CSE/Mathematics 451 Homework Four –Part Two

1. Hermite Interpolation
(a) Show that the following interpolation problem yields a unique cubic polynomial.

(b) Solve the interpolation problem

2. Compute f(x) = sin x at x = 0, ⇡/4, ⇡/3, ⇡/2 and produce its interpolation polynomial using Vandermonde, Newton, and Lagrange forms.

Notice that the spacing of points is uneven. You can use MATLAB to do the computations as long as you specify what you did.

3. Determine whether the following function is a cubic spline

4. Determine whether the following function is a quadratic spline

Is it a cubic spline? Explain briefly.
5. Consider Newton’s method for finding pa for a > 0 which is just the Newton iteration applied to
f(x) = x2 " a.

As shown in class, that yields the iteration

Assume the initial guess x0 satisfies x0 > 0 and let ↵ = pa.
(a) Show that

(b) Using the result of (a), show that if x0 > 0 then for n # 1

and that

(c) Using the result of (b), show that