Chapter 3  Interpolation and Polynomial Approximation


Function Interpolation by Polynomial or Piecewise Polynomial


   Lagrange interpolating polynomial :

If  are  distinct numbers and  is a function whose values are given at these numbers, then there exists a unique polynomial  of degree at most  with the property that

            for each

This polynomial is give by


, for each


   Natural and clamped cubic splines

Given a function  defined on , and a set of numbers, called nodes, , a cubic spline interpolant, , for  is a function that satisfies the following conditions:

a.       is a cubic polynomial, denoted , on the subinterval , where


b. ;

c. ;

d. ;

e. ;

f.  One of the following set of boundary conditions is satisfied:

i    free or nature boundary;

ii and  clamped boundary.




【1】         R. L. Burden and J. D. Faires, Numerical Analysis, PWS, Boston, 1993.