M371 - Alexiades

**Problem Set 5: Interpolation and Numerical Differentiation**

Consider the four data points:

They are assumed to be values of some function f(x).
1. *Interpolation*
a. Construct the 3rd degree interpolating polynomial P_{3}(x) using the Lagrange basis.
b. Construct the 3rd degree interpolating polynomial P_{3}(x) using the Newton basis.
c. Show that they produce the same polynomial.
d. Plot the data and the interpolant on the same plot.
e. Use Matlab's ** pchip** interpolant, evaluate it at x=1.5 and compare with P_{3}(1.5).
2. *Derivatives*
Approximate f**'**(1) (the first derivative of f at x=1)
a. by a forward difference.
b. by a backward difference.
c. by a centered difference.
d. which one would you "trust" most ? why ?
e. Another approximation of f**'**(1) would be P**'**_{3}(1)
(the derivative of the cubic interpolant you constructed above).
Find the value.
3.a. Construct the quadratic polynomial P_{2}(x) interpolating the first 3 points.
b. P**'**_{2}(1) is yet another approximation of f'(1).
Does it agree with any of the above ?
c. Try to find the slope of the **pchip** interpolant at x=1.
Compare with the other approximations to f'(1).