interpolation1 - Interpolation Lagrange...

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Interpolation
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Lagrange Interpolation . . . . . (x ,f ) (x 2 ,f 2 ) (x 3 ,f 3 ) (x 4 ,f 4 ) (x 5 ,f 5 ) f(x) (x 1 ,f 1 ) P(x)
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Introduction The interpolation problem Given values of an unknown function f(x) at values x = x 0 , x 1 , …, x n , find approximate values of f(x) between these given values Polynomial interpolation – Find n th-order polynomial p n (x) that approximates the function f(x) and provides exact agreement at the n node points: ) ( ) ( ), ( ) ( ), ( ) ( 1 1 0 0 n n n n n x f x p x f x p x f x p = = = L The polynomial p n (x) is unique Interpolation: evaluate p n (x) for x 0 <= x <= x n Extrapolation: evaluate p n (x) for x 0 > x > x n
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Purposes for Interpolation • Plotting smooth curve through discrete data points • Quick and easy evaluation of mathematical function • Replacing “difficult” function by “easy” one • “Reading between the lines” of table • Differentiating or integrating tabular data
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Functions for Interpolation Some families of functions commonly used for interpolation include ° Polynomials ° Piecewise polynomials ° Trigonometric functions ° Exponentials ° Rational functions We focus on interpolation by polynomials and piecewise polynomials.
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Polynomial Interpolation ) ( ... ) ( ) ( ) ( 2 2 1 1 x u a x u a x u a x P m m m + + + = ) 1 ( Y A V = V = u x u x u x u x u x u x u x u x u x n n n n n n 1 1 2 1 1 1 2 2 2 2 1 2 ( ) ( ) ... ( ) ( ) ( ) ... ( ) ... ... ... ... ( ) ( ) ... ( ) Vandermonde matrix n=m m i y x P i i m ,..., 2 , 1 ) ( , = = A Y = = a a a y y y n n 1 2 1 2 ... , ... Y V A 1 - = Extrapolation
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Polynomial Interpolation Simplest and commonest type of interpolation uses polynomials. Unique polynomial of degree at most n – 1 passes through n data points ( t i , y i ), i = 1, . . . , n , where t i are distinct. There are many ways to represent or compute polynomial, but in theory all must give same result.
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Lagrange
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Lagrange Interpolation For given set of data points ( t i , y i ), i = 1, . . ., n , Lagrange basis functions are given by ) x )...(x x )(x x )...(x x )(x x (x ) x )...(x x )(x x )...(x x )(x x (x (x) l n j 1 j j 1 j j 2 j 1 j n 1 j 1 j 2 1 j - - - - - - - - - - = + - + - ) x (x n - which means that matrix of linear system Ax = y is identity. Thus, Lagrange polynomial interpolating data points ( t i , y i ) is given by p n –1 ( t ) = y 1 l 1 ( t ) + y 2 l 2 ( t ) + ... + y n l n ( t ). ) x (x k j k j k 1, k - = = = π
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Examples: Lagrange Interpolation 1- Use Lagrange interpolation to find interpolating polynomial for three data points (–2, –27), (0, –1), 1, 0). Lagrange polynomial of degree two interpolating three points ( t 1 , y 1 ), ( t 2 , y 2 ), ( t 3 , y 3 ) is .
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