Germund Dahlquist

A special stability problem for linear multistep methods

The trapezoidal formula has the smallest truncation error among all linear multistep methods with a certain stability property. For this method error bounds are derived which are valid under rather general conditions. In order to make sure that the error remains bounded ast → ∞, even though the product of the Lipschitz constant and the step-size is quite large, one needs not to assume much more than that the integral curve is uniformly asymptotically stable in the sense of Liapunov.

G-stability is equivalent toA-stability

In 1975 the author showed that a norm (Liapunov function) can be constructed for the stability and error analysis of a linear multistep method (and the related one-leg method) for the solution of stiff non-linear systems, provided that the system satisfies a monotonicity condition and the method possesses a property calledG-stability.In this paper it is shown thatG-stability is equivalent toA-stability. More generally, a Liapunov function exists if the stability region of the method contains a circle (half-plane), provided that the system satisfies a monotonicity condition related to this circle (half-plane). In the general case this condition depends on the stepsize.

ON ACCURACY AND UNCONDITIONAL STABILITY OF LINEAR MULTISTEP METHODS FOR SECOND ORDER DIFFERENTIAL EQUATIONS

Linear multistep methods for solution of the equationy″=f(t, y) are studied by means of the test equationy″=−ω2y, with ω real. It is shown that the order of accuracy cannot exceed 2 for an unconditionally stable method.

Bounds for the error of linear systems of equations using the theory of moments

Abstract Consider the system, of linear equations Ax = b where A is an n × n real symmetric, positive definite matrix and b is a known vector. Suppose we are given an approximation to x, ξ, and we wish to determine upper and lower bounds for ∥ x − ξ ∥ where ∥ ··· ∥ indicates the euclidean norm. Given the sequence of vectors {ri}ik = 0, where ri = Ari − 1 and r0 = b − Aξ, it is shown how to construct a sequence of upper and lower bounds for ∥ x − ξ ∥ using the theory of moments. In addition, consider the Jacobi algorithm for solving the system x = Mx + b, viz., xi + 1 = Mxi + b. It is shown that by examining δi = xi + 1 − xi, it is possible to construct upper and lower bounds for ∥ xi − x ∥.

Post-Secondary Education

Since 1970 the number of U.S. college and university students choosing to major in mathematics has declined sharply. There are several sources of data describing this decline and many conjectures about the causes. The situation is summarized in two sections of the paper below.

33 years of numerical instability, Part I

BIT has played and plays a great role in the development of concepts concerning numerical (in)stability in initial value problems forODEs and related questions. This development is here seen through the looking-glass of the author, who experienced much of its pains and pleasures. The article is based on a talk given in 1981 at the Zürich symposium to commemorate the tenth anniversary of the death of the eminent Swiss numerical analyst, Heinz Rutishauser. The presentation is mainly chronological with a few digressions. Part I ends at the beginning of the stiff epoch.

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.

Positive Functions and some Applications to Stability Questions for Numerical Methods

Publisher Summary This chapter discusses positive functions and some applications to stability questions for numerical methods. Related classes of functions have been studied in several branches of mathematics. Functions mapping the upper half-plane into itself, sometimes called Pick functions, are important in the moment problem or in spectral theory for operators in Hilbert space. The class P is regarded as a convenient normal form for such problems. It is a matter of taste whether, in a particular application, one should transform a problem concerning a related class to a class P problem or one should formulate P class properties in terms of the function class of the original statement of the problem. The spirit of the notion of stiff stability can be obtained by formulating that the region of absolute stability should contain the union of two domains of this type. The properties of positive functions give hints for a neat formulation of stability and accuracy requirements as a system of inequalities and equations.

On matrix majorants and minorants, with applications to differential equations☆

Abstract Some tools of linear algebra are collected and developed for potential use in the analysis of stiff differential equations. Bounds for the triangular factors of a large matrix are given in terms of the triangular factors of an associated “minorant” matrix of lower order. Minorants are also used to produce estimates of solutions of systems of ordinary differential equations, which may be sharper than those obtained by the use of logarithmic norms.

On the uniform power-boundedness of a family of matrices and the applications to one-leg and linear multistep methods

SummaryWe study the difference equations obtained when a linear multistep method is applied to the scalar test equationdy/dt=λy and constant stepsizeh. LetS be the region of the absolute stability of the method, and letD be a closed subset ofS (on the Riemann sphere