# Lagrange interpolation formula

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A formula for obtaining a polynomial of degree (the Lagrange interpolation polynomial) that interpolates a given function at nodes : (1)

When the are equidistant, that is, , using the notation one can reduce (1) to the form (2) In the expression (2), called the Lagrange interpolation formula for equidistant nodes, the coefficients of the are called the Lagrange coefficients.

If has a derivative of order on the interval , if all interpolation nodes lie in this interval and if for any point one defines then a point exists such that where If the absolute value of the derivative is bounded on by a constant and if the interpolation nodes are chosen such that the roots of the Chebyshev polynomial of degree are mapped into these points under a linear mapping from onto , then for any one has If the interpolation nodes are complex numbers and lie in some domain bounded by a piecewise-smooth contour , and if is a single-valued analytic function defined on the closure of , then the Lagrange interpolation formula has the form where The Lagrange interpolation formula for interpolation by means of trigonometric polynomials is: which is a trigonometric polynomial of order having prescribed values at the given nodes .

The formula was proposed by J.L. Lagrange in 1795.

How to Cite This Entry:
Lagrange interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lagrange_interpolation_formula&oldid=17497
This article was adapted from an original article by L.D. KudryavtsevM.K. Samarin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article