Typing linear algebra: A biproduct-oriented approach
This work addresses formal code generation for linear algebra applications, offering a novel theoretical framework that could enhance computer algebra systems, though it appears incremental in applying existing category theory concepts.
The authors tackled the problem of generating fast code for linear algebra by developing an index-free, calculational approach using category theory with biproducts, shifting from indexed matrices to a type-level perspective and deriving laws for algorithm implementation, including matrix multiplication variants and blockwise extensions like Gaussian elimination.
Interested in formalizing the generation of fast running code for linear algebra applications, the authors show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category with biproducts. This shifts the traditional view of matrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming. The derivation of fusion, cancellation and abide laws from the biproduct equations makes it easy to calculate algorithms implementing matrix multiplication, the central operation of matrix algebra, ranging from its divide-and-conquer version to its vectorization implementation. From errant attempts to learn how particular products and coproducts emerge from biproducts, not only blocked matrix algebra is rediscovered but also a way of extending other operations (e.g. Gaussian elimination) blockwise, in a calculational style, is found. The prospect of building biproduct-based type checkers for computer algebra systems such as MatlabTM is also considered.