=================== UMSA HW Assignment - Zhang ====================

Calculating pi by Monte Carlo.

1) Write a program to simulate the children's game. Let the circle be
centered at the origin, with radius 1. The sides of the square are
hence given by x=+/-1 and y=+/-1. You will need a random number
generator. Use the one from Prof. Park, or see me (Zhang).

2). Write a program to simulate the grown-up's version of the pi game,
as discussed in class. Recall that the idea is to generate points
(x,y) uniformly distributed inside the square (see 1) by a Markov
random walk process. Start your random walk in an arbitrary position
inside the square. In each step, the walker first attempts to move to
a new position randomly inside a (smaller) square centered at its
current position.  Let the sides of this smaller square always be
parallel with the x- and y-axes, with lengths 0.8. Then a decision is
made on whether to accept the attempted move, which completes one
step.  Discard the first few hundred steps before starting your
measurement (of pi), since the random walk requires some time to
equilibrate.

Compute pi in 1) and 2), and compare with exact results. Study the
accuracy of your results as a function of the number of samples
(points). Given an equal number of samples in the measurement, which
of the two approaches tends to give the more accurate result? Why?
(You may need to run each several times to obtain a statistical
measure.)

As usual, you are encouraged to work together. Please turn in
numerical results, answers to questions, and brief descriptions
explaining your programs. Also attach the source codes.