Sequential Monte Carlo for linear systems - a practical summary

This paper has been written in response to many requests for a practical guide to the use of the technique of sequential Monte Carlo in the fast numerical solving of large systems of linear equations. This method, which I have used with considerable success to solve such problems, improving the tricks of the trade as I learned more about it, has suffered from some neglect through the mathematical difficulty, for some of those who are more interested in using the tool than in thinking about it, of some of the theoretical aspects of rigorously proving its validity, which - at this juncture - is no longer in question. I hope that I have now closed this gap in the related literature.