#### 43.2.1 Overview

Many practical iterative methods of solving large n-by-n sparse linear systems involve projecting an approximate solution for x onto a subspace of {\bf R}^n. If we define a m-dimensional subspace {\cal K} as the subspace of approximations to the solution x, then m constraints must be imposed to determine the next approximation. These m constraints define another m-dimensional subspace denoted by {\cal L}. The subspace dimension m is typically chosen to be much smaller than n in order to reduce the computational effort needed to generate the next approximate solution vector. The many iterative algorithms which exist differ mainly in their choice of {\cal K} and {\cal L}.