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.
See Also
All Wikipedia text
is available under the
terms of the GNU Free Documentation License