Typing linear algebra: A biproduct-oriented approach

Abstract

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 MATLAB™ is also considered.Keywords: Linear algebra, categories of matrices, algebra of programming

Gaussian elimination is not optimal, revisited

“…Although in [5] and the previous definition hint on how the dual object would correspond to the traditional notion of transposition, in [5] one also proves the following results for (standard) biproducts projections/injections pairs:…”

“…We build upon the work in [5], but we chose not to depict matrices as arrows with right to left orientation. Although it makes sense to draw arrows backwards, due to the flow of matrix calculations, in a setting where we want to mingle matrices (linear functions) with common (non-linear) functions we opted to make morphism direction evolve.…”

“…We denote such matrices by π. An example well studied in [5] are the π 1 = [1|0] and π 2 = [0|1] projections that appear naturally in the case of matrix block operations.…”

“…The vectorization operation is well studied in [5], and the diagram in (15) summarizes what we need to use in this paper.…”

“…The text is mainly an adaptation of results from [4,5] that we develop showing why when equipped with the direct sum (⊕, 0) monoid the category defined lacks closing and thus not Cartesian closed, thus not Turing complete. In section 3 we improve the translation of the traditional specification of the GE in terms of elementary matrices into the biproduct framework, derive new lemmas (18), and novel notation to specify the column and line application of GE, equations (19) and (20), used in achieving the main results.…”