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
where
, 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).
References:
【1】
R. L. Burden and J. D. Faires, Numerical Analysis, PWS,