Fast algorithms for the characteristic polynomial
Theoretical Computer Science
Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Matrix multiplication via arithmetic progressions
Journal of Symbolic Computation - Special issue on computational algebraic complexity
Processor efficient parallel solution of linear systems over an abstract field
SPAA '91 Proceedings of the third annual ACM symposium on Parallel algorithms and architectures
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
On computing determinants of matrices without divisions
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
Processor-efficient parallel matrix inversion over abstract fields: two extensions
PASCO '97 Proceedings of the second international symposium on Parallel symbolic computation
Modern computer algebra
On Wiedemann's Method of Solving Sparse Linear Systems
AAECC-9 Proceedings of the 9th International Symposium, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
On computing the determinant and Smith form of an integer matrix
FOCS '00 Proceedings of the 41st Annual Symposium on Foundations of Computer Science
Tellegen's principle into practice
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
High-order lifting and integrality certification
Journal of Symbolic Computation - Special issue: International symposium on symbolic and algebraic computation (ISSAC 2002)
On the complexity of computing determinants
Computational Complexity
The shifted number system for fast linear algebra on integer matrices
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
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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.