John A. Wisniewski

A Trace Minimization Algorithm for the Generalized Eigenvalue Problem

An algorithm for computing a few of the smallest (or largest) eigenvalues and associated eigenvectors of the large sparse generalized eigenvalue problem Ax = ABx is presented. The matrices A and B are assumed to be symmetric, and haphazardly sparse, with B being positive definite. The problem is treated as one of constrained optimization and an inverse iteration is developed which requires the solution of linear algebraic systems only to the accuracy demanded by a given subspace. The rate of convergence of the method is established, and a technique for improving it is discussed. Numerical experiments and comparisons with other methods are presented.

Algorithm 594: Software for Relative Error Analysis

This package performs an automatic round-off error analysis of numerical algorithms, which is useful in determining the numerical propert ies (stability) of such algorithms. The approach taken with this software differs f rom tha t of Miller [4, 5] by dealing with relative as opposed to absolute error analysis. This is the natural approach taken for dealing with floating-point ari thmetic. In addition, this software gives an accurate analysis of an algori thm for some fixed data, ra ther than searches for data for which an approximate analysis indicates tha t the algorithm performs poorly. I t uses one of two methods, e i ther an exact method [1] or a heuristic algori thm [1-3]. The following paragraphs summarize the previous work in [2], and it is assumed tha t the reader is familiar with the concepts and techniques presented there. To perform the error analysis, the algori thm being analyzed is represented as a directed graph with each numeric quant i ty being a node in this graph. Directed arcs lead from operands to their results. Each nodes value is subject to error, called local relative error. This error affects those subsequent nodes tha t are computed using the contaminated node as an operand. All of the local relat ive errors together are used to produce a total relative error for each node. F rom the

Module Positioning Algorithms for Rectilinear Macrocell Assemblies

A completely hierarchical approach to integrated circuit design begins by partitioning a design problem into subproblems which are based on functional boundaries. It is desirable to produce a final layout which is compact, yet preserves the functional decomposition. Allowing the physical macrocells to have arbitrary rectilinear shapes permits this goal to be achieved but introduces many levels of complexity into the modeling of the assembly. To support macrocells with rectilinear shapes, a directed graph, referred to as an adjacency graph is used to model the positional relationship of the components in the assembly. Algorithms are presented for constructing the adjacency graphs, identifying the cycles present in the adjacency graph, converting the graph to an acyclic graph, and for establishing the component and channel positions based on a critical path analysis. These algorithms are implemented in Pascal on a DECSYSTEM-20.

A mathematical programming updating method using modified Givens transformations and applied to LP problems

An efficient and numerically stable method is presented for the problem of updating an orthogonal decomposition of a matrix of column (or row) vectors. The fundamental idea is to add a column (or row) analogous to adding an additional row of data in a linear least squares problem. A column (or row) is dropped by a formal scaling with the imaginary unit, √-1, followed by least squares addition of the column (or row). The elimination process for the procedure is successive application of the Givens transformation in modified (more efficient) form. These ideas are illustrated with an implementation of the revised simplex method. The algorithm is a general purpose one that does not account for any particular structure or sparsity in the equations. Some suggested computational tests for determining signs of various controlling parameters in the revised simplex algorithm are mentioned. A simple means of constructing test cases and some sample computing times are presented.

Parallel Algorithms for Network Routing Problems and Recurrences

In this paper, we consider the parallel solution of recurrences, and linear systems in the regular algebra of Carre. These problems are equivalent to solving the shortest path problem in graph theory, and they also arise in the analysis of Fortran programs. Our methods for solving linear systems in the regular algebra are analogues of well-known methods for solving systems of linear algebraic equations. A parallel version of Dijkstra’s method, which has no linear algebraic analogue, is presented. Considerations for choosing an algorithm when the problem is large and sparse are also discussed.

IC Design Capability Conversion from Mainframe to Workstation Environment

Over the last eighteen months, Sandia National Laboratories has converted its IC design facility from a mainframe to a workstation environment. Although workstations were advertised as turnkey systems, augmentation of vendor software with Sandia-developed software greatly enhanced Sandias specific IC design capabilities. This article examines advantages of the workstation approach for sophisticated CAD tool users, discussing complete hardware, software, and networking capacities and capabilities.

Some experiments with computing the complex absolute value

Portability for the numerical analyst not only includes writing codes which will compile and execute on a variety of machines, but also writing codes which are robust and accurate as well. This is particularly important for the computation of the complex absolute value. This computation, the computation of the 2-norm of a 2-vector, also occurs in the formation of Givens transformations [3] which are used in the solution of the symmetric eigenvalue problem, the computation of the singular value decomposition, and in the solution of linear least squares problems.

Computer-Aided Design: Leading or responding?

It is shown that past and current CAD algorithm development for IC design has focused on meeting the challenges of the day, rather than on the needs of the perceivable future. It is argued that new tools required by an emerging technology should be available as soon as the technology is available. Consequently, work on the tools must begin while the technology is being developed. This is not an unrealistic goal, since all major mainframe computers have their operating systems and compilers under development while prototype hardware is being built in the laboratory.