Differentiation of Kaltofen's division-free determinant algorithm

Kaltofen has proposed a new approach in [8] 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 [1], the determinant algorithm, actually a straight-line program, leads to an algorithm with the same complexity for computing the adjoint of a matrix [8]. 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.