A concise introduction to computational methods for solving large linear systems of equations. This is the only textbook that treats iteration methods for symmetric linear systems from a polynomial point of view. This particular feature enables readers to understand the convergence behaviour and subtle differences of the various schemes, which are useful tools for the design of powerful preconditioners. First published nearly fifteen years ago, this book continues to be useful to the mathematical, scientific and engineering communities as a presentation of what appear to be the most efficient methods for symmetric linear systems of equations. To help potential users of numerical iteration algorithms design schemes for their particular needs, the author provides MATLAB (R) code on a supplementary web page to serve as a guideline. The code not only solves the linear system but also computes the underlying residual polynomials, illustrating the convergence behaviour of the given linear system.