F. R. de Hoog

An Improved Method for Numerical Inversion of Laplace Transforms

An improved procedure for numerical inversion of Laplace transforms is proposed based on accelerating the convergence of the Fourier series obtained from the inversion integral using the trapezoidal rule. When the full complex series is used, at each time-value the epsilon-algorithm computes a .(trigonometric) Pade approximation which gives better results than existing acceleration methods. The quotient-difference algorithm is used to compute the coefficients of the corresponding continued fraction, which is evaluated at each time-value, greatly improving efficiency. The convergence of the continued fraction can in turn be accelerated, leading to a further improvement in accuracy.

On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems

The use of algebraic eigenvalues to approximate the eigenvalues of Sturm-Liouville operators is known to be satisfactory only when approximations to the fundamental and the first few harmonics are required. In this paper, we show how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.ZusammenfassungDie Benutzung algebraischer Eigenwerte zur näherungsweisen Berechnung der Eigenwerte von Sturm-Liouville-Operatoren ist bekanntlich nur für die Grundschwingung und einige weitere Harmonische zufriedenstellend. In dieser Arbeit zeigen wir, wie man den asymptotischen Fehler, der bei verwandten aber einfachen Sturm-Liouville-Operatoren auftritt, dazu benutzen kann, um gewisse Klassen algebraischer Eigenwerte so zu korrigieren, daß die gleichmäßig gute Approximationen liefern.

A note on downdating the Cholesky factorization

We analyse and compare three algorithms for “downdating” the Cholesky factorization of a positive definite matrix. Although the algorithms are closely related, their numerical properties differ. Two algorithms are stable in a certain “mixed” sense while the other is unstable. In addition to comparing the numerical properties of the algorithms, we compare their computational complexity and their suitability for implementation on parallel or vector computers.

QR factorization of toeplitz matrices

SummaryThis paper presents a new algorithm for computing theQR factorization of anm×n Toeplitz matrix inO(mn) operations. The algorithm exploits the procedure for the rank-1 modification and the fact that both principal (m−1)×(n−1) submatrices of the Toeplitz matrix are identical. An efficient parallel implementation of the algorithm is possible.

The application of compressive sampling to radio astronomy - I. Deconvolution

Compressive sampling is a new paradigm for sampling, based on sparseness of signals or signal representations. It is much less restrictive than Nyquist-Shannon sampling theory and thus explains and systematises the widespread experience that methods such as the Hogbom CLEAN can violate the Nyquist-Shannon sampling requirements. In this paper, a CS-based deconvolution method for extended sources is introduced. This method can reconstruct both point sources and extended sources (using the isotropic undecimated wavelet transform as a basis function for the reconstruction step). We compare this CS-based deconvolution method with two CLEANbased deconvolution methods: the Hogbom CLEAN and the multiscale CLEAN. This new method shows the best performance in deconvolving extended sources for both uniform and natural weighting of the sampled visibilities. Both visual and numerical results of the comparison are provided.

An approximation theory for boundary value problems on infinite intervals

Boundary value problems for ordinary differential equations on infinite intervals are often solved by restricting the problem to a large but finite interval and imposing certain supplementary boundary conditions at the far end. The success of this procedure depends on the proper choice of these conditions. For a rather general class of problems we give a characterization of all possible supplementary boundary conditions which work, examine the rate of convergence of the solution of the “finite” problem to that of the original “infinite” problem as the interval length of the finite problem tends to infinity, and describe the supplementary boundary conditions for which this rate is optimal.ZusammenfassungEin mögliches Verfahren zur numerischen Lösung von Randwertproblemen auf unendlichen Intervallen besteht darin, das unendliche Intervall durch ein endliches zu ersetzen und zusätzliche Randbedingungen am entfernten Intervallende aufzuerlegen, in denen das asymptotische Verhalten der Lösung zum Ausdruck kommt. In dieser Arbeit wird für eine recht allgemeine Klasse von Differentialgleichungen die Menge aller zusätzlichen Randbedingungen charakterisiert, für welche die Lösung des „endlichen” Problems bei wachsender Intervallänge gegen die Lösung des „unendlichen” Problems konvergiert. Weiters wird die Konvergenzgeschwindigkeit abgeschätzt, und es werden die „optimalen” Randbedingungen beschrieben, die zu einer möglichst schnellen Konvergenz führen.

On the Stability of the Bareiss and Related Toeplitz Factorization Algorithms

This paper contains a numerical stability analysis of factorization algorithms for computing the Cholesky decomposition of symmetric positive definite matrices of displacement rank 2. The algorithms in the class can be expressed as sequences of elementary downdating steps. The stability of the factorization algorithms follows directly from the numerical properties of algorithms for realizing elementary downdating operations. It is shown that the Bareiss algorithm for factorizing a symmetric positive definite Toeplitz matrix is in the class and hence the Bareiss algorithm is stable. Some numerical experiments that compare behavior of the Bareiss algorithm and the Levinson algorithm are presented. These experiments indicate that generally (when the reflection coefficients are not all of the same sign) the Levinson algorithm can give much larger residuals than the Bareiss algorithm.

On the correction of finite difference eigenvalue approximations for sturm-liouville problems with general boundary conditions

When finite difference and finite element methods are used to approximate continuous (differential) eigenvalue problems, the resulting algebraic eigenvalues only yield accurate estimates for the fundamental and first few harmonics. One way around this difficulty would be to estimate the error between the differential and algebraic eigenvalues by some independent procedure and then use it to correct the algebraic eigenvalues. Such an estimate has been derived by Paine, de Hoog and Anderssen for the Liouville normal form with Dirichlet boundary conditions. In this paper, we extend their result to the Liouville normal form with general boundary conditions.

On dichotomy and well conditioning in BVP

We investigate the relationships between the stability bounds of the problem on the one hand and the growth behaviour of the fundamental solution on the other hand. It is shown that if these stability bounds are moderate (i.e. if the problem is well conditioned) then the homogeneous solution space is dichotomic, which means that it can be split into a subspace of nondecreasing and a complementary subspace of nonincreasing modes. This is done by carefully examining the Green’s functions. If these exhibit an exponential behaviour then the solution space is also exponentially dichotomic. On the other hand, we also show that (exponential) dichotomy implies moderate stability constants, i.e. well conditioning. From this it follows that both concepts are more or less equivalent.

A numerical study of similarity solutions for combined forced and free convection

SummaryThe similarity equations for combined forced and free convection flow over a horizontal plate when the wall temperature is inversely proportional to the square root of the distance from the leading edge are solved by introducing a scaling similar to that for the Blasius equation. The technique is also applied to the local similarity equations for the case of a constant wall temperature.