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Unformatted text preview: Interpolation and Lagrange Polynomials - (3.1) 1. Polynomial Interpolation: Problem: Given n 1 pairs of data points x i , y i , i 0,1,..., n , we want to find a polynomial P k x of lowest possible degree for which P k x i y i , i 0,1,..., n . The polynomial P k x is said to interpolate the data x i , y i , i 0,1,..., n and is called an interpolating polynomial. Graphically, P k x is an approximation to f x and satisfies the conditions : P k x i f x i , i 0,1,..., n .-10-5 5 10-3-2-1 1 2 3 x y f x , -.-.- y P k x Obviously, for this example P k x is not a good approximation to f x though P k x satisfies the conditions: P k x i y i for i 0,1,2. Note that the differences between a k th degree Taylor polynomial and a k th degree interpolating polynomial are: a. P Taylor x f x at only x x and P interpolating x f x , at x x , x 1 , ..., x n . b. P Taylor requires knowledge of f U , f UU , ... but P interpolating requires f x , f x 1 , ..., f x n . 2. Lagrange Interpolating Polynomials: a. Lagrange Polynomials: For k 0,1,..., n , define L n , k x i 0, i p k n x " x i x k " x i x " x ... x " x k " 1 x " x k 1 ... x " x n x k " x ... x k " x k " 1 x k " x k 1 ... x k " x n . L n , k x are called Lagrange polynomials . For example, let x i i , i 0,1,2,3. L 3,1 x x x " 2 x " 3 1 " 1 " 2 1 " 3 1 2 x x " 2 x " 3 Observe that Lagrange Polynomials have the following properties: i. L n , k x k 1 L n , k x k x k " x ... x k " x k " 1 x k " x k 1 ......
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- Spring '08