  This is the first in a series of papers which deal with the development of
novel methods for solving a system of linear algebraic equations with a time
complexity lower than existing algorithms. The NxN system of linear equations,
Ax = b, is often solved iteratively by minimizing the corresponding quadratic
form using well known optimization techniques. The simplest of these is the
steepest descent method, whose approach to the solution is usually quite rapid
at the beginning but slows down drastically after a few iterations. This paper
investigates possible approaches which can reduce or avoid this slowing down.
The two approaches used here involve random movement of the point between
iterations, and possible matrix transformations between iterations. This paper
reports the results of computational experiments and shows the remarkable
improvement in performance of the steepest descent method that is possible. The
approaches described here do not give a practical algorithm right away.
However, they set the stage for developing practical algorithms based on SD
alone, which will be presented in later publications.
