On the fractal characteristics of a stabilised Newton method

In this report, we present a complete theory for the fractal that is obtained when applying Newton's Method to find the roots of a complex cubic. We show that a modified Newton's Method improves convergence and does not yield a fractal, but basins of attraction with smooth borders. Extensions to higher-order polynomials and the numerical relevance of this fractal analysis are discussed

Numerical integration of asymptotic solutions of ordinary differential equations

Classical asymptotic analysis of ordinary differential equations derives approximate solutions that are numerically stable. However, the analysis also leads to tedious expansions in powers of the relevant parameter for a particular problem. The expansions are replaced with integrals that can be evaluated by numerical integration. The resulting numerical solutions retain the linear independence that is the main advantage of asymptotic solutions. Examples, including the Falkner-Skan equation from laminar boundary layer theory, illustrate the method of asymptotic analysis with numerical integration

Grey-box model identification via evolutionary computing

This paper presents an evolutionary grey-box model identification methodology that makes the best use of a priori knowledge on a clear-box model with a global structural representation of the physical system under study, whilst incorporating accurate blackbox models for immeasurable and local nonlinearities of a practical system. The evolutionary technique is applied to building dominant structural identification with local parametric tuning without the need of a differentiable performance index in the presence of noisy data. It is shown that the evolutionary technique provides an excellent fitting performance and is capable of accommodating multiple objectives such as to examine the relationships between model complexity and fitting accuracy during the model building process. Validation results show that the proposed method offers robust, uncluttered and accurate models for two practical systems. It is expected that this type of grey-box models will accommodate many practical engineering systems for a better modelling accuracy

An improved Newton iteration for the generalized inverse of a matrix, with applications

The purpose here is to clarify and illustrate the potential for the use of variants of Newton's method of solving problems of practical interest on highly personal computers. The authors show how to accelerate the method substantially and how to modify it successfully to cope with ill-conditioned matrices. The authors conclude that Newton's method can be of value for some interesting computations, especially in parallel and other computing environments in which matrix products are especially easy to work with

Convergence of Newton's method for a single real equation

Newton's method for finding the zeroes of a single real function is investigated in some detail. Convergence is generally checked using the Contraction Mapping Theorem which yields sufficient but not necessary conditions for convergence of the general single point iteration method. The resulting convergence intervals are frequently considerably smaller than actual convergence zones. For a specific single point iteration method, such as Newton's method, better estimates of regions of convergence should be possible. A technique is described which, under certain conditions (frequently satisfied by well behaved functions) gives much larger zones where convergence is guaranteed

Buckling of imperfect cylinders under axial compression

Donnell equations, Newton method, and numerical solution applied to buckling of imperfect cylinders under axial compressio

NEWSUMT: A FORTRAN program for inequality constrained function minimization, users guide

A computer program written in FORTRAN subroutine form for the solution of linear and nonlinear constrained and unconstrained function minimization problems is presented. The algorithm is the sequence of unconstrained minimizations using the Newton's method for unconstrained function minimizations. The use of NEWSUMT and the definition of all parameters are described

Mathematical description and program documentation for CLASSY, an adaptive maximum likelihood clustering method

Discussed in this report is the clustering algorithm CLASSY, including detailed descriptions of its general structure and mathematical background and of the various major subroutines. The report provides a development of the logic and equations used with specific reference to program variables. Some comments on timing and proposed optimization techniques are included