Chapter 3 Interpolation and Polynomial Approximation
Interpolation for Monotonically Increasing Discrete Nodes
Lagrange interpolation polynomial：
In the software, given , with being monotically increasing, then there exists a unique polynomial of degree at most with the property that
, for each
This polynomial is give by
where
, for each
Hermite interpolation polynomial：
In the software, given , with being monotically increasing, and being equal to the slope of the light blue dash bar associated with each , then there exists a unique polynomial of degree at most agreeing with and at , where
where ,
and
denotes the th Lagrange coefficient polynomail of degree .
Natural cubic spline and Clamped cubic spline：
In the software, given , with being monotically increasing, 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）,
where and are equal to the slope of the light blue
dash bar associated with and .
References:
【1】
R. L. Burden and J. D. Faires, Numerical Analysis, PWS,