Solving Systems of Non-Linear Equations by Broyden's Method with Projected Updates

We introduce a modification of Broyden's method for finding a zero of n nonlinear equations in n unknowns when analytic derivatives are not available. The method retains the local Q-superlinear convergence of Broyden's method and has the additional property that if any or all of the equations are linear, it locates a zero of these equations in n+1 or fewer iterations. Limited computational experience suggests that our modification often improves upon Eroyden's method.

Flexible language constructs for large parallel programs

The goal of the research described is to develop flexible language constructs for writing large data parallel numerical programs for distributed memory (MIMD) multiprocessors. Previously, several models have been developed to support synchronization and communication. Models for global synchronization include SIMD (Single Instruction Multiple Data), SPMD (Single Program Multiple Data), and sequential programs annotated with data distribution statements. The two primary models for communication include implicit communication based on shared memory and explicit communication based on messages. None of these models by themselves seem sufficient to permit the natural and efficient expression of the variety of algorithms that occur in large scientific computations. An overview of a new language that combines many of these programming models in a clean manner is given. This is done in a modular fashion such that different models can be combined to support large programs. Within a module, the selection of a model depends on the algorithm and its efficiency requirements. An overview of the language and discussion of some of the critical implementation details is given

A comparison of the NOGAPS and GFDN dynamical track prediction models during the 1997 western North Pacific typhoon season

The performance of both the U.S. Navy (NOGAPS) and regional (GFDN) dynamical track prediction models during the 1997 western North Pacific typhoon season is documented. In the context of the Systematic Approach of Carr and Elsberry, a knowledge base of six conceptual models (summary in Table 8.1) is proposed that associates recurring tropical cyclone (TC) forecast track errors with various types of TC and environmental structures. Twenty-one storms of the 27 analyzed have periods in which at least one significant track error source was identified (summary in Table 8.3). More situations (23) were identified in the NOGAPS forecasts than in the GFDN forecasts (14). Individual case studies are presented to illustrate recurring scenarios with poor performance in either the NOGAPS model, GFDN model, or both. Use of these conceptual models and their supporting case studies may allow the JTWC forecaster to better understand how the NOGAPS model and GFDN model may perform in specified synoptic environments. It is hoped that the JTWC forecaster can use the information in this study to provide more accurate TC tracks by rejecting inappropriate model guidance during future typhoon seasons in the western North Pacific. In addition, this study may provide feedback to dynamical model producer as to situations in which large track errors have occurred, in hopes that the model might be improved in the futurehttp://archive.org/details/acomparisonofnog109458124Lieutenant, United States NavyApproved for public release; distribution is unlimited

On the Q-Superlinear Convergence of Self-Scaling Quasi-Newton Methods ; CU-CS-144-78

Self-scaling updates have been proposed by Luenberger, Oren, and Spedicato for use in quasi-Newton minimization algorithms. Their departure from other updates is that the intermediate update Hi=ɤi Hi to the inverse Hessian approximation is performed before each regular update. In recent computational tests by Brodlie and Shanno and Phua, they performed less well than the BFGS update except on problems with a singular Hessian at the solution. In this paper we examine the self-scaling updates in an attempt to explain this behavior. We find that for the self-scaling BFGS update to retain the Q-superlinear convergence of the normal BFGS on problems with a non-dingular Hessian at the solution, it is necessary that ɤi converge to 1; a somewhat stronger condition is sufficient. This indicates that asymptotically, use of the scaling parameter is unlikely to be advantageous on non-singular problems. On the other hand, on problmes with a singular Hessian at the solution, where only linear convergenceis expected in general, ɤi does not necessarily converge to 1, so that the self-scaling update may differ from the BFGS even asymptotically

Determining Feasibility of a Set of Nonlinear Inequality Constraints ; CU-CS-172-80

We present a new class of algorithms for determining whether there exists a point x ɛ Rn satisfying the nonlinear inequality constraints ci(x) ≤ 0, i=1, …, m, subject perhaps to satisfying linear inequality constraints lj¬(x) ≤0, j=1, …,k which are known to be feasible. Our algorithm consists of solving a sequence of linearly constrained optimization problems, using a sequence of objective functions ф(x,p) which are at least twice continuously differentiable, and which are generated by monotonically increasing the value of the non-negative parameter p. It is shown that in almost all cases, once p reaches or exceeds some finite value, that the solution to the linearily constrained optimization problem either is a feasible point, or establishes the infeasibility of the set of constraints. Computational results are presented in which the algorithm performs satisfactorily on feasible and on infeasible systems

Developing Modular Software for Unconstrained Optimization ; CU-CS-148-79

In this paper we support the use of modular software in constructing medium to large size numerical algorithms or systems of algorithms. First we discuss the advantages of modular numerical software, which include ease of development, testing, use and modification. Then we suggest a way of progressing from the initial high-level design of the modular system to the computer code through a series of descriptions which also serve as an important aid to understanding the system. As an example we use our recently developed system of quasi-Newton algorithms for unconstrained optimization. Our final pre-code description stage causes us to be interested in programming languages which allow user specification and efficient execution of basic data operations