The Lagrange interpolating polynomial is the Polynomial of degree which passes through the points
, , ..., . It is given by

(1) |

(2) |

(3) |

For points,

(4) | |||

(5) |

Note that the function passes through the points , as can be seen for the case ,

(6) | |||

(7) | |||

(8) |

Generalizing to arbitrary ,

(9) |

The Lagrange interpolating polynomials can also be written using

(10) | |||

(11) | |||

(12) |

so

(13) |

Lagrange interpolating polynomials give no error estimate. A more conceptually straightforward method for calculating them is Neville's Algorithm.

**References**

Abramowitz, M. and Stegun, C. A. (Eds.).
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 878-879 and 883, 1972.

Beyer, W. H. (Ed.) *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, p. 439, 1987.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. ``Polynomial Interpolation and Extrapolation''
and ``Coefficients of the Interpolating Polynomial.'' §3.1 and 3.5 in
*Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed.* Cambridge, England:
Cambridge University Press, pp. 102-104 and 113-116, 1992.

© 1996-9

1999-05-26