Sven-Åke Gustafson

Convergence acceleration from the point of view of linear programming

The best possible bounds for the sum of an alternating series with completely monotonic terms, when 2N terms have been computed, are determined. It is shown that their difference decreases exponentially withN. Various generalizations are indicated. The optimal application of Eulers transformation is also discussed. The error of that method also decreases exponentially, though the logarithmic decrement is only about 2/3 compared with the best possible error bounds.

Rapid computation of general interpolation formulas and mechanical quadrature rules

Let ƒ have <italic>n</italic> continuous derivatives on a closed interval [<italic>a, b</italic>] and let <italic>L</italic> be a linear functional. The attempt is made to approximate <italic>L</italic>(ƒ) with <italic>L</italic>(<italic>Q</italic>) where <italic>Q</italic> is a polynomial, approximating ƒ. Algorithms are developed for rapid computation of <italic>L</italic>(<italic>Q</italic>) for a wide class of selections of <italic>Q</italic> which includes the Lagrangian and Hermitian rules as special cases.

Numerical computation of an integral appearing in the Fröman-Fröman phase-integral formula for calculation of quantal matrix elements without the use of wave functions

Abstract In this paper we present an efficient method for the numerical treatment of an integral appearing in the Froman-Froman phase-integral formula. Realistic error bounds are developed. The stability and convergence of the method are verified for a large class of integrands.

Incompressible flow in porous media with two moving boundaries

Abstract The industrial process considered is production of oil through a horizontal well perforated into an oil-zone which is bounded below by water and bounded above by gas. This process is modelled by a two-dimensional, incompressible, porous-medium flow into a point sink. The interfluid boundaries are then moving boundaries. The conditions at each moving boundary are expressed, with appropriate assumptions, as equations between the fluid potential and the vertical displacement of the boundary. There is one static and one dynamic condition at each boundary. Boundary-fitted orthogonal coordinates are then introduced. The Laplace equation for the vertical displacement and the Poisson equation for the velocity potential, together with the static boundary conditions, can then be solved analytically, in terms of the vertical displacement at each boundary, with time as a parameter. The vertical displacement at each boundary must be calculated at each time-step by solving the equations expressing the dynamic boundary conditions. These are coupled, non-linear integro-differential equations. The solution is expressed as an infinite trigonometric series, whose coefficients are determined as the solution of an infinite system of ordinary differential equations. Convergence acceleration is applied to this series and hence only the first few terms need to be calculated.

Investigating semi-infinite programs using penalty functions and Lagrangian methods

In this paper the relations between semi-infinite programs and optimisation problems with finitely many variables and constraints are reviewed. Two classes of convex semi-infinite programs are defined, one based on the fact that a convex set may be represented as the intersection of closed halfspaces, while the other class is defined using the representation of the elements of a convex set as convex combinations of points and directions. Extension to nonconvex problems is given. A common technique of solving a semi-infinite program computationally is to derive necessary conditions for optimality in the form of a nonlinear system of equations with finitely many equations and unknowns. In the three-phase algorithm, this system is constructed from the optimal solution of a discretised version of the given semi-infinite program. i.e. a problem with finitely many variables and constraints. The system is solved numerically, often by means of some linearisation method. One option is to use a direct analog of the familiar SOLVER method.

Convergence acceleration by means of numerical quadrature

By regarding a series as a Stieltjes integral to which classical numerical methods are applied, very accurate expressions for the sum are obtained. Convergence and stability are investigated in some cases of practical importance. When certain conditions are satisfied, realistic and strict error bounds for the sum can be found. Some generalizations are also indicated.

Computing fourier integrals by means of near-optimal quadrature rules of the Lagrangian type

As shown inGustafson-Dahlquist [3] the computation of a large class of slowly convergent Fourier integrals can be cast into the problem of evaluating a certain Stieltjes integral over the interval [0, 1]. If the integrator can be shown to be non-decreasing, generalized Gauss rules can be used but this calls for the solution of a generally ill-conditioned non-linear system. In this paper we report experimental results, which illustrate the fact, that if the abscissae are chosen according to a simple strategy and then the weights are computed from a linear system, the resulting rules of Lagrangian type are almost optimal in a certain sense.ZusammenfassungWieGustafson-Dahlquist [3] gezeigt haben, kann für eine große Klasse von Funktionen, die langsam abnehmen, das Problem das Fourierintegral zu berechnen, so umformuliert werden, daß man ein Stieltjesintegral über das Intervall [0, 1] auswerten soll. Wenn man zeigen kann, daß der Integrand nicht-abnehmend ist, können verallgemeinerte Quadraturformeln vom Gaußschen Typus benutzt werden, aber dann muß man ein Gleichungssystem lösen, daß nicht-linear und häufig instabil ist. In diesem Aufsatz präsentieren wir experimentelle Ergebnisse die illustrieren, daß wenn man die Abszissen gemäß einer einfachen Regel wählt und dann die Gewichte von einem linearen System berechnet, Lagrangesche Formeln erhalten werden, die in einer gewissen Meinung fast-optimal sind.

Stable convergence acceleration using Laplace transforms

SummaryWe consider the general class of power series where the terms may be expressed as the Laplace transforms of known functions. The sum of the series can then be evaluated efficiently and accurately by means of quadrature schemes, recently published by Frank Stenger. The method works also far outside the region of convergence as will be illustrated by numerical examples.