A function fx is known at n points x0 x1 xn 1 What is the d

A function f(x) is known at n points x_0, x_1, ..., x_n -1. What is the degree L_n of the Lagrange interpolating polynomial P_n(x) that interpolates the function at these n point? If the values of the derivative f\'(x_j) are also known, we can form the Hermite interpolant of f(x). What is the degree H_n of the Hermite interpolating polynomial Q_n(x) that interpolates the function at these n points?

Solution

The general rule to recall: The highest possible degree of the interpolating polynomial is one less than the Number of constraints required to form them.

a. The degree is (atmost) (n-1).

Reason: Total no of constraints is n. Indeed, to form the lagrange interpolant from set of n points we form (n-1) products and the interpolating polynomial is the sum of these (n-1) degree polynomials. Hence the degree is atmost (n-1).

b. To develop the Hermite interpolant where n no. of points are given and the k-the order derivatives are given then total no. of constraints is n(k+1). Hence the degree is n(k+1)-1.

Here k=1.

Hence the required degree is (atmost) 2n-1.

#Note

Lagrange interpolating polynomial is the special case of Hermite interpolating polynomial where k=0. i.e

No derivative has matched. Hence the degree using b. is (0+1)n-1=n-1 which is precisely the answer to a.