On computing the determinant and Smith form of an integer matrix

Abstract

Abstract

On the complexity of computing determinants

“…For the classical matrix multiplication exponent ω = 3, the bit complexity of integer matrix determinants is thus proportional to n η+o(1) as follows: η = 3 + 1 2 (Eberly et al 2000;Kaltofen 1992Kaltofen , 2002, η = 3 + 1 3 (Theorem 4.2 on page 111), η = 3 + 1 5 (line 4 in Table 6.1 on page 119), η = 3 (Storjohann 2004). Together with the algorithms discussed in Section 1 on page 94 that perform well on propitious inputs, such a multitude of results poses a problem for the practitioner: which of the methods can yield faster procedures in computer algebra systems?…”

“…For the determinant of an n × n integer matrix A, an algorithm with (n 3.5 log A 1.5 ) 1+o(1) bit operations is given by Eberly et al (2000). 1 Their algorithm computes the Smith normal form via the binary search technique of Villard (2000).…”

An output-sensitive variant of the baby steps/giant steps determinant algorithm

“…Suppose that the dimension of the matrix is n × n and that the maximum bit length of all entries is b. The algorithm by [10] requires (n 3.5 b 2.5 ) 1+o (1) fixed precision, that is, bit operations. Here and in the following we use the exponent "+o(1)" to capture missing polylogarithmic factors O((log n) C 1 (log b) C 2 ), where C1, C2 are constants ("soft-O").…”

“…Both algorithms use randomization and the algorithm in [10] is Monte Carlo-always fast and probably correct-and the one in [20] is Las Vegas-always correct and probably fast. Both algorithms can be speeded by asymptotically fast subcubic matrix multiplication algorithmsà la Strassen [8,7,14].…”

“…In our survey [21] we list Gaussian elimination combined with Chinese remaindering, Hensel lifting [1] and the above cited result [10] as propitious. For example, if the the determinant ∆ is a small integer, Chinese remaindering can employ what is know as "early termination."…”