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Neville Interpolating Polynomial

Tool to find a curve equation via the Neville-Aikten algorithm. The Neville interpolating polynomial method is a polynomial approximation to obtain the equation of a curve by knowing some coordinates of it.

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Neville Interpolating Polynomial -

Tag(s) : Mathematics

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# Neville Interpolating Polynomial

## Interpolation of Polynomial by Neville

Tool to find a curve equation via the Neville-Aikten algorithm. The Neville interpolating polynomial method is a polynomial approximation to obtain the equation of a curve by knowing some coordinates of it.

### How to find the equation of a curve using Neville algorithm?

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

### What are the limits for Interpolating with Neville?

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