interpolation2 - Interpolation Interpolation Error Lagrange...

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Interpolation
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• Lagrange form of interpolating polynomial. (Has a simple form and useful for the error estimation.) Defining the Lagrange polynomial by Lagrange form of interpolating polynomial is written Derive an interpolating polynomial for points, Interpolation Error Theorem : ( Interpolation Error ) If a function f is continuous on [a,b] and has n+1 continuous derivatives on (a,b), then for 8 x 2 [a,b], 9 ξ (x) 2 (a,b), such that
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• Newton form of interpolating polynomial is written namely, • Interpolation error in Newton form can be derived as follows:
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Important Theorem Theorem: Let be n times continuously differentiable on , and let be distinct points in .then there exists a number such that f [ ] b a , n x x x ......... .......... , , 1 0 [ ] b a , [ ] b a c , [ ] ( 29 ! ......... .......... , , 1 0 n c f x x x f x n n = x
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French Mathematician Charles Hermite 1822 - 1901
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Hermite Interpolation Hermite interpolation allows us to find a ploynomial that matched both function value and some of the derivative values
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Runge's phenomenon Carle David Tolmé Runge (August 30, 1856 – January 3, 1927) was a German mathematician, physicist, and spectroscopist
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Nonconvergence Polynomial interpolating an underlying continuous function at equally spaced points may not converge to function as number of data points (and hence polynomial degree) increases , as illustrated by Runge’s function.
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