On the complexity of computing determinants

Abstract

Abstract. We present new baby steps/giant steps algorithms of asymptotically fast running time for dense matrix problems. Our algorithms compute the determinant, characteristic polynomial, Frobenius normal form and Smith normal form of a dense n × n matrix A with integer entries in (n 3.2 log A ) 1+o(1) and (n 2.697263 log A ) 1+o(1) bit operations; here A denotes the largest entry in absolute value and the exponent adjustment by "+o(1)" captures additional factors C 1 (log n) C 2 (loglog A ) C 3 for positive real constants C 1 , C 2 , C 3 . The bit complexity (n 3.2 log A ) 1+o(1) results from using the classical cubic matrix multiplication algorithm. Our algorithms are randomized, and we can certify that the output is the determinant of A in a Las Vegas fashion. The second category of problems deals with the setting where the matrix A has elements from an abstract commutative ring, that is, when no divisions in the domain of entries are possible. We present algorithms that deterministically compute the determinant, characteristic polynomial and adjoint of A with n 3.2+o(1) and O(n 2.697263 ) ring additions, subtractions and multiplications.

Sampling the feasible sets of SDPs and volume approximation

“…Then, with O(m 2.697 ) operations we compute its characteristic polynomial [26] and in O(m) we decide if it has negative roots, for example by solving [41] or using fast sub-resultant algorithms [31,34]. For the bit complexity bound, the construction costs O B (nm 2 (τ + σ ) and computation of the characteristic polynomial O B (m 2.697+1 (τ + σ )) using a randomized algorithm [26]. We test for negative roots, and thus eigenvalues, in O B (m 2 n(τ + σ )) [34].…”

Computing a Lattice Basis Revisited

“…It remains to analyze the complexity. Since BB t ∈ Z d×d is nonsingular and W = det(BB t )(BB t ) −1 B, it is folklore to compute W deterministically in time O(md θ−1 • B(d log B )): we can efficiently compute det(BB t ) and (BB t ) −1 by calculating their residues modulo small primes (say, ≤ 2 d B 2d ) and then recovering the final results using Chinese Remainder Theorem; this approach is classical (see, e.g., [KV04,Vil03]). The operands during the computation have size O(d log B ).…”

PPP-Completeness with Connections to Cryptography

“…We define the circuit ℐ R : bd(R) → {0, 1} such that ℐ R defines a bijective map between bd(R) and bd([r]). The circuit ℐ R first computes the Smith Normal Form of the basis B = UDV, which can be done by a circuit that has size polynomial in |bd(B)| [KV05], then uses the index function of the set (D) ∩ Z n , as defined in Lemma 2.1, to map each element of R to a number in [r]. Claim 3.3.…”