The calculation of the distance to a nearby defective matrix

In this paper a new fast algorithm for the computation of the distance of a matrix to a nearby defective matrix is presented. The problem is formulated following Alam & Bora (Linear Algebra Appl., 396 (2005), pp.~273--301) and reduces to finding when a parameter-dependent matrix is singular subject to a constraint. The solution is achieved by an extension of the Implicit Determinant Method introduced by Spence & Poulton (J. Comput. Phys., 204 (2005), pp.~65--81). Numerical results for several examples illustrate the performance of the algorithm.Comment: 12 page

Equity Within and Between Nations

Income inequality, Food Security and Poverty, International Development, Productivity Analysis,

Regional science at the turn of the century: Reflections on its epistemological status

As a contribution to the current debate on the state-of-the art of regional science, this paper presents some reflections on the epistemological and methodological status of the discipline as we approach the turn of the century. First of all, and contrary to the widely held view that quantitative approach is seriously 'in crisis', it is argued that the discipline is going through a period of intensive, but constructive, theoretical development. To support this assertion, the authors suggest that it is important to abandon a hidden source of prejudice: the tendency to evaluate the present situation in terms of an outdated conception of the discipline. Modern quantitative geography and regional science is a vast and varied scientific field, which has radically evolved under the pressure of changing theoretical paradigms and technological advance. It has little to do with the old regional science of the 60s. The first part of the paper reviews this evolution: 1. from the original goal of applying to geography the tools of classical science, such as statistics, optimization and modelling (whose use was made possible in the 60s by the availability of the new "number crunching" computers) 2. to the present informatization (and hence quantification) of all branches of regional science, based on PCs and the Net, used as tools not just for computation, but for data handling, representation, visualization and communication). An attempt is made to fit all of these efforts, those with a long tradition (modelling, O.R., gaming simulation, statistics etc.), as well as the more recent approaches (expert systems, G.I.S., hypermedia, virtual reality, A.I.) into a single framework, stressing the specific aims of each and identifying existing - or potential - interconnections. In the second part of the paper we focus on the new frontiers of regional science and quantitative geography with particular reference to the processes of analysis and planning. It is suggested that: 1. the goal of analysis is shifting from simulation (the explicitation in terms of the "scientific method") of the mental processes involved in problem-solving, to the replication of the human ability to "formulate problems". This implies that creativity, and related aspects such as learning, and expertise, will come increasingly within the scope of research in regional science 2. progress in planning will be limited unless we will be able to go beyond the misleading counterposition between the formalised "rational" approach and the intuitive design approach. A fruitful way to cope with planning in a complex world is to integrate the two strategies and, in doing so, to tap into wider sources of knowledge. In other words, it is important to learn the 'art' of using the tools of geographical science.

Domain Decomposition preconditioning for high-frequency Helmholtz problems with absorption

In this paper we give new results on domain decomposition preconditioners for GMRES when computing piecewise-linear finite-element approximations of the Helmholtz equation $-\Delta u - (k^2+ {\rm i} \varepsilon)u = f$, with absorption parameter $\varepsilon \in \mathbb{R}$. Multigrid approximations of this equation with $\varepsilon \not= 0$ are commonly used as preconditioners for the pure Helmholtz case ($\varepsilon = 0$). However a rigorous theory for such (so-called "shifted Laplace") preconditioners, either for the pure Helmholtz equation, or even the absorptive equation ($\varepsilon \not=0$), is still missing. We present a new theory for the absorptive equation that provides rates of convergence for (left- or right-) preconditioned GMRES, via estimates of the norm and field of values of the preconditioned matrix. This theory uses a $k$- and $\varepsilon$-explicit coercivity result for the underlying sesquilinear form and shows, for example, that if $|\varepsilon|\sim k^2$, then classical overlapping additive Schwarz will perform optimally for the absorptive problem, provided the subdomain and coarse mesh diameters are carefully chosen. Extensive numerical experiments are given that support the theoretical results. The theory for the absorptive case gives insight into how its domain decomposition approximations perform as preconditioners for the pure Helmholtz case $\varepsilon = 0$. At the end of the paper we propose a (scalable) multilevel preconditioner for the pure Helmholtz problem that has an empirical computation time complexity of about $\mathcal{O}(n^{4/3})$ for solving finite element systems of size $n=\mathcal{O}(k^3)$, where we have chosen the mesh diameter $h \sim k^{-3/2}$ to avoid the pollution effect. Experiments on problems with $h\sim k^{-1}$, i.e. a fixed number of grid points per wavelength, are also given