Gaussian elimination as an iterative algorithm

Gaussian elimination (GE) for solving an $n \times n$ linear system of equations $Ax=b$ is the archetypical direct method of numerical linear algebra, as opposed to iterative. In this note we want to point out that GE has an iterative side too

Radical Artificial Intelligence: A Postmodern Approach

The dynamic response of end-clamped monolithic beams and sandwich beams has been measured by loading the beams at mid-span using metal foam projectiles. The AISI 304 stainless-steel sandwich beams comprise two identical face sheets and either prismatic Y-frame or corrugated cores. The resistance to shock loading is quantified by the permanent transverse deflection at mid-span of the beams as a function of projectile momentum. The prismatic cores are aligned either longitudinally along the beam length or transversely. It is found that the sandwich beams with a longitudinal core orientation have a higher shock resistance than the monolithic beams of equal mass. In contrast, the performance of the sandwich beams with a transverse core orientation is very similar to that of the monolithic beams. Three-dimensional finite element (FE) simulations are in good agreement with the measured responses. The FE calculations indicate that strain concentrations in the sandwich beams occur at joints within the cores and between the core and face sheets; the level of maximum strain is similar for the Y-frame and corrugated core beams for a given value of projectile momentum. The experimental and FE results taken together reveal that Y-frame and corrugated core sandwich beams of equal mass have similar dynamic performances in terms of rear-face deflection, degree of core compression and level of strain within the beam

Connectionist Inference Models

The performance of symbolic inference tasks has long been a challenge to connectionists. In this paper, we present an extended survey of this area. Existing connectionist inference systems are reviewed, with particular reference to how they perform variable binding and rule-based reasoning, and whether they involve distributed or localist representations. The benefits and disadvantages of different representations and systems are outlined, and conclusions drawn regarding the capabilities of connectionist inference systems when compared with symbolic inference systems or when used for cognitive modeling

An extension of Chebfun to two dimensions

An object-oriented MATLAB system is described that extends the capabilities of Chebfun to smooth functions of two variables defined on rectangles. Functions are approximated to essentially machine precision by using iterative Gaussian elimination with complete pivoting to form “chebfun2” objects representing low rank approximations. Operations such as integration, differentiation, function evaluation, and transforms are particularly efficient. Global optimization, the singular value decomposition, and rootfinding are also extended to chebfun2 objects. Numerical applications are presented

Computing with functions in spherical and polar geometries I. The sphere

A collection of algorithms is described for numerically computing with smooth functions defined on the unit sphere. Functions are approximated to essentially machine precision by using a structure-preserving iterative variant of Gaussian elimination together with the double Fourier sphere method. We show that this procedure allows for stable differentiation, reduces the oversampling of functions near the poles, and converges for certain analytic functions. Operations such as function evaluation, differentiation, and integration are particularly efficient and can be computed by essentially one-dimensional algorithms. A highlight is an optimal complexity direct solver for Poisson's equation on the sphere using a spectral method. Without parallelization, we solve Poisson's equation with $100$ million degrees of freedom in one minute on a standard laptop. Numerical results are presented throughout. In a companion paper (part II) we extend the ideas presented here to computing with functions on the disk.Comment: 23 page

Digital Signal Processing

Contains an introduction and reports on twenty research projects.National Science Foundation (Grant ECS 84-07285)U.S. Navy - Office of Naval Research (Contract N00014-81-K-0742)National Science Foundation FellowshipSanders Associates, Inc.U.S. Air Force - Office of Scientific Research (Contract F19628-85-K-0028)Canada, Bell Northern Research ScholarshipCanada, Fonds pour la Formation de Chercheurs et l'Aide a la Recherche Postgraduate FellowshipCanada, Natural Science and Engineering Research Council Postgraduate FellowshipU.S. Navy - Office of Naval Research (Contract N00014-81-K-0472)Fanny and John Hertz Foundation FellowshipCenter for Advanced Television StudiesAmoco Foundation FellowshipU.S. Air Force - Office of Scientific Research (Contract F19628-85-K-0028