By Claude Brezinski
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Additional resources for Algorithmes d'acceleration de la convergence: etude numerique
Xr −1 ] − [x2 . . xr −1 xr ] . 18) They are invariant against permutation of the arguments which can be seen from the explicit formula 18 2 Interpolation r f (xk ) . i =k (x k − x i ) [x1 x2 . . 21) · · · + [xn xn−1 . . x0 ](x − x0 )(x − x1 ) · · · (x − xn−1 ), and the function q(x) = [x xn · · · x0 ](x − x0 ) · · · (x − xn ). 22) Obviously q(xi ) = 0, i = 0 · · · n, hence p(x) is the interpolating polynomial. 3 Interpolation Error The error of the interpolation can be estimated with the following theorem: If f (x) is n +1 times differentiable then for each x there exists ξ within the smallest interval containing x as well as all of the xi with n q(x) = (x − xi ) i=0 f (n+1) (ξ ) .
34) The most important case is the cubic spline which is given in the interval xi ≤ x < xi+1 by pi (x) = αi + βi (x − xi ) + γi (x − xi )2 + δi (x − xi )3 . 35) We want to have a smooth interpolation and assume that the interpolating function and their first two derivatives are continuous. Hence we have for the inner boundaries 22 2 Interpolation i = 0, . . , n − 1, pi (xi+1 ) = pi+1 (xi+1 ), pi (xi+1 ) = pi+1 (xi+1 ), pi (xi+1 ) = pi+1 (xi+1 ). 38) We have to specify boundary conditions at x0 and xn .
3 Spline Interpolation Polynomials are not well suited for interpolation over a larger range. Often spline functions are superior which are piecewise defined polynomials [6, 7]. The simplest case is a linear spline which just connects the sampling points by straight lines: yi+1 − yi (x − xi ), xi+1 − xi s(x) = pi (x) where xi ≤ x < xi+1 . 34) The most important case is the cubic spline which is given in the interval xi ≤ x < xi+1 by pi (x) = αi + βi (x − xi ) + γi (x − xi )2 + δi (x − xi )3 . 35) We want to have a smooth interpolation and assume that the interpolating function and their first two derivatives are continuous.