Solving Transcendental Equations using Genetic Algorithms

In the following research project, it was attempted to develope an application to solve transcendental equations using Genetic Algorithms (GAs). A draft paper was written which was presented at DITECH 2002, technical festival of NSIT. A well structured paper with proper results is in the pipeline.

Abstract

Traditionally, many different analytical/iterative methods (such as Newton Raphson, Bisection method, etc.) are used for solving transcendental equations. But these methods fail or work inefficiently in many cases. Infact, none of this method seeks to find all solutions of a given transcendental equation in a given range. This is a big drawback because, many physical problems yielding transcendental equations have more than a single solution (eg. z*tan(z)=b, One dimensional transient Heat conduction Problem with one adiabatic and one convective boundary condition). In such cases, exhaustive search remains the only option when using traditional root finding methods.

Apart from this, these methods suffer from many other drawbacks, examples: these methods fail in case of misbehaved, discrete or discontinuous functions, etc The paper studies these cases in detail.

It is unbelievable, that even after existence of all these limitations, popular softwares such as Maple, Mathematica, MATLAB, etc. seek to find only ONE root, that too by traditional methods (Newton Raphson). Due to the same reason, they are many a times unable to find the root.

Therefore use of genetic algorithms to solve transcendental equations is suggested. Two schemes are being worked upon. In the first case, to ensure no loss of generality, only Genetic Algorithms are being used to find roots. In the second case, a hybrid algorithm is being used. Genetic algorithms are being used to search for the regions where the root may lie and then traditional methods are being used to find the root. This way, the roots can be found more accurately in lesser time. But at the same time, the method partially suffers from the same disadvantages as traditional methods. The first method though slower, but is a general method and shall work well in most cases.

Implementation: The equation at hand may have many roots i.e. a multimodal fitness landscape, hence multimodality has been discussed in detail in the paper (including literature survey, improvements and frontiers) and methods to ensure identification of all roots are developed. Primarily, 'The Clearing Procedure' of Petrowski is being used. Experimental results have been included (carried out in MATLAB).

A complete model to implement the problem has been prescribed and parrallelization of the algorithm is also being attempted.

Acknowledgement

A special thanks to Dr. Alan Petrowski, National Institute of Telecommunication, France, without whose help and guidance, this paper couldn't have been put together. Apart from him, I thank the following people for giving me suggestions and answering my queries through email.

• Jean-Philippe Rennard, Field of Alife, (http://www.rennard.org/alife/)

• Roger Wainwright, University of Tulsa, USA

• David Andre, University of California, Berkley, USA

• Everett Carter, California, USA