Representing linear algebra algorithms in code: The FLAME application programming interfaces

In this article, we present a number of Application Program Interfaces (APIs) for coding lin-ear algebra algorithms. On the surface, these APIs for the MATLAB M-script and C program-ming languages appear to be simple, almost trivial, extensions of those languages. Yet with them, the task of programming and maintaining families of algorithms for a broad spectrum of linear algebra operations is greatly simplified. In combination with our Formal Linear Alge-bra Methods Environment (FLAME) approach to deriving such families of algorithms, dozens of algorithms for a single linear algebra operation can be derived, verified to be correct, im-plemented, and tested, often in a matter of minutes per algorithm. Since the algorithms are expressed in code much like they are explained in a classroom setting, these APIs become not just a tool for implementing libraries, but also a valuable tool for teaching the algorithms that are incorporated in the libraries. In combination with an extension of the Parallel Lin-ear Algebra Package (PLAPACK) API, the approach presents a migratory path from algorithm to MATLAB implementation to high-performance sequential implementation to parallel im-plementation. Finally, the APIs are being used to create a repository of algorithms and im-plementations for linear algebra operations, the FLAME Interface REpository (FIRE), whic