Showing papers by "Harald Niederreiter published in 1993"

A new efficient factorization algorithm for polynomials over small finite fields

Abstract: We present a new deterministic factorization algorithm for polynomials over a finite prime fieldF p . As in other factorization algorithms for polynomials over finite fields such as the Berlekamp algorithm, the key step is the “linearization” of the factorization problem, i.e., the reduction of the problem to a system of linear equations. The theoretical justification for our algorithm is based on a study of the differential equationy (p −1)+y p =0 of orderp−1 in the rational function fieldF p(x). In the casep=2 the new algorithm is more efficient than the Berlekamp algorithm since there is no set-up cost for the coefficient matrix of the system of linear equations.

Kronecker-type sequences and nonarchimedean diophantine approximations

Abstract: xn = ({nα1}, . . . , {nαs}) , n = 0, 1, . . . , where {u} is the fractional part of u ∈ R. It is well known that the sequence x0,x1, . . . is uniformly distributed in I = [0, 1] if and only if 1, α1, . . . , αs are linearly independent over Q, and that the finer quantitative description of the distribution behavior of this sequence depends on the diophantine approximation character of the point (α1, . . . , αs); compare with [6]. In this paper we study sequences of points in I that are obtained by a construction reminiscent of that of classical Kronecker sequences, but which operates in a function field setting. This construction was introduced in Niederreiter [17, Chapter 4], and the resulting sequences have attractive distribution properties. The detailed investigation of these Kronecker-type sequences that we carry out in the present work leads to interesting connections with nonarchimedean diophantine approximations. The construction belongs to the framework of the theory of (t,m, s)-nets and (t, s)-sequences, which are point sets and sequences, respectively, with special uniformity properties. We follow [17] in the notation and terminology. For a point set P consisting of N arbitrary points y0,y1, . . . ,yN−1 in I and for an arbitrary subset B of I, let A(B;P ) be the number of n with 0 ≤ n ≤ N − 1 for which yn ∈ B. Let an integer b ≥ 2 be fixed, and let λs denote the s-dimensional Lebesgue measure. A subinterval E of I = [0, 1) of the form

Good parameters for a class of node sets in quasi-Monte Carlo integration

Abstract: For 2 < s < 12 we determine good parameters in a general construction of node sets for s-dimensional quasi-Monte Carlo integration recently introduced by the third author. Some of the parameters represent optimal choices in this construction and lead to improvements on node sets obtained by earlier techniques.

Quasi-Monte Carlo Methods with Modified Vertex Weights

Abstract: Quasi-Monte Carlo methods for integration over the s-dimensional unit cube in which only one vertex is a quadrature point are here modified by distributing the corresponding weight between all the vertices One particular rule of this kind, namely the one that integrates all multilinear functions exactly, is shown to be optimal, in the sense that among all vertex-modified rules that integrate constants exactly it yields the least value of an L 2 version of the discrepancy The sum of the squares of the vertex weights in that rule, the “vertex variance”, is shown to be related to the quality of the underlying quasi-Monte Carlo method Numerical experiments confirm in a dramatic way the important role played by the vertex variance, even for the unmodified quasi-Monte Carlo method