Matrix Bruhat decompositions with a remark on the QR (GR) algorithm

AbstractIn a simple and systematic way we present matrix Bruhat decompositions of two kinds: basic and modified. We show that it is the modified Bruhat decomposition that governs the eigenvalue disorder in the QR (GR) algorithm. This paper can be considered as a commentary on a previous observation about the QR algorithm made by Wilkinson

Convergence of LR algorithm for a one-point spectrum tridiagonal matrix

We proved convergence for the basic LR algorithm on a real unreduced tridiagonal matrix with a one-point spectrum - the Jordan form is one big Jordan block. First we develop properties of eigenvector matrices. We also show how to deal with the singular case

News on modeling of walking robot critical positions

The principal objective of our study is to emphasize the strategies for the walking robot mathematical model to traverse an uneven terrain, respecting the hypothesis of environment model defined by us. The multiple aspects on axiomatic systems, with possible application to environment’s mathematical model axiomatization, open an interesting new way of research and is exposed in the first part of the paper. Our study on the walking robot begins with formulation of improved mathematical model for physical identification derived from geometrical identification of critical position in particular case of waking robot leg. The physical identification of the critical position is analyzed in the paper on our concrete case of walking robot leg mathematical model. The specialized algorithm performed by us is used for verification of the theory. The new directions of research, opened by our analyses in this area, are described

A comparison of the LR and QR transformations for finding the eigenvalues for real nonsymmetric matrices

The LR and QR algorithms, two of the best available iterative methods for finding the eigenvalues of a nonsymmetric matrix associated with a system of linear homogeneous equations, are studied. These algorithms are discussed as they apply to the determination of the eigenvalues of real nonsymmetric matrices. A comparison of the speed and accuracy of these transformations is made. A detailed discussion of the criterion for convergence and the numerical difficulties which may occur in the computation of multiple and complex conjugate eigenvalues are included. The results of this study indicate that the QR algorithm is the more successful method for finding the eigenvalues of a real nonsymmetric matrix --Abstract, page ii

Berechnung und Anwendungen Approximativer Randbasen

This thesis addresses some of the algorithmic and numerical challenges associated with the computation of approximate border bases, a generalisation of border bases, in the context of the oil and gas industry. The concept of approximate border bases was introduced by D. Heldt, M. Kreuzer, S. Pokutta and H. Poulisse in "Approximate computation of zero-dimensional polynomial ideals" as an effective mean to derive physically relevant polynomial models from measured data. The main advantages of this approach compared to alternative techniques currently in use in the (hydrocarbon) industry are its power to derive polynomial models without additional a priori knowledge about the underlying physical system and its robustness with respect to noise in the measured input data. The so-called Approximate Vanishing Ideal (AVI) algorithm which can be used to compute approximate border bases and which was also introduced by D. Heldt et al. in the paper mentioned above served as a starting point for the research which is conducted in this thesis. A central aim of this work is to broaden the applicability of the AVI algorithm to additional areas in the oil and gas industry, like seismic imaging and the compact representation of unconventional geological structures. For this purpose several new algorithms are developed, among others the so-called Approximate Buchberger Möller (ABM) algorithm and the Extended-ABM algorithm. The numerical aspects and the runtime of the methods are analysed in detail - based on a solid foundation of the underlying mathematical and algorithmic concepts that are also provided in this thesis. It is shown that the worst case runtime of the ABM algorithm is cubic in the number of input points, which is a significant improvement over the biquadratic worst case runtime of the AVI algorithm. Furthermore, we show that the ABM algorithm allows us to exercise more direct control over the essential properties of the computed approximate border basis than the AVI algorithm. The improved runtime and the additional control turn out to be the key enablers for the new industrial applications that are proposed here. As a conclusion to the work on the computation of approximate border bases, a detailed comparison between the approach in this thesis and some other state of the art algorithms is given. Furthermore, this work also addresses one important shortcoming of approximate border bases, namely that central concepts from exact algebra such as syzygies could so far not be translated to the setting of approximate border bases. One way to mitigate this problem is to construct a "close by" exact border bases for a given approximate one. Here we present and discuss two new algorithmic approaches that allow us to compute such close by exact border bases. In the first one, we establish a link between this task, referred to as the rational recovery problem, and the problem of simultaneously quasi-diagonalising a set of complex matrices. As simultaneous quasi-diagonalisation is not a standard topic in numerical linear algebra there are hardly any off-the-shelf algorithms and implementations available that are both fast and numerically adequate for our purposes. To bridge this gap we introduce and study a new algorithm that is based on a variant of the classical Jacobi eigenvalue algorithm, which also works for non-symmetric matrices. As a second solution of the rational recovery problem, we motivate and discuss how to compute a close by exact border basis via the minimisation of a sum of squares expression, that is formed from the polynomials in the given approximate border basis. Finally, several applications of the newly developed algorithms are presented. Those include production modelling of oil and gas fields, reconstruction of the subsurface velocities for simple subsurface geometries, the compact representation of unconventional oil and gas bodies via algebraic surfaces and the stable numerical approximation of the roots of zero-dimensional polynomial ideals

Informatikai algoritmusok 3.

A könyv a Magyar Tudományos Akadémia támogatásával készült