On the complexity of polynomial matrix computations

Abstract

We study the link between the complexity of polynomial matrix multiplication and the complexity of solving other basic linear algebra problems on polynomial matrices. By polynomial matrices we mean n × n matrices in K[x] of degree bounded by d, with K a commutative field. Under the straight-line program model we show that multiplication is reducible to the problem of computing the coefficient of degree d of the determinant. Conversely, we propose algorithms for minimal approximant computation and column reduction that are based on polynomial matrix multiplication; for the determinant, the straight-line program we give also relies on matrix product over K [x] and provides an alternative to the determinant algorithm of [16,17]. We further show that all these problems can be solved in particular in O˜(n ω d) operations in K. Here the "soft O" notation O˜in-dicates some missing log(nd) factors and ω is the exponent of matrix multiplication over K.

Probabilistic analysis of Wiedemann's algorithm for minimal polynomial computation

“…), where, for chosen block size b, U, V are uniformly random matrices of shape b × n and n × b, respectively. A block Berlekamp/Massey algorithm is then used to compute the matrix minimal generating polynomial of B (Kaltofen and Yuhasz, 2013;Giorgi et al, 2003), and from it the minimal scalar generating polynomial. All of the algorithms based on these random projections rely on preservation of some properties, including at least the minimal generating polynomial.…”

Subquadratic Binary Field Multiplier in Double Polynomial System

“…This can be achieve within a complexity of O(n 3 k 2 ) binary operations with Algorithm WeakPopovForm of (Mulders and Storjohann, 2003) or with an asymptotic complexity of O(n 3 k log k) binary operations with Algorithm ColumnReduction of (Giorgi et al, 2003).…”

On the complexity of computing determinants

“…One is a sophisticated generalization of the Berlekamp/Massey algorithm (Coppersmith 1994;Dickinson et al 1974;Rissanen 1972). Another generalizes the theory of Padé approximation (Beckermann & Labahn 1994;Forney, Jr. 1975;Giorgi et al 2003;Van Barel & Bultheel 1992). The interpretation of the Berlekamp/Massey algorithm as a specialization of the extended Euclidean algorithm (Dornstetter 1987;Sugiyama et al 1975) can be carried over to matrix polynomials (Coppersmith 1994;Thomé 2002) (see also Section 3 below).…”

“…Remark 4.1. As we have seen in Section 2.1 there are several alternatives for carrying out Step 3 (Beckermann & Labahn 1994;Coppersmith 1994;Dickinson et al 1974;Forney, Jr. 1975;Giorgi et al 2003;Kaltofen 1995;Rissanen 1972;Thomé 2002;Van Barel & Bultheel 1992). In Step 4 we require that det F A,Y X (λ) = det(λI − A).…”