Homework 2 -- Due Tuesday, 25 January

For this problem, you will implement a 2D finite difference method for 
solving Poisson's equation -Delta u = f  using the 5 point stencil on 
an n by n mesh and compare 1D and 2D data decompositions.

1.) Analyze the communication and computation costs for using a 
one-dimensional data distribution and using a two-dimensional data 
distribution.  How does the ratio scale with n and p?

2.) Implement the matrix-vector product for both data decompositions.  
You can test your result by taking some smooth function (say 
sin(pi x) sin(pi y), evaluating at all the grid points, and seeing if 
you get something close to the Laplacia)

3.) Solve the system -Delta u = f , u = 0 on the boundary with
f = -pi * pi * sin(pi x) * sin( pi y ) using conjugate gradient 
iterations (explained in class on 1/20)

4.) Perform a scaling study, with fixed work per processor (for a range 
of different work) and for a fixed (very big) problem size with variable 
number of processors.

Some notes to consider:
1.) You might consider learning how to use gnuplot to make timing plots 
(there are Python bindings installed for Gnuplot -- you can import 
Gnuplot and learn about it, or you can print out files from one node)
2.) There is a simple graphics package available as part of MPI.  It can 
plot points and lines, and may be appropriate for use.
3.) There are now 6 debian unstable machines -- you should add 
california and kimball to your hosts file.

In either case, you can perform your scaling studies by writing an 
appropriate loop and having your code print a table (gnuplot file, latex 
table, whatever).  Be smart -- it saves time.