Perturbation, extraction and refinement of invariant pairs for matrix polynomials

Generalizing the notion of an eigenvector, invariant subspaces are frequently used in the context of linear eigenvalue problems, leading to conceptually elegant and numerically stable formulations in applications that require the computation of several eigenvalues and/or eigenvectors. Similar benefits can be expected for polynomial eigenvalue problems, for which the concept of an invariant subspace needs to be replaced by the concept of an invariant pair. Little has been known so far about numerical aspects of such invariant pairs. The aim of this paper is to fill this gap. The behavior of invariant pairs under perturbations of the matrix polynomial is studied and a first-order perturbation expansion is given. From a computational point of view, we investigate how to best extract invariant pairs from a linearization of the matrix polynomial. Moreover, we describe efficient refinement procedures directly based on the polynomial formulation. Numerical experiments with matrix polynomials from a number of applications demonstrate the effectiveness of our extraction and refinement procedures.

Cost733cat: a database of weather and circulation type classifications

A new database of weather and circulation type catalogs is presented comprising 17 automated classification methods and five subjective classifications. It was compiled within COST Action 733 "Harmonisation and Applications of Weather Type Classifications for European regions" in order to evaluate different methods for weather and circulation type classification. This paper gives a technical description of the included methods using a new conceptual categorization for classification methods reflecting the strategy for the definition of types. Methods using predefined types include manual and threshold based classifications while methods producing types derived from the input data include those based on eigenvector techniques, leader algorithms and optimization algorithms. In order to allow direct comparisons between the methods, the circulation input data and the methods' configuration were harmonized for producing a subset of standard catalogs of the automated methods. The harmonization includes the data source, the climatic parameters used, the classification period as well as the spatial domain and the number of types. Frequency based characteristics of the resulting catalogs are presented, including variation of class sizes, persistence, seasonal and inter-annual variability as well as trends of the annual frequency time series. The methodological concept of the classifications is partly reflected by these properties of the resulting catalogs. It is shown that the types of subjective classifications compared to automated methods show higher persistence, inter-annual variation and long-term trends. Among the automated classifications optimization methods show a tendency for longer persistence and higher seasonal variation. However, it is also concluded that the distance metric used and the data preprocessing play at least an equally important role for the properties of the resulting classification compared to the algorithm used for type definition and assignment.

Comments, with reply, on "Computational algorithms for pole assignment" by N.K. Nichols

Some points of the paper by N.K. Nichols (see ibid., vol.AC-31, p.643-5, 1986), concerning the robust pole assignment of linear multiinput systems, are clarified. It is stressed that the minimization of the condition number of the closed-loop eigenvector matrix does not necessarily lead to robustness of the pole assignment. It is shown why the computational method, which Nichols claims is robust, is in fact numerically unstable with respect to the determination of the gain matrix. In replying, Nichols presents arguments to support the choice of the conditioning of the closed-loop poles as a measure of robustness and to show that the methods of J Kautsky, N. K. Nichols and P. VanDooren (1985) are stable in the sense that they produce accurate solutions to well-conditioned problems.