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Lagrange polynomial

Lagrange polynomials (named after their inventor Joseph Louis Lagrange) are polynomials used for polynomial interpolation. Rather than solving the huge Vandermonde matrix equation obtained in the polynomial interpolation by Gauss-Jordan elimination, we may prepare the polynomial by just inserting the function values into a combination of equations created using the variable e.g. <math>t</math> and the given points and respective values. Given a data set:

<math>(t_1,y_1), (t_2,y_2), ... (t_n,y_n)</math>

where no two <math>t_i</math> are the same, we assume the <math>y_n</math>:s are values of a function, <math>f</math>, at some certain <math>t</math>-points named <math>f(t_i) = f_i</math>. We know from Weierstrass' theorem[?] that there exists a unique polynomial of degree <math>n-1</math> that pass through all these points, and we write it thusly:

<math>P_n(x) = \sum_{i=1}^{n+1} f_i p_i(t)</math>

<math>p_i(t) = \frac{(t-t_1) \ldots (t-t_{i-1})(t-t_{i+1}) \ldots (t-t_{n+1})}{(t_i-t_1) \ldots (t_i-t_{i-1})(t_i-t_{i+1}) \ldots (t_i-t_{n+1})}</math>

Lagrange polynomials can be used for e.g. deriving formulas for numerical integration. For calculating the value in some given point <math>t</math> not in the data set from the data set, newton polynomials are commonly preferred.

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