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arXiv · 0804.1021

Differentiation of Kaltofen's division-free determinant algorithm

Gilles Villard
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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.

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Gilles Villard. 2008-04-07. Differentiation of Kaltofen's division-free determinant algorithm. https://doi.org/10.1016/j.jsc.2010.08.012

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