Unformatted text preview: Polynomial Interpolation
Givenf0 , f1 , . . . , fN , find a polynomial P (x) of degree N so that P (xi ) = fi . This problem has a unique solution and f (N +1) () x f (x) - P (x) = (x - x0 )(x - x1 ) (x - xN ) (N + 1)! for some x0 x xN . To show the polynomial is unique, write P (x) = c0 +c1 (x-x0 )+c2 (x-x0 )(x-x1 )+ +cN (x-x0 )(x-x1 ) (x-xN ) Then f0 = P (x0 ) = c0 determines c0 , f1 = P (x1 ) = c0 +c1 (x1 -x0 ) determines c1 , etc. Lagrangian Interpolation
N = 3 example: P (x) = (x - x0 )(x - x2 )(x - x3 ) (x - x1 )(x - x2 )(x - x3 ) f0 + f1 + (x0 - x1 )(x0 - x2 )(x0 - x3 ) (x1 - x0 )(x1 - x2 )(x1 - x3 ) (x - x0 )(x - x1 )(x - x3 ) (x - x0 )(x - x1 )(x - x2 ) f2 + f3 (x2 - x0 )(x2 - x1 )(x2 - x3 ) (x3 - x0 )(x3 - x1 )(x3 - x2 ) General formula:
N P (x) =
k=0 (x - xj ) fk j=k (xk - xj ) ...
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- Fall '11
- Numerical Analysis, Polynomial interpolation, ) fk j=k