Neville Interpolating Polynomial

dCode implement the method of Neville for Polynomial interpolation to find an equation by knowing some of its points \( (x_i, y_i) \).

Example: Points (0,0),(2,4),(4,16) can be interpolated to find the original equation : x^2

The interpolated polynomial is calculated by the Neville algorithm for n distinct points. (This algorithm can be represented as a pyramid, at each step a term disappears until having a single final result).

- Create polynomials \( P_i \) of degree 0 for each point \( x_i, y_i \) with \( i = 1,2,...,n \), this is equivalent to \( P_i (x) = y_i \).

Example: \( P_1 = 0 \), \( P_2 = 4 \), \( P_3 = 16 \)

- For each consecutive \( P_i \) and \( P_j \) calculate $$ P_{ij}(x) = \frac{(x_j-x)P_i(x) + (x-x_i)P_j(x)}{x_j-x_i} $$

Example: \( P_{12} = \frac{(2-x)0 + (x-0)4}{2-0} = 2x \), \( P_{23} = \frac{(4-x)4 + (x-2)16}{4-2} = \frac{16-4x+16x-32}{2} = 6x-8 \)

- Repeat this last step until having a single polynomial.

Example: \( P_{1(2)3} = \frac{(4-x)(2x) + (x-0)(6x-8)}{4-0} = \frac{8x-2x^2 + 6x^2 -8x}{4} = x^2 \)

Calculations are costful so the program is limited to 25 points with distinct x-coordinate in the set Q.