for matrix adjoint computation

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. Kaltofen has proposed in [8] a new approach for computing matrix determinants. This approach has brought breakthrough ideas for improving the complexity estimate for the problem of computing the determinant without divisions over an abstract ring [8, 11]. The same ideas also lead to the currently best known bit complexity estimates for some problems on integer matrices such as the problem of computing the characteristic polynomial [11].

A general goal concerning fundamental linear algebra problems is to reduce the complexity estimates to essentially the same as that of multiplying two matrices (plus possibly a cost related to the input and output sizes). Among the bottlenecks one usually finds the questions of designing a recursive approach and mastering the sizes of the intermediately computed data. In this talk we are interested in two special cases of lattice basis reduction. We consider bases given by square matrices over K[x] or Z, with, respectively, the notion of reduced form and LLL reduction. Our purpose is to introduce basic tools for understanding how to generalize the Lehmer and Knuth-Schönhage gcd algorithms for basis reduction. Over K[x] this generalization is a key ingredient for giving a