The PVM system is a programming environment for the development and execution of large concurrent or parallel applications that consist of many interacting, but relatively independent, components. It is intended to operate on a collection of heterogeneous computing elements interconnected by one or more networks. The participating processors may be scalar machines, multiprocessors, or special-purpose computers, enabling application components to execute on the architecture most appropriate to the algorithm. PVM provides a straightforward and general interface that permits the description of various types of algorithms (and their interactions), while the underlying infrastructure permits the execution of applications on a virtual computing environment that supports multiple parallel computation models. PVM contains facilities for concurrent, sequential, or conditional execution of application components, is portable to a variety of architectures, and supports certain forms of error detection and recovery.

PVM: a framework for parallel distributed computing

Mathematica--A System for Doing Mathematics by Computer.

A number of problems in probability and statistics can be addressed using the multivariate normal (Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the $n$ -dimensional setting, however, it requires the inversion of an $n \times n$ covariance matrix, $C$ , as well as the evaluation of its determinant, $\det (C)$ . In many cases, such as regression using Gaussian processes, the covariance matrix is of the form $C = \sigma ^2 I + K$ , where $K$ is computed using a specified covariance kernel which depends on the data and additional parameters (hyperparameters). The matrix $C$ is typically dense, causing standard direct methods for inversion and determinant evaluation to require $\mathcal {O}(n^3)$ work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix $C$ can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an $\mathcal {O} (n\,\log^2\, n)$ algorithm for inversion. More importantly, we show that this factorization enables the evaluation of the determinant $\det (C)$ , permitting the direct calculation of probabilities in high dimensions under fairly broad assumptions on the kernel defining $K$ . Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels.

Fast Direct Methods for Gaussian Processes

Mixed-mode oscillations (MMOs) are trajectories of a dynamical system in which there is an alternation between oscillations of distinct large and small amplitudes. MMOs have been observed and studied for over thirty years in chemical, physical, and biological systems. Few attempts have been made thus far to classify different patterns of MMOs, in contrast to the classification of the related phenomena of bursting oscillations. This paper gives a survey of different types of MMOs, concentrating its analysis on MMOs whose small-amplitude oscillations are produced by a local, multiple-time-scale “mechanism.” Recent work gives substantially improved insight into the mathematical properties of these mechanisms. In this survey, we unify diverse observations about MMOs and establish a systematic framework for studying their properties. Numerical methods for computing different types of invariant manifolds and their intersections are an important aspect of the analysis described in this paper.

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Mixed-Mode Oscillations with Multiple Time Scales

The National Institute of Standards and Technology is preparing a Digital Library of Mathematical Functions (DLMF) to provide useful data about special functions for a wide audience. The initial products will be a published handbook and companion Web site, both scheduled for completion in 2003. More than 50 mathematicians, physicists and computer scientists from around the world are participating in the work. The data to be covered include mathematical formulas, graphs, references, methods of computation, and links to software. Special features of the Web site include 3D interactive graphics and an equation search capability. The information technology tools that are being used are, of necessity, ones that are widely available now, even though better tools are in active development. For example, LaTeX files are being used as the common source for both the handbook and the Web site. This is the technology of choice for presentation of mathematics in print but it is not well suited to equation search, for example, or for input to computer algebra systems. These and other problems, and some partially successful work-arounds, are discussed in this paper and in the companion paper by Miller and Youssef.

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NIST Digital Library of Mathematical Functions

Modern developments in theoretical and applied science depend on knowledge of the properties of mathematical functions, from elementary trigonometric functions to the multitude of special functions. These functions appear whenever natural phenomena are studied, engineering problems are formulated, and numerical simulations are performed. They also crop up in statistics, financial models, and economic analysis. Using them effectively requires practitioners to have ready access to a reliable collection of their properties. This handbook results from a 10-year project conducted by the National Institute of Standards and Technology with an international group of expert authors and validators. Printed in full color, it is destined to replace its predecessor, the classic but long-outdated Handbook of Mathematical Functions, edited by Abramowitz and Stegun. Included with every copy of the book is a CD with a searchable PDF of each chapter.

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NIST Handbook of Mathematical Functions

Higher transcendental functions continue to play varied and important roles in investigations by engineers, mathematicians, scientists and statisticians. The purpose of this paper is to assist in locating useful approximations and software for the numerical generation of these functions, and to offer some suggestions for future developments in this field. March 1994 December 200

Daniel W. Lozier

Papers

NIST Handbook of Mathematical Functions

NIST Digital Library of Mathematical Functions

Numerical evaluation of special functions

The NIST Handbook of Mathematical Functions

Numerical solution of linear difference equations