Showing papers in "Siam Review in 1985"
TL;DR: The paper is intended to give a survey on the state-of-the-art of extrapolation methods for initial value problems in ordinary differential equation systems.
Abstract: The paper is intended to give a survey on the state-of-the-art of extrapolation methods for initial value problems in ordinary differential equation systems. All basic discretizations, which are su...
222 citations
TL;DR: A survey of theories for the generation and maintenance of spatial pattern in reaction-diffusion equations and their generalizations is presented in this paper, with special emphasis on nonlocal interaction, as manifested by the inclusion of terms involving higher derivatives or integrals.
Abstract: A survey is presented of theories for the generation and maintenance of spatial pattern in reaction-diffusion equations and their generalizations. Applications are selected from the biological sciences and physical chemistry. Special emphasis is placed on nonlocal interaction, as manifested by the inclusion of terms involving higher derivatives or integrals. It is stressed that traditional ideas of spatial pattern generation can usefully be extended to the study of pattern in general descriptive (“aspect”) variables, particularly in understanding ecological diversity and heterogeneity.
193 citations
TL;DR: This paper reviews three topics in queueing network theory: queue length processes, sojourn times, and flow processes, and compares continuous-time processes with embedded processes.
Abstract: In this paper we review three topics in queueing network theory: queue length processes, sojourn times, and flow processes. In the discussion of the queue length processes we present results for the continuous-time process and several embedded processes. Then we compare continuous-time processes with embedded processes. In considerable generality we present results for mean sojourn times and discuss the distributions of sojourn times. In the discussion of flow processes we present results for various queueing systems. Our bibliography of over 300 references, while not exhaustive, does cover the major papers for the topics considered.
169 citations
TL;DR: In this paper, Richardson extrapolation, overrelaxation, bordering by a singularity condition and appending the usual linear model by a quadratic term are described in detail.
Abstract: The paper begins with a review of the convergence theory for Newton’s method near simple and other regular singularities, followed by a brief discussion of the inherent difficulty of singular problems and the effect of rounding errors on the achievable solution accuracy. Then several ways to accelerate the slow convergence of Newton’s method are described in some detail. These are: Richardson extrapolation, overrelaxation, bordering by a singularity condition and appending the usual linear model by a quadratic term. The last two approaches require limited second derivative information which can be obtained analytically, by differencing or through secant updating. Numerical comparisons reported in the final sections lead to the conclusion that the collection of the additional information pays off in terms of speed and accuracy.
122 citations
80 citations
TL;DR: In this paper, it was shown that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed, and how this stability is related to the use of the boundary conditions.
Abstract: The ordinary differential equations occurring in linear boundary value problems characteristically have both stable and unstable solution modes. Therefore a stable numerical algorithm should avoid both forward and backward integration of solutions on large intervals. It is shown that most methods (like multiple shooting, collocation, invariant imbedding and difference methods) derive their stability from the fact that they all decouple the continuous or the discrete problem sooner or later (for instance when solving a linear system). This decoupling is related to the dichotomy of the ordinary differential equations. In fact it turns out that the inherent initial value instability is an important prerequisite for a stable utilization of the decoupled representations from which the solutions are computed. How this stability is related to the use of the boundary conditions is also investigated.
62 citations
62 citations
49 citations
42 citations
TL;DR: In this paper, Levinson's algorithm is developed in the context of mean-square estimation and is applied to a variety of topics related to Wiener filtering and spectral estimation, including prediction theory, Wold's decomposition, lattice filters, autoregressive processes, the method of maximum entropy, and the general class of extrapolating spectra.
Abstract: Levinson’s algorithm is developed in the context of mean-square estimation and is applied to a variety of topics related to Wiener filtering and spectral estimation. The study includes the innovations approach to prediction theory, Wold’s decomposition, lattice filters, autoregressive processes, the method of maximum entropy, and the general class of extrapolating spectra.
42 citations
TL;DR: In this paper, the integral of the mean curvature vector over a closed surface was shown to vanish, and two methods for proving that the integral vanishes were presented, respectively.
Abstract: We present two methods for proving that the integral of the mean curvature vector over a closed surface vanishes.
TL;DR: The probability of winning a simple game of competing Poisson processes turns out to be equal to the well-known Bessel function integral $J(x,y)$ (cf. Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962).
Abstract: The probability of winning a simple game of competing Poisson processes turns out to be equal to the well-known Bessel function integral $J(x,y)$ (cf. Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962). Several properties of J, some of which seem to be new, follow quite easily from this probabilistic interpretation. The results are applied to the random telegraph process as considered by Kac [Rocky Mountain J. Math., 4 (1974), pp. 497–509].
TL;DR: The classical Huygens' principle implies that distortionless wave propagation is possible only in odd dimensions as mentioned in this paper, and it is known that radially symmetric wave propagation can be achieved only in dimensions one and three.
Abstract: The classical Huygens’ principle implies that distortionless wave propagation is possible only in odd dimensions. A little known clarifications of this principle, due to Duffin and Courant, states that radially symmetric wave propagation is possible only in dimensions one and three. This paper presents an elementary proof of this result.
TL;DR: In this article, the probability of a player winning a game of racquetball is calculated as a function of the probability p of his winning a rally when he serves and the probability q of her opponent serves.
Abstract: The probability of a player winning a game of racquetball is calculated as a function of the probability p of his winning a rally when he serves and the probability q of his winning a rally when his opponent serves.
TL;DR: In this paper, the formulae for calculating ellipsoidal approximations to the earth's shape are analyzed, and numerical convergence and error analyses are done, as well as an investigation of the propagation of observational errors.
Abstract: The formulae for calculating ellipsoidal approximations to the earth’s shape are analyzed. Numerical convergence and error analyses are done, as well as an investigation of the propagation of observational errors.