Deriving dense linear algebra libraries

Starting in the late 1960s computer scientists including Dijkstra and Hoare advocated goal-oriented programming and the formal derivation of algorithms. The chief impediment to realizing this for loop-based programs was that a priori determination of loop-invariants, a prerequisite for developing loops, was a task too complex for any but the simplest of operations. Around 2000, these techniques were for the first time successfully applied to the domain of high-performance dense linear algebra libraries. This has led to a multitude of papers (mostly published in the ACM Transactions for Mathematical Software), a system for the mechanical derivation of algorithms, and a high-performance linear algebra library, ${tt libflame}$ , that includes more than a thousand variants of algorithms for more than a hundred linear algebra operations. To our knowledge, this success story has unfolded with limited awareness on the part the formal methods community. This paper reports on ten years of experience and is meant to raise that awareness.