Differentiation of Kaltofen's division-free determinant algorithm

TL;DR

This paper analyzes Kaltofen's determinant algorithm, clarifies its adjoint computation, and presents an explicit implementation, especially for polynomial matrices, enhancing understanding and practical application of the method.

Contribution

It provides an explicit implementation of Kaltofen's adjoint algorithm without automatic differentiation, applied to polynomial matrices, and clarifies its theoretical foundation.

Findings

The adjoint algorithm can be implemented explicitly.

The method is effective for polynomial matrices.

Complexity matches that of the original determinant algorithm.

Abstract

Kaltofen has proposed a new approach in [Kaltofen 1992] for computing matrix determinants. The algorithm is based on a baby steps/giant steps construction of Krylov subspaces, and computes the determinant as the constant term of a characteristic polynomial. For matrices over an abstract field and by the results of Baur and Strassen 1983, the determinant algorithm, actually a straight-line program, leads to an algorithm with the same complexity for computing the adjoint of a matrix [Kaltofen 1992]. However, the latter is obtained by the reverse mode of automatic differentiation and somehow is not ``explicit''. We study this adjoint algorithm, show how it can be implemented (without resorting to an automatic transformation), and demonstrate its use on polynomial matrices.