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)