David J. Leeming

A combined Remes-differential correction algorithm for rational approximation

Abstract : In this paper a hybrid Remes-differential correction algorithm for computing best uniform rational approximants on a compact subset of the real line is developed. This algorithm differs from the classical multiple exchange Remes algorithm in two crucial aspects. First of all, the solving of a nonlinear system to find a best approximation on a given reference set in each iteration of the Remes algorithm is replaced with the differential correction algorithm to compute the desired best approximation on the reference set. Secondly, the exchange procedure itself has been modified to eliminate the possibility of cycling that can occur in the usual exchange procedure. This second modification is necessary to guarantee the convergence of this algorithm on a finite set without the usual normal and sufficiently dense assumptions that exist in other studies. (Author)

Approximation with reciprocals of polynomials on compact sets

Abstract : A theory best uniform approximation by reciprocals of polynomials on compact sets is developed. In particular an existence theorem is shown to hold in this case. (Author)

A combined remes-differential correction algorithm for rational approximation: experimental results

Abstract It is our contention that the recently developed Remes-difcor algorithm is the best general purpose algorithm available for computing best uniform rational approximations, and should be widely used as a library routine. The purpose of this paper is to support this contention by theoretical arguments and by a numerical comparison of the Remes-difcor algorithm, the differential correction algorithm, and the widely-used Remes algorithm. The Remes-difcor algorithm is shown to be more robust than the Remes algorithm and faster than the differential correction algorithm. The three algorithms are briefly described and discussed, and the experimental results for 70 examples are presented in six tables.

Uniform reciprocal approximation subject to linear constraints

Abstract In this paper a general theory is presented for uniform approximation by reciprocals of elements of a linear subspace subject to linear constraints. In fact, the main results in the known theory of constrained linear approximation have analogues in this non-linear setting.

Approximation on [0, ∞) by reciprocals of polynomials with nonnegative coefficients☆

Abstract A complete theory of best uniform approximation to positive functions decaying to zero on [0, ∞) by reciprocals of polynomials with nonnegative coefficients is presented.

The coefficients of sinh xt/sin tand the Bernoulli polynomials

For a set of polynomials defined by the generating function sinh xt/sin twe are able to show how they are related to the Bernoulli polynomials of odd degree. We also provide an easy method of generating the Bernoulli numbers.

Full length article: The exact number of real roots of the Bernoulli polynomials

We extend previous work by Inkeri, Leeming and Delange on the number of real roots of the Bernoulli polynomials. By these earlier methods the number of real roots could not be determined exactly in many cases. We introduce a new method that enables us to determine precisely the number of real roots in virtually every case, with rare exceptions of approximately one in 1.5x10^8.