for a sparse matrix 
. There
are two classes of methods for solving linear systems: direct
and iterative methods. Gaussian elimination is a direct method,
but in this assignment we are going to use an iterative method. This
is because iterative methods work well with sparse matrices and are
easy to parallelize. The basic idea of an iterative method is that we
repeat a set of steps and each step generates a better approximation
of the final answer.
The particular iterative technique we want you to use in the assignment
is called Jacobi iteration. The basic idea is that we can first
guess at a solution for the 
values (the 
in our application).
We can then determine how bad the guess is by calculating 
and
seeing how far from 0 it is at each point. This difference is
called the residual.
The idea of the iterative technique is that we use this residual to improve
our next guess of . In particular we use the iteration:
We keep repeating this step deriving new s until our error is
small.
To make this work well we first need to adjust each row of the matrix
so that the diagonal is 
. We can do this by multiplying each
row
and value 
by 
.