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Newton Interpolating Polynomial

Tool to find the equation of a curve via Newton's algorithm. Newtonian interpolation is a polynomial approximation allowing to obtain the Lagrange polynomial as equation of the curve by knowing its points.

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Newton Interpolating Polynomial -

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Newton Interpolating Polynomial

Newtonian Interpolation Calculator


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Extrapolation


Tool to find the equation of a curve via Newton's algorithm. Newtonian interpolation is a polynomial approximation allowing to obtain the Lagrange polynomial as equation of the curve by knowing its points.

Answers to Questions

How to find the equation of a curve using Newton algorithm?

dCode allows to use Newton's method for Polynomial Interpolation in order to find the equation of the polynomial (identical to Lagrange) in the Newton form.

From $ n + 1 $ known points $ (x_i, y_i) $, the Newton form of the polynomial is equal to $$ P(x)= [y_0] + [y_0,y_1] (x-x_0) + \ldots + [y_0,\ldots ,y_n] (x-x_0) \ldots (x-x_{n-1}) $$

with the notation $ [y_i] $ for divided difference.

Example: Curve whose points (1,3) and (2,5) are known. $$ P(x) = [y_0] + [y_0,y_1] (x-x_0) \\ = 3 + \left(\frac{3}{1-2}+\frac{5}{2-1}\right) (x-1) = 3+2(x-1) = 2x+1 $$

What are the limits for Interpolating with Newton?

Newton Divided Differences are noted $ [y_i] $ and computed by the formula $$ [y_0,\dots ,y_k]=\sum_{j=0}^k {\frac{y_j}{\prod_{0\leq i\leq k,\,i\neq j}(x_j-x_i)}} $$

NB: If $ k = 0 $, then the product $ \prod(x_j-x_i) = 1 $ (empty product)

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