A Levinson-like algorithm for symmetric strongly nonsingular higher order semiseparable plus band matrices

In this paper, we will derive a solver for a symmetric strongly nonsingular higher order generator representable semiseparable plus band matrix. The solver we will derive is based on the Levinson algorithm, which is used for solving strongly nonsingular Toeplitz systems.     

A numerical example showing that deflations based on rotation leads to higher relative accuracy in QR algorithm

We present a numerical example illustrating that the deflation procedure in Francis's implicitly shifted QR algorithm can be improved by a deflation criteria based on the QR decomposition of the upper Hessenberg matrix. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spatially adaptive PIV interrogation based on data ensemble

This paper proposes a specific application of the approach recently proposed by the authors to achieve an autonomous and robust adaptive interrogation method for PIV data sets with the focus on the determination of mean velocity fields. Under circumstances such as suboptimal flow seeding distribution and large variations in the velocity field properties, neither multigrid techniques nor adaptive interrogation with criteria based on instantaneous conditions offer enough robustness for the flow field analysis. A method based on the data ensemble to select the adaptive interrogation parameters, namely, the window size, aspect ratio, orientation, and overlap factor is followed in this study. Interrogation windows are sized, shaped and spatially distributed on the basis of the average seeding density and the gradient of the velocity vector field. Compared to the instantaneous approach, the ensemble-based criterion adapts the windows in a more robust way especially for the implementation of non-isotropic windows (stretching and orientation), which yields a higher spatial resolution. If the procedure is applied recursively, the number of correlation samples can be optimized to satisfy a prescribed level of window overlap ratio. The relevance and applicability of the method are illustrated by an application to a shock-wave-boundary layer interaction problem. Furthermore, the application to a transonic airfoil wake verifies by means of a dual-resolution experiment that the spatial resolution in the wake can be increased by using non-isotropic interrogation windows.     

A small note on the scaling of symmetric positive definite semiseparable matrices

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix satisfies property A, one can easily construct a diagonal matrix such that has the lowest condition number over all matrices , for any choice of diagonal matrix . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form. *The research was partially supported by the Research Council K.U. Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann–Hilbert problems, random matrices and Padé–Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The scientific responsibility rests with the authors. The second author participates in the SCCM program, Gates 2B, Stanford University, CA, USA and is also partially supported by the NSF. The first author visited the second one with a grant by the Fund for Scientific Research–Flanders (Belgium).     

A reflection on the implicitly restarted Arnoldi method for computing eigenvalues near a vertical line

In this article, we will study the link between a method for computing eigenvalues closest to the imaginary axis and the implicitly restarted Arnoldi method. The extension to computing eigenvalues closest to a vertical line is straightforward, by incorporating a shift. Without loss of generality we will restrict ourselves here to computing eigenvalues closest to the imaginary axis.In a recent publication, Meerbergen and Spence discussed a new approach for detecting purely imaginary eigenvalues corresponding to Hopf bifurcations, which is of interest for the stability of dynamical systems. The novel method is based on inverse iteration (inverse power method) applied on a Lyapunov-like eigenvalue problem. To reduce the computational overhead significantly a projection was added.This method can also be used for computing eigenvalues of a matrix pencil near a vertical line in the complex plane. We will prove in this paper that the combination of inverse iteration with the projection step is equivalent to Sorensen’s implicitly restarted Arnoldi method utilizing well-chosen shifts.     

A unitary similarity transform of a normal matrix to complex symmetric form

In this work a new unitary similarity transformation of a normal matrix to complex symmetric form will be discussed. A constructive proof as well as some properties and examples will be given.     

An implicit QR algorithm for symmetric semiseparable matrices

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n³) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

The analysis of volatile siloxanes in waste activated sludge

The increasing presence of siloxanes in waste activated sludge (WAS) considerably hampers the energy use of the biogas obtained during the anaerobic digestion of the sludge when concentrations exceed critical limits. To prevent the occurrence of unacceptable operating conditions, it is hence necessary to have a reliable analysis method for determining the siloxane content of the sludge. This paper describes and validates such a method, consisting of the extraction of the siloxanes using n-hexane and a subsequent analysis of the extract using GC-FID. The validation procedure confirms the excellent recovery and repeatability of the proposed method.     

A note on a symmetrical set covering problem: The lottery problem

In a lottery, n numbers are drawn from a set of m numbers. On a lottery ticket we fill out n numbers. Consider the following problem: what is the minimum number of tickets so that there is at least one ticket with at least p matching numbers? We provide a set-covering formulation for this problem and characterize its LP solution. The existence of many symmetrical alternative solutions, makes this a very difficult problem to solve, as our computational results indicate.