### Abstract

We formalized three algorithms for polynomial interpolation over arbitrary
fields: Lagrange's explicit expression, the recursive algorithm of Neville
and Aitken, and the Newton interpolation in combination with an efficient
implementation of divided differences. Variants of these algorithms for
integer polynomials are also available, where sometimes the interpolation
can fail; e.g., there is no linear integer polynomial

BSD License*p*such that*p(0) = 0*and*p(2) = 1*. Moreover, for the Newton interpolation for integer polynomials, we proved that all intermediate results that are computed during the algorithm must be integers. This admits an early failure detection in the implementation. Finally, we proved the uniqueness of polynomial interpolation.The development also contains improved code equations to speed up the division of integers in target languages.

### Depends On

### Used by

- Three_Circles
- Formal_Puiseux_Series
- Gauss_Sums
- Count_Complex_Roots
- Deep_Learning
- Berlekamp_Zassenhaus
- Polynomial_Factorization