Valerie A. Miller

Successive overrelaxation methods for solving the rank deficient linear squares problem

Abstract We develop successive overrelaxation (SOR) methods for finding the least squares solution of minimal norm to the system Ax = b , (∗) where A is an m × n matrix of rank r . The methods are obtained by first augmenting the system (∗) to a block 4 × 4 consistent system of linear equations. The augmented coefficient matrix is then split by a subproper SOR splitting. An interval for the relaxation parameter in which the subproper SOR iteration matrix is semiconvergent is determined along with the optimal relaxation parameter which minimizes the modulus of the controlling eigenvalue of the SOR matrix. Since the scheme computes at first only a solution to the augmented system, it is subsequently shown how to transform such a solution to the unique solution of minimal 2-norm. Analysis of the practical implementation of the algorithms developed here will be given elsewhere.

The rank of a graph after vertex addition

Abstract Let r(G) denote the rank of the adjacency matrix of a graph G. When a vertex and its incident edges are deleted from G, the rank of the resultant graph cannot exceed r(G) and can decrease by at most 2. The problem of determining the exact effect of adding a single vertex to a graph is more difficult, since the number of edges that can be added with this vertex is variable. The rank of the new graph cannot decrease and it can increase by at most 2. We obtain results examining several cases of vertex addition.

Adaptive quadrature on a message-passing multiprocessor

Abstract We present an adaptive quadrature algorithm for message-passing multiprocessors. One node processor is specially designated to collect partial sums and dynamically balance the work-load. Four different redistribution strategies are considered. Results and timings for several different functions are given.

A workshop on computer graphics for undergraduate faculty

A week long workshop in computer graphics, co-sponsored by Georgia State University (GSU) and the National Science Foundation (NSF grant No. USE-8954402 ) was held at GSU August 20-24,1990. The twenty four participants in the workshop were faculty from undergraduate institutions and the instructors were the authors of this paper. In the following we will describe the demographics of the faculty group, the subject areas covered, the format of the workshop, the developed course materials, and the results. This workshop can possibly be used as the prototype for future such workshops.

Introduction: SIGGRAPH 2000 Educators Program

missions that embraced the basic tenets not only of teaching computer graphics and using computer graphics to facilitate learning across the curriculum, but also of innovation, creativity, diversity and inspiration. To say we were successful in our solicitation would be an understatement. The SIGGRAPH 2000 Educators Program was filled with all of these things and more. Each of our submissions was reviewed by highly qualified individuals from academia and industry. After the review process, each submission was considered by the SIGGRAPH 2000 Educators Program Jury—a panel of artists and scientists, academics and industry persons. After the program was selected, the following two papers were chosen to be submitted to Leonardo for publication. These papers were selected because the jury felt that they best captured the vast interdisciplinary nature of art and technology and the wondrous learning environments that can be created by the blurring of the standard curricular lines. We hope that Leonardo readers find these papers as innovative, creative and inspiring as we did.

The use of fractals in visualizing iterative techniques from numerical analysis (abstract)

With the advent of inexpensive and yet powerful personal computers with good graphics capability, we can now use sophisticated computer graphics as both an educational and research tool in mathematics. In this project we are interested in applying these Mathematical Visualization techniques to specific areas of the mathematics curriculum and research. Our initial focus is on comparing different numerical analysis methods; specifically to compare different techniques for determining the root of a single unknown in a single equation. Since many numerical analysis methods require an initial guess to a solution and then perform iterative computations to converge to a root of the equation, the goal is to compare the rates of convergence for the different techniques for the same type of equation. The four methods we are using are as follows: Halleys, Secant, Newtons, and Aitkenss method. We can compare the different methods by computing the same map of initial starting points versus the rate of convergence. We take a region of space, which contains the roots, and then for each point in the region, we use that point as the initial guess, determine the number of iterations, N, to convergence (if any) and then assign a color value to this N, thereby generating a fractal image. This way the different methods can be visually compared. Since the number of iterations needed for convergence is not necessarily directly related to the computer time required, e.g. one method may require a large number of very rapid iterations while a second method might require fewer but more expensive iterations, we also generate images which have been adjusted for the expected cost of each iteration.