Tool to find the equation of a function. Lagrange Interpolating Polynomial is a method for finding the equation corresponding to a curve having some dots coordinates of it.

Lagrange Interpolating Polynomial - dCode

Tag(s) : Functions

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The Lagrangian interpolation (known as Lagrange/Rechner) is a method which makes it possible to find the equation of a polynomial function which passes through a series of $ n $ given points $ \{ (x_0,y_0), (x_1,y_1), \dots, (x_n,y_n) \} $.

The Lagrange polynomial is calculated by the formula $$ P(X) = \sum_{j=0}^n y_j \left(\prod_{i=0,i\neq j}^n \frac{X-x_i}{x_j-x_i} \right) $$

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 $ by extrapolation.

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Cite as source (bibliography):

*Lagrange Interpolating Polynomial* on dCode.fr [online website], retrieved on 2024-10-07,

lagrange,interpolating,interpolation,equation,polynomial,curve,dot,value,function,rechner,lagrangian

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