Systematic Generation of Algorithms for Iterative Methods

TL;DR

This paper extends the FLAME methodology to systematically generate provably correct algorithms for iterative methods like Conjugate Gradient from formal matrix descriptions.

Contribution

It introduces an extension of the FLAME methodology to automate the derivation of iterative algorithms, previously applied only to direct methods.

Findings

Generated algorithms are provably correct.

The methodology can be fully automated.

Applicable to iterative methods like Conjugate Gradient.

Abstract

The FLAME methodology makes it possible to derive provably correct algorithms from a formal description of a linear algebra problem. So far, the methodology has been successfully used to automate the derivation of direct algorithms such as the Cholesky decomposition and the solution of Sylvester equations. In this thesis, we present an extension of the FLAME methodology to tackle iterative methods such as Conjugate Gradient. As a starting point, we use a formal description of the iterative method in matrix form. The result is a family of provably correct pseudocode algorithms. We argue that all the intermediate steps are sufficiently systematic to be fully automated.