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)