### Are these methods cubically convergent?

In class, I propsed viewing Newton's method x_{n+1} = x_n - f(x_n)/f'(x_n) = g(x_n) as functional iteration. If the functional iteration converges to a point, r, then from the definition of g, r must be a root of f. To show that the above Newton's method converges quadratically, it sufficed to know that g, g', and g'' exist and are continuous and g(r) = g'(r) = 0. Analogously, if x_n converges to r and g(r) = g'(r) = g''(r) = 0 then the method converges cubically. Using this, and Maple, verify that Olver's method converges cubically:
x_{n+1} = x_n - f(x_n)/f'(x_n) - 1/2 f''(x_n) f(x_n)^2/f'(x_n)^3
(This is problem 9 on page 117.) Does it always converge cubically? If not, when does it fail?

### Section 4.3

Written problems: 1, 2, 7, 8, 10, 11, 13b, 14

### Verifying convergences

We define three different versions of the first derivative of f:
Df_1(x) = ( f(x+h)-f(x) )/h
Df_2(x) = ( f(x+h)-f(x-h) )/(2h)
Df_3(x) = 4/3 ( f(x+h/2)-f(x-h/2) )/h - 1/3 ( f(x+h)-f(x-h) )/(2 h)
Choose a function and for each of the above finite-difference definitions of the first derivative, demonstrate that the errors are decreasing like the appropriate power of h.

Note, for some functions, like quadratic functions, these finite-difference approximations appear to be exactly correct. Prove this, for one of the above approximations and some sufficiently low-degree polynomial.

### Using polynomial interpolation instead of finite-differences

Use maple to help out with the grunt-work on this problem.
Instead of using finite-differences to calculate the derivative, assume that you're given the function at the points x and x+h. Find the interpolating polynomial for these points and find its derivative at x.
Now assume you're given the function at the points x-h and x+h. Again, find the interpolating polynomial for these points and find its derivative at x.
Now assume you're given the function at the points x-h, x-h/2, x+h/2, x+h. Find the interpolating polynomial for these points and find its derivative at x

### Section 5.1

Written problems: 2, 3, 7, 8, 12
Computer problems: 1, 3, 4