On the Berlekamp/Massey algorithm and counting singular Hankel matrices over a finite field

We derive an explicit count for the number of singular n×n$n\times n$ Hankel (Toeplitz) matrices whose entries range over a finite field with q$q$ elements by observing the execution of the Berlekamp/Massey algorithm on its elements. Our method yields explicit counts also when some entries above or on the anti-diagonal (diagonal) are fixed. For example, the number of singular n×n$n\times n$ Toeplitz matrices with 0’s on the diagonal is q²ⁿ⁻³+qⁿ⁻¹−qⁿ⁻²$q^{2n-3}+q^{n-1}-q^{n-2}$.     

高阶Sylvester矩阵方程的解析通解

给出了矩阵方程Am1 VJm1+…+A1VJ+A0V=Bm2 WJm2+…+B1WJ+BoW的3种完全解析参数通解.这些解由一组参数向量给出,这些参数向量提供了问题的全部自由度.求解算法不要求矩阵J具有不同的特征值,或者和A(s)的特征值不同.这些通解仅包含数值矩阵计算,为工程应用计算提供了方便.算例说明本文所给方程通...     

On partitions of functions on finite sets

Let D and R be finite sets with cardinality n and m respectivelyR ^D be the set of all functions from D into R, and G and H be permutation groups acting on D and R respectively. Two functions f and g in R ^D are said to be related if there exists a σ in G and a τ in H with f(σd) = τg(d) for every d in D. Since the relation is an equivalence relation, R ^D is partitioned into disjoint classes. Roughly, by using the cycle indices of G and H, de Bruijn's theorem determines the number of equivalence classes, and Pólya's theorem, with H being the identity group, gives the function counting series, Pólya-de Bruijn's theorem has many applications (for instance, see Pólya and Read [6]). The theorem and its applications, basically, centered around the partitions of functions. Here, we present an algorithm to determine which functions in R ^D belong to the same equivalent class. Our algorithm does not use the cycle indices of G and H (to compute the cycle index of a given group, in general, is difficult), but it uses the generators of G and H, and the m-nary numbers to code the functions in R ^D . Our algorithm also gives the function counting series and the number of equivalence classes. An important application is that for each positive integer n, we use our algorithm and the symmetric group S _n to determine all isomorphic and nonisomorphic graphs and directed graphs with n vertices.     

On the complexity of computing determinants

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 and bit operations; here denotes the largest entry in absolute value and the exponent adjustment by +o(1) captures additional factors for positive real constants C₁, C₂, C₃. The bit complexity 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.To B. David Saunders on the occasion of his 60th birthday     

On solving the Lyapunov and Stein equations for a companion matrix

When the matrix A is in companion form, the essential step in solving the Lyapunov equation PA + A^TP = −Q involves a linear n × n system for the first column of the solution matrix P. The complex dependence on the data matrices A and Q renders this system unsuitable for actual computation. In this paper we derive an equivalent system which exhibits simpler dependence on A and Q as well as improved complexity and robustness characteristics. A similar results is obtained also for the Stein equation P − A^TPA = Q.