THE APPLICATION OF NONLINEAR PROGRAMMING TO THE OPTIMIZATION OF MULTISTEP METHODS FOR THE NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS

There are two aspects of this study that should be of special interest to researchers in the field of numerical methods for the solution of ordinary differential equations. These aspects are (1) the systematic use of the concepts and methods of nonlinear programming to optimize multistep methods for the solution of ordinary differential equations, (2) the relaxation of the usual conditions for stepsizes approaching zero, thus emphasizing the essential requirements for multistep methods and allowing for greater improvements in the important aspects of performance throughout an interval of stepsizes.; Chapter I is an introduction to and summary statement of the research and results obtained.; Chapter II is a set of notes on the optimization problem as conceived and developed in 1976 and contains some topological results and formulas regarding the roots of polynomials. These results are especially directed toward application to optimization in which the characteristic polynomial defining the stability of multistep methods is used. Chapter II has its own table of contents.; Chapter III consists of some additional notes for the development of specific formulas used in the multistep methods optimizer.; Chapter IV is documentation describing the general design of the multistep methods optimizer.; Chapter V describes experimental results using the multistep methods optimizer. Substantial improvement in corrector formulas over the comparable Adams-Moulton formulas were obtained for an entire interval of stepsizes. Alternatively, the optimized corrector formulas are a substantial improvement over the Adams-Moulton methods for a fixed stepsize and a class of differential equations. A "least sixth power" optimized corrector obtained was further tested successfully on a ("wobbly") variable-coefficient linear ordinary differential equation.; A listing of the multistep methods optimizer and a listing of the multistep methods tester are provided in the addendum.