Kaltofen's division-free determinant algorithm differentiated for matrix adjoint computation

TL;DR

This paper presents a new, explicit algorithm for computing the matrix adjoint by differentiating Kaltofen's division-free determinant algorithm, avoiding automatic differentiation and improving implementation clarity.

Contribution

It introduces a directly implementable adjoint algorithm derived from Kaltofen's method through manual program differentiation techniques.

Findings

The new algorithm is explicitly described and implementable.

It reduces the cost of division avoidance using lazy polynomial evaluation.

The approach provides an alternative to automatic differentiation for matrix adjoint computation.

Abstract

Kaltofen has proposed a new approach in 1992 for computing matrix determinants without divisions. 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 ring, by the results of Baur and Strassen, the determinant algorithm, actually a straight-line program, leads to an algorithm with the same complexity for computing the adjoint of a matrix. However, the latter adjoint algorithm is obtained by the reverse mode of automatic differentiation, hence somehow is not "explicit". We present an alternative (still closely related) algorithm for the adjoint thatcan be implemented directly, we mean without resorting to an automatic transformation. The algorithm is deduced by applying program differentiation techniques "by hand" to Kaltofen's method, and is completely decribed. As subproblem, we study the differentiation of programs that compute minimum polynomials of lineraly generated sequences, and we use a lazy polynomial evaluation mechanism for reducing the cost of Strassen's avoidance of divisions in our case.