William F. Moss

Decay rates for inverses of band matrices

Spectral theory and classical approximation theory are used to give a new proof of the exponential decay of the entries of the inverse of band matrices. The rate of decay of A 1 can be bounded in terms of the (essential) spectrum of AA* for general A and in terms of the (essential) spectrum of A for positive definite A. In the positive definite case the bound can be attained. These results are used to establish the exponential decay for a class of generalized eigenvalue problems and to establish exponential decay for certain sparse but nonbanded matrices. We also establish decay rates for certain generalized inverses.

Backward error analysis for a pole assignment algorithm

Of the six or so pole assignment algorithms currently available, several have been claimed to be numerically stable, but no proofs have been published to date. It is shown, by performing a backward error analysis, that one of these algorithms, due to Petkov, Christov, and Konstantinov [IEEE Trans. Automat. Control, AC–29 (1984), pp. 1045–10481 is numerically stable.

Using parallel banded linear system solvers in generalized eigenvalue problems

Subspace iteration is a reliable and cost effective method for solving positive definite banded symmetric generalized eigenproblems, especially in the case of large scale problems. This paper discusses an algorithm that makes use of two parallel banded solvers in subspace iteration. A shift is introduced to decompose the banded linear systems into relatively independent subsystems and to accelerate the iterations. With this shift, an eigenproblem is mapped efficiently into the memories of a multiprocessor and a high speedup is obtained for parallel implementations. An optimal shift is a shift that balances total computation and communication costs. Under certain conditions, we show how to estimate an optimal shift analytically using the decay rate for the inverse of a banded matrix, and how to improve this estimate. Computational results on iPSC/2 and iPSC/860 multiprocessors are presented.

Backward error analysis for a pole assignment algorithm II: the complex case

In a previous paper [SIAM J. Matrix Anal. Appl., 10 (1989), pp. 446–456], Cox and Moss proved that the pole assignment algorithm of Petkov, Christov, and Konstantinov [IEEE Trans. Automat. Control, AC-29 (1984), pp. 1045–1048] is numerically stable for the real case. In this paper, a modified version of the algorithm of Petkov, Christov, and Konstantinov for the complex case is analyzed and the full algorithm (real and complex) is shown to be numerically stable.

Fundamental solutions of degenerate or singular elliptic equations

Abstract Exitence of fundamental solutions is established for a class of degenerate or singular, second-order, linear elliptic partial differential equations. This class contains, for example, Tricomis equation in the upper half-plane which arises in the study of aerodynamics; the equation of Weinsteins generalized axially symmetric potential theory which arises in the study of fluid dynamics and elasticity; and Schrodingers equation with a singular potential which arises in quantum mechanics.

Improved numerical modelling of competitive counterion binding in polyelectrolyte solutions

Abstract In a previous study of the system Mg2+/Na+/polyion, polyion = polygalacturonate, the relative suppression of the binding of Na+ by the presence of Mg2+ was found to be nearly linearly related to the ratio P b,Mg P b,Na , where Pb,Mg and Pb,Na are the numbers of counterions per unit charge associated with or bound to the polyion, in the pure magnesium and sodium salts of the polyelectrolyte, as calculated from the relationships based on the mathematical formulation of the Poisson-Boltzmann (PB) model using the method of Peterlin and Dolar. Subsequent numerical studies involved polyions which had smaller interchange distances and the Peterlin and Dolar approach was not able to model these systems. This work describes a shooting method that successfully treats a wider range of polyion systems. The dependence of competitive counterion binding properties on representative polyelectrolyte parameters and concentration predicted by the model are also provided.

Special session - enhancing student learning using SCALE-UP format

SCALE-UP (Student-Centered Activities for Large Enrollment Undergraduate Programs) is a specialized active learning format that relies largely upon social interaction among students, instructor, and learning assistants. The instructor and learning assistants serve as facilitators of guided inquiry by asking students leading questions as they work through class assignments. On-going, real time formative assessments ensure that the instructor is constantly aware of which students are mastering the material and which are struggling. The SCALE-UP format is currently used at our institution in the General Engineering program (all sections of the first year courses), Math Sciences (all sections of first year calculus, and one section of second year calculus), Civil Engineering (one section of engineering statics), Mechanical Engineering (all sections of engineering statics and dynamics), as well as in courses in Horticulture, Nursing, English, and Computer Science. This special session focuses on strategies for successful implementation of this pedagogical innovation. These include development of student activities, formative assessments, training for instructors and learning assistants, and the physical features of the learning environment.

Numerical and experimental studies of territorial binding of counterions in polyelectrolyte solutions including the added salt case

Abstract This paper provides an extension of a procedure for calculating counterion binding ratios in a polyelectrolyte solution based on the Poisson-Boltzmann formulation of territorial counterion binding to include the presence of excess salt of the counterion. It is an extension of an earlier version which dealt with two counterions with different charge numbers without excess salt. The relative binding parameter of sodium to poly(galacturonic acid) as a function of the degree of neutralization in the presence of excess sodium chloride is determined by 23 Na n.m.r. and compared with the numerical results from the Poisson-Boltzmann and the Manning condensation theories. Agreement is found with the Poisson-Boltzmann results at degrees of neutralization greater than the critical condensation point predicted by the Manning theory. Qualitative elements of both theories appear in the experimental results.

Nonlinear eigenvalue approximation

SummaryFor each λ in some domainD in the complex plane, letF(λ) be a linear, compact operator on a Banach spaceX and letF be holomorphic in λ. Assuming that there is a ξ so thatI−F(ξ) is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue ξ. In the Newton method the smallest eigenvalue of the operator pencil [I−F(λ),F′(λ)] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I−F(ξ),I] with algebraic multiplicitym. For fixed λ leth(λ) denote the arithmetic mean of them eigenvalues of the pencil [I−F(λ),I] which are closest to 0. Thenh is holomorphic in a neighborhood of ξ andh(ξ)=0. Under suitable hypotheses the classical Mullers method applied toh converges locally with order approximately 1.84.

Collocation for an integral equation arising in duct acoustics

Abstract A model has been developed for spinning mode acoustic radiation from the inlet of an air-craft engine. Consider the region bounded by the z -axis and the curve PABCDEF in Fig. 2. The model inlet is the solid of revolution obtained by rotating this region about the z -axis. The circular disk S 1 generated by rotating the line segment C 1 = 0 B separates the interior of the inlet from its exterior. The interior acoustic pressure consists of a pure azimuthal mode for a hardwall boundary condition. The interior and exterior acoustic pressures and their normal derivatives are matched on SI. A hardwall boundary condition is applied on the surface S 2 generated by rotating the curve C 2 = BCDEF The governing boundary value problem for the Helmholtz equation is first converted into an integral equation for the unknown acoustic pressure on S 1 + S 2, and then the azimuthal dependence is integrated out yielding a one-dimensional integral equation over C 1 + C 2. We approximate the pressure on C 1 by a truncated interior modal expansion and on C 2 by a linear spline.