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.

A fraction free Matrix Berlekamp/Massey algorithm

“…The algorithm requires that the matrix sequence be a normalizable remainder sequence. These normalizable remainder sequences are similar to the sequences in [Kaltofen and Villard 2004]. Normalizable here requires the non-singularity of the matrix leading coefficients in the remainders (discrepancies), not that the degrees of all quotient polynomials are 1 in the "normal polynomial remainder sequences" in the literature.…”

“…However, when the input sequence has a higher dimension, then the input sequences must be a normalizable remainder sequence. Our concept of a normalizable remainder sequence is related to the results of Kaltofen and Villard [2004].…”

“…Whenever a nonzero discrepancy requires a degree increase, the discrepancy must be nonsingular. Such degree increases correspond to the division step of the half-gcd algorithm of Kaltofen and Villard [2004]. The determininant of the discrepancy is calculated using the diagonalization algorithm of Nakos et al [1997].…”

“…To relax the requirement of normalizable remainder sequences, a new algorithm with different proofs is needed and currently unknown. The use of the gap variable γ that represents the degree gap between successive generator degrees, allows us to remove the gap condition present in Kaltofen and Villard [2004].…”

Computation of the Adjoint Matrix

“…The best method for computing the adjoint matrix of an order n matrix in an arbitrary commutative ring requires O(n β+1/3 log n log log n) operations (see [1] and [2]). For a commutative domain the complexity of the best method is 6γn β /(2 β − 2) + o(n β ) (see [3]).…”