Lagrange Interpolating Polynomial

Lagrange polynomials (also called Lagrange/Rechner) are computed using the formula:

$$ P(X)=\sum_{j=0}^n y_j \left(\prod_{i=0,i\neq j}^n \frac{X-x_i}{x_j-x_i} \right) $$

with $ P(X) $ the Lagrange polynomial and the dots $ (x_0, y_0),\dots,(x_n, y_n) $ and $ x_i $ distinct.

From the points whose coordinates are known, the lagrange polynomial calculator can thus predict other points based on the assumption that the curve formed by these points is derived from a polynomial equation.

dCode allows to use the Lagrangian method for interpolating a Polynomial and finds back the original equation using known points (x,y) values.

Example: By the knowledge of the points $ (x,y) $ : $ (0,0),(2,4),(4,16) $ the Polynomial Lagrangian Interpolation method allows to find back the équation $ y = x^2 $. Calculations details step by step : $$ P(x) = 0 \times \frac{(x-2)}{(0-2)} \frac{(x-4)}{(0-4)} + 4 \times \frac{(x-0)}{(2-0)} \frac{(x-4)}{(2-4)} + 16 \times \frac{(x-0)}{(4-0)} \frac{(x-2)}{(4-2)} \\ = 4 \times \frac{x}{2}\frac{(x-4)}{(-2)} + 16 \times \frac{x}{4}\frac{(x-2)}{2} \\ = -x(x-4)+2x(x-2) \\ = -x^2+4x+2x^2-4x \\ = x^2
$$ Once deducted, the interpolating function $ f(x) = x^2 $ allows to estimate the value for $ x = 3 $, here $ f(x) = 9 $.

The Lagrange interpolation method allows a good approximation of polynomial functions.

There are other interpolation formulas (rather than Lagrange/Rechner) such as Neville interpolation also available online on dCode.

Since the complexity of the calculations increases with the number of points, the program is automatically limited (with distinct x-values in the rational number set Q).

From a list of numbers, the Lagrange interpolation allows to find an equation for $ f(x) $. Using this equation with a new value of $ x $, it is possible to calculate the image of $ x $ by $ f $.