In mathematics, Neville's algorithm is an algorithm used for polynomial interpolation. Given n + 1 points, there is a unique polynomial of degree n which goes through the given points. Neville's algorithm evaluates this polynomial. Neville's algorithm is based on the Newton form of the interpolating polynomial and the recursion relation for the divided differences. It is similar to Aitken's algorithm, which is nowadays not used.
The algorithm
Given a set of n+1 data points (xi, yi) where no two xi are the same, the interpolating polynomial is the polynomial p of degree at most n with the property
- p(xi) = yi for all i = 0,…,n
This polynomial exists and it is unique. Neville's algorithm evaluates the polynomial at some point x. Let pi,j denote the polynomial of degree j − i which goes through the points (xk, yk) for k = i, i + 1, …, j. The pi,j satisfy the recurrence relation
-
<math> p_{i,i}(x) = y_i, \, </math> <math> 0 \le i \le n, \, </math> <math> p_{i,j}(x) = \frac{(x-x_j)p_{i,j-1}(x) + (x_i-x)p_{i+1,j}(x)}{x_i-x_j}, \, </math> <math> 0\le i < j \le n. \, </math>
This recurrence can calculate p0,n(x), which is the value being sought. This is Neville's algorithm. For instance, for n = 4, one can use the recurrence to fill the triangular tableau below from the left to the right.
-
<math> p_{0,0}(x) = y_0 \, </math> <math> p_{0,1}(x) \, </math> <math> p_{1,1}(x) = y_1 \, </math> <math> p_{0,2}(x) \, </math> <math> p_{1,2}(x) \, </math> <math> p_{0,3}(x) \, </math> <math> p_{2,2}(x) = y_2 \, </math> <math> p_{1,3}(x) \, </math> <math> p_{0,4}(x) \, </math> <math> p_{2,3}(x) \, </math> <math> p_{1,4}(x) \, </math> <math> p_{3,3}(x) = y_3 \, </math> <math> p_{2,4}(x) \, </math> <math> p_{3,4}(x) \, </math> <math> p_{4,4}(x) = y_4 \, </math>
This process yields p0,4(x), the value of the polynomial going through the n + 1 data points (xi, yi) at the point x. This algorithm needs O(n2) elementary arithmetical operations.
References
- Press, William; Saul Teukolsky, William Vetterling and Brian Flannery (1992). "§3.1 Polynomial Interpolation and Extrapolation", Numerical Recipes in C. The Art of Scientific Computing, 2nd edition, Cambridge University Press. ISBN 978-0-521-43108-8, doi:10.2277/0521431085.
