large matrix, small rank
For the problem of computing the rank of a matrix we have a complexity result and a practical implementation, both of which apply best to the case of a matrix whose rank is substantially smaller than its order. First, matrix rank can be computed in essentially optimal time when the rank is sufficiently small, specifically when the rank is less than the matrix order to the two-thirds power. Second, we have built software infrastructure to improve efficiency of matrix operations over a small finite field. Using this we have computed the 3-ranks of the first 7 matrices of the sequence D(3,k). These are adjacency matrices of strongly regular graphs defined by a construction of Dickson based on operations in a semifield of order 3 2k . In particular, D(3,7) is a 4,782,969 by 4,782,969 matrix and it's rank computation uses four days of CPU time. On the basis of the first 7 values, we conjecture that the sequence of ranks of the D(3,k) satisfies a linear recurrence of degree three.
matrix p-rank