8.6 9. Rational Interpolation (notes) 2
a39 a38 a36 a37 References In addition to the text, the following are useful references for this topic. 1. Isaacson and Keller, Analysis of Numerical Methods, Wiley Press, 1966. 2. Bartle, The Elements of Real Analysis, Wiley, Second Edition, 1976. 3. Higham, Accuracy and Stability of Numerical Algorithms, SIAM, Second Edition, 2002. 4. Dahlquist and Bjorck, Numerical Methods, Prentice-Hall, 1974. 5. Ueberhuber, Numerical Computation, Volume 1, Springer, 1995. 3

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a39 a38 a36 a37 Polynomial Interpolation Find p n ( x ) P n | y i = p n ( x i ) 0 i n . n + 1 parameters and n + 1 constraints global, local nonsmooth, or local smooth polynomial interpolation polynomials and their derivatives are cheap to evaluate many representations of polynomials, i.e., parameterizations efficient interpolation algorithms exist and can be adapted to many circumstances: quadrature, differentiation, integration. accuracy of approximation achieved and achievable may be a problem accuracy must ultimately be considered in terms of the application exploiting the interpolating polynomial. 4
a39 a38 a36 a37 Monomial Form, Existence and Uniqueness Assume the polynomial, p n ( x ) , is taken in terms of monomials, x i p n ( x ) = α 0 + α 1 x + α 2 x 2 + · · · + α n x n Consider the constraints y 0 = α 0 + α 1 x 0 + α 2 x 2 0 + · · · + α n x n 0 y 1 = α 0 + α 1 x 1 + α 2 x 2 1 + · · · + α n x n 1 . . . y n = α 0 + α 1 x n + α 2 x 2 n + · · · + α n x n n 5

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a39 a38 a36 a37 Monomial Form, Existence and Uniqueness 1 x 0 x 2 0 . . . x n 0 1 x 1 x 2 1 . . . x n 1 . . . . . . . . . 1 x n 1 x 2 n 1 . . . x n n 1 1 x n x 2 n . . . x n n α 0 α 1 . . . α n 1 α n = y 0 y 1 . . . y n 1 y n V T a = y 6
a39 a38 a36 a37 Example ( x, y ) = braceleftbig (1 , 10) (2 , 26) (3 , 58) (4 , 112) bracerightbig 4 distinct x i implies cubic p 3 ( x ) = α 0 + α 1 x + α 2 x 2 + α 3 x 3 α 0 + α 1 1 + α 2 1 + α 3 1 = 10 α 0 + α 1 2 + α 2 4 + α 3 8 = 26 α 0 + α 1 3 + α 2 9 + α 3 27 = 58 α 0 + α 1 4 + α 2 16 + α 3 64 = 112 7

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a39 a38 a36 a37 Example 1 1 1 1 1 2 4 8 1 3 9 27 1 4 16 64 α 0 α 1 α 2 α 3 = 10 26 58 112
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set1 - \$ Set 1 Polynomial Interpolation Part 1 Kyle A...

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