  Recent work in theoretical computer science and scientific computing has
focused on nearly-linear-time algorithms for solving systems of linear
equations. While introducing several novel theoretical perspectives, this work
has yet to lead to practical algorithms. In an effort to bridge this gap, we
describe in this paper two related results. Our first and main result is a
simple algorithm to approximate the solution to a set of linear equations
defined by a Laplacian (for a graph $G$ with $n$ nodes and $m \le n^2$ edges)
constraint matrix. The algorithm is a non-recursive algorithm; even though it
runs in $O(n^2 \cdot \polylog(n))$ time rather than $O(m \cdot polylog(n))$
time (given an oracle for the so-called statistical leverage scores), it is
extremely simple; and it can be used to compute an approximate solution with a
direct solver. In light of this result, our second result is a straightforward
connection between the concept of graph resistance (which has proven useful in
recent algorithms for linear equation solvers) and the concept of statistical
leverage (which has proven useful in numerically-implementable randomized
algorithms for large matrix problems and which has a natural data-analytic
interpretation).
