  Over the past two decades, topological data analysis has emerged as a field
of applied mathematics with new applications and algorithmic developments
appearing rapidly. Two fundamental computations in this field are persistent
homology and zigzag homology. In this paper, we show how these computations in
the most general case reduce to finding a canonical form of a matrix associated
with a type A quiver representation, which in turn can be computed using
factorizations of associated matrices. We show how to use arbitrary induced
maps on homology for computation, providing a framework that goes beyond the
capabilities of existing software for topological data analysis. Furthermore,
this framework offers multiple opportunities for parallelization which have not
been previously exploited. We provide several examples of the utility of this
framework, demonstrate parallel speedups, and report on significant
improvements in comparison to existing software.
