Updating Matrix Polynomials

Abstract

Given a square matrix $M = (u_{ij})_{n \times n}$ and an m-order matrix polynomial $f_m(M) = \sum _{k=0}^{m} a_k M^k = a_0 I + a_1M + a_2 M^2 + \cdots + a_m M^m$, if M is a dense matrix and is perturbed to become $M'$ at a single entry, say $u_{pq}$, a straightforward re-calculation of $f_{m}(M')$ would require $O(n^{\omega } \cdot \alpha (m))$ arithmetic operations, where $\omega < 2.3728639$ and $\alpha (m)$ depends on the strategy of computing $M'^k, 1 \le k \le m$ appearing in $f_m(M')$, using the fastest square matrix multiplication algorithm by François Le Gall (ISSAC’14). In this paper, we assume that M is a dense matrix and that $f_{m}(M)$ is known while no other additional information is available. From the perspective of the naive (a.k.a., standard row-by-column) matrix multiplication, we discuss the update of matrix polynomials. First, we present O(n)-, $O(n^2)$- and $O(n^2)$-operations update algorithms for 2-order, 3-order and 4-order matrix polynomials, respectively. Furthermore, we discuss the update of high-order matrix polynomials with a sparse coefficient vector and as a result, propose a combinatorial heuristic updating method based on directed Steiner tree in a directed acyclic graph.