Editing Lagrange polynomial (section)

==Notes==
[[File:Runge's phenomenon in Lagrange polynomials.svg|thumb|upright=1.5|Example of interpolation divergence for a set of Lagrange polynomials.]]

The Lagrange form of the interpolation polynomial shows the linear character of polynomial interpolation and the uniqueness of the interpolation polynomial.  Therefore, it is preferred in proofs and theoretical arguments.  Uniqueness can also be seen from the invertibility of the Vandermonde matrix, due to the non-vanishing of the [[Vandermonde determinant]].

But, as can be seen from the construction, each time a node ''x''<sub>''k''</sub> changes, all Lagrange basis polynomials have to be recalculated. A better form of the interpolation polynomial for practical (or computational) purposes is the barycentric form of the Lagrange interpolation (see below) or [[Newton polynomial]]s. <!-- Using [[Horner scheme|nested multiplication]] amounts to the same idea. -->

Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as [[Runge's phenomenon]]; the problem may be eliminated by choosing interpolation points at [[Chebyshev nodes]].<ref>{{cite book|title=Scientific Computing with MATLAB|volume=2|series=Texts in computational science and engineering |author-link1= Alfio Quarteroni |first1=Alfio|last1=Quarteroni|first2=Fausto|last2=Saleri|publisher=Springer|year=2003|isbn=978-3-540-44363-6|page=66|url=https://books.google.com/books?id=fE1W5jsU4zoC&pg=PA66}}.</ref>

The Lagrange basis polynomials can be used in [[numerical integration]] to derive the [[Newton–Cotes formulas]].
<!-- Lagrange interpolation is often used in [[digital signal processing]] of audio for the implementation of fractional delay [[finite impulse response|FIR]] filters (e.g., to precisely tune [[digital waveguide synthesis|digital waveguides]] in [[physical modelling synthesis]]). -->