Homework 3: Root finding

Due Feb. 15

1. Write a program to find the roots of one-dimensional equations using a) Bisection method and b) Newton-Raphson method.

   Write your code in a flexible manner so that you can use it later for other problems. Choose your stopping criteria carefully. You should balance the desire to obtain as accurate a result as possible with the amount of computation involved.

2. Test your implementations on the following equations:
   1. $x^2 = a$ (choose an $a$ and solve for $x$).

   2. Kepler's equation:
      
      $$M = E - e \sin(E).$$
      
      
      Choose $M = 1.5$ and $e = 0.5$ and solve for $E$ in the interval $0 \le E \le 2 \pi$. Try it again for $M = 1.5$ and $e = 0.9$, but be careful where you start the Newton-Raphson.

   For each attempt, plot the ``trajectory'' of the iterations by displaying each iterate on a plot of $x$ vs. $f(x)$ and connecting successive points. That is, connect the point $(x_0, f(x_0))$ to $(x_1, f(x_1))$, and so on.

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