Polynomial Interpolation

Problem 17.4.

Given the data 

(a) Calculate f(3.4) using Newton's interpolating polynomials of order 1 through 3. Choose the sequence of the points for your estimates to attain the best possible accuracy. That is, the points should be centered around and as close as possible to the unknown. 

(b) Repeat (a) but use the Lagrange polynomial. 

Solution:

(a) Newton's interpolating polynomial. Ordering of points:

Note that based purely on the distance from the unknown (nearer to 3.4), the fourth point would be (2,5). However, because it provides better balance and is located only a little bit farther from the unknown, the point at (5,0) is chosen.

Set up divided difference table to ease the evaluation of each order of polynomial interpolation. 

First divided differences:

Second divided differences:

Third divided difference:

Thus, the divided difference table is

First order (linear) interpolation:

Second order (quadratic) interpolation:

Third order (cubic) interpolation:

(b) Lagrange interpolating polynomial.

First order interpolation:

Second order interpolation:

Third order interpolation: