@MISC{Ogita_exactdeterminant, author = {T. Ogita}, title = {Exact Determinant of Integer Matrices}, year = {} }

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Abstract

Computing a determinant of a given matrix has several applications. For example, it is used for computational geometry (cf. e.g. Emiris and Pan (2001)), in which it is essential to check the sign of the determinant. Moreover, it is also used for robust control methods (cf. e.g. Smagina and Brewer (2002)). In this paper, we restrict A to be an integer matrix. Then the determinant of A is also an integer. Such a case has been discussed in Pan and Yu (2001); Kaltofen and Villard (2004); Emiris and Pan (2005) and the literatures cited there. Of course, we can efficiently obtain the approximation d ̃ of det(A) via LU factorization of A by floating-point arithmetic. The aim of the paper is to obtain the exact value t of det(A), where t is some integer. For this purpose, we try to compute a rigorous enclosure [d] = [d, d] such that d ≤ det(A) ≤ d with dde = bdc = t, which implies det(A) = t, by use of verified numerical computations. It is much more difficult than to simply compute d̃, since we need to obtain a verified and accurate result of the determinant. Let u be the unit roundoff. In IEEE standard 754 double precision, u = 2−53. Let κ(A): = ‖A ‖ · ‖A−1 ‖ be the condition number of A. In this paper, we propose an algo-rithm of computing an accurate and verified enclosure of det(A) with A being extremely ill-conditioned, i.e. κ(A) À u−1. The proposed algorithm uses an accurate matrix fac-torization presented in Ogita (2009), which is based on standard numerical algorithms and accurate dot product. References Brönnimann, H., C. Burnikel and S. Pion. Interval arithmetic yields efficient dynamic filters for computational geometry. Discrete Applied Mathematics, 109(1–2):25–47, 2001. Emiris, I. Z. and V. Y. Pan. Improved algorithms for computing determinants and resultants. Journal of Complexity, 21(1):43–71, 2005. Kaltofen, E. and G. Villard. Computing the sign or the value of the determinant of an integer matrix, a complexity survey. J. Comput. Appl. Math., 162(1):133–146, 2004. Ogita, T. Accurate matrix factorization: Inverse LU and inverse QR factorizations. sub-mitted for publication, 2009.