The objective of this research is to systematize the derivation of algorithms in the field of iterative linear system solving: iterative methods, preconditioners, multigrid. The approach is by extending the Formal Linear Algebra Methods Environment (FLAME), a system originally developed for deriving dense matrix algorithms.
The merit of this research lies firstly in the fact that it facilitates experimentation, since it makes derivation of new algorithms essentially simpler than the lengthy induction arguments that are traditionally necessary. Secondly, it will lead to algorithms being proved correct by the very mechanism of derivation. Finally, a complete systematization of FLAME may take the form of a symbolic system, where algorithm implementations are derived mechanically, steered by the user but otherwise autonomously, from a specification of their properties rather than from an algorithmic description.
The impact of this research will be on the computational community, since it lowers the threshold to exploring new algorithmic strategies, and on software developers, since it makes it easier to derive correct implementations of algorithms. Additionally, it will impact the way the subject of iterative linear system solving is taught, since FLAME worksheets offer a simpler and more insightful description of algorithms than is used traditionally.
