# Lagrange interpolation formula

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.

#### References

[1] | I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian) |

[2] | N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) |

#### Comments

#### References

[a1] | P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126 |

[a2] | L.W. Johnson, R.D. Riess, "Numerical analysis" , Addison-Wesley (1977) |

[a3] | G.M. Phillips, P.J. Taylor, "Theory and applications of numerical analysis" , Acad. Press (1973) |

**How to Cite This Entry:**

Lagrange interpolation formula.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Lagrange_interpolation_formula&oldid=17497