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

Kaltofen has proposed a new approach in Kaltofen (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 the characteristic polynomial. For matrices over an abstract ring, 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. However, the latter adjoint algorithm is obtained by the reverse mode of automatic differentiation, and hence is in some way not ''explicit''. We present an alternative (still closely related) algorithm for obtaining the adjoint that can be implemented directly, without resorting to an automatic transformation. The algorithm is deduced partly by applying program differentiation techniques ''by hand'' to Kaltofen's method, and is completely described. As a subproblem, we study the differentiation of the computation of minimum polynomials of linearly generated sequences, and we use a lazy polynomial evaluation mechanism for reducing the cost of Strassen's avoidance of divisions in our case.

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