On the diagonal approximation of full matrices

AbstractIn this paper the construction of diagonal matrices, in some sense approximating the inverse of a given square matrix, is described. The matrices are constructed using the well-known computer algebra system Maple. The techniques we show are applicable to square matrices in general. Results are given for use in Parallel diagonal-implicit Runge-Kutta (PDIRK) methods. For an s-stage Radau IIA corrector we conjecture s! possibilities for the diagonal matrices

Udvariatlanság, hatalom és genderidentitás

This paper consists of two parts. In the first part we give a review of a good multigrid method for solving the steady Euler equations of gas dynamics on a locally refined mesh. The method is selfadaptive and makes use of unstructured grids that can be considered as parts of a nested sequence of structured grids. It is briefly described and applied to some steady Euler-flow problems. The method appears to be much more accurate and efficient than the corresponding multigrid method that applies global refinements only. In the second part of the paper, vectorisation of the code is treated. To enable this vectorisation, index arrays are introduced and added to the quad-tree type data-structure that is applied in the scalar case. Speed-up factors are given for the same test cases as considered in the first part of the paper. The results are most satisfactory. 1 A solution-adaptive multigrid method for the steady Euler equations In this first section we describe a self-adaptive multigrid me..

Systematic computations on Mertens’conjecture and Dirichlet’s divisor problem by vectorized sieving

In this paper we present two vectorized numerical sieve algorithms for the number theoretical functions µ(n) and τ(n). These sieve algorithms are generalizations of Eratosthenes ’ sieve for finding prime numbers. We show algorithms for fast systematic computations on Mertens ’ conjecture and Dirichlet’s divisor problem. We have implemented the algorithm for Mertens ’ conjecture on a Cray C90 and performed a systematic computation of extremes of M(x) / √ x up to 10 13. We established the bounds −0.513 <M(x) / √ x< 0.571, valid for 200 <x≤10 13.

Specification of PSIDE

PSIDE is a code for solving implicit differential equations on parallel computers. It is an implementation of the four-stage Radau IIA method. The nonlinear systems are solved by a modified Newton process, in which every Newton iterate itself is computed by an iteration process. This process is constructed such that the four stage values can be computed simultaneously. We describe here how PSIDE is set up as a modular system and what control strategies have been chosen