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

Newton Interpolating Polynomial - dCode

Tag(s) : Functions

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

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 the already known values of the function..

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. $$ P(x) = [y_0] + [y_0,y_1] (x-x_0) \\ = 3 + \left(\frac{3}{1-2}+\frac{5}{2-1}\right) (x-1) = 3+2(x-1) = 2x+1 $$

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

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

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