**Homework 5 **

1) Rootfinding:

Use a rootfinding routine of your choice (i.e. the bisection, Newton, or secant
method out of Pang's book) to numerically solve the transcendental equation for
an Ising ferromagnet, **m=tanh(m/t) **. Make a table and a plot of
**m(t) **with **t** **= 0.05, ..., 1.0** in
increments of **0.05**.** (30 points) **

2) Monte Carlo Integration:

Modify Pang's program 'mcds.f' (or the equivalent program in C) to integrate
the function **f(x,y,z)=y*z*z*exp(x*y*z)** numerically with respect to **x**,
**y**, and **z**. All lower integration limits are 0, and all upper
limits are 1. Then integrate the same function with all lower limits equal to 1
and all upper limits equal to 1.5. Please hand in a printout of your program
for this second integral. (It might be useful to check your result for the
second integral with Mathematica). **(30 points) **

3) Multi-Dimensional Monte Carlo
Integration:

Solve Pang's problem 9.3. The integrand is symmetric and decays exponentially
for large x, y, and z. Please choose integration limits 0 and 10, and a weight
function of the form **x*exp(-x)*y*exp(-y)*z*exp(-z)**. Compare your result
to the exact analytical solution. **(40 points) **