Obtaining parameter estimates for nonlinear regression model using gauss-newton and gradient-based methods present some complex analytical challenges. In this paper we investigated the effectiveness and simplicity of Particle Swarm Optimization (PSO) and Genetic Algorithm (GA) on five nonlinear regression models with varying level of complexities. We developed function in R-programming for each models and performed 30 independent runs for at least 100 iterations for both PSO and GA. We evaluated PSO and GA performance in view of computation time, residual error produced and compared our results with values published online. Based on the results obtained, PSO significantly outperform GA in view of computation time and quality of parameter estimates. Even so, GA required fewer iterations and produced fairly accurate results. Further investigation shows that PSO and GA are both competitive, effective, simple to implement, and can be considered reliable for obtaining the parameter estimates of different nonlinear regression tasks.
Most recently, higher order problems are being addressed by decomposing it into system of lower order problems. However, it was discovered that methods with high order and strong stability were able to approximate the resulting systems accurately as the problems become unstable in the region of the new transformation field. This research actually sought for methods of solution of higher order problems without any need for system transformation. The method is proposed for the direct solution of fourth order ordinary differential equations. The fundamental basis is sought from the combination of Shifted Chebyshev Orthogonal Polynomial and the Hermite Orthogonal Polynomial, these polynomial functions are then used to obtain the method using the concept of interpolation and collocation. The proposed method is found to be consistent and zero-stable, which then implies convergence. From the numerical results obtained, the efficiency of the method was obtained and its superiority strength was also established when comparison was made with existing.
This paper reviews the theory of matrices and determinants. Matrix and determinant are nowadays considered inseparable to some extent, but the determinant was discovered over two centuries before the term matrix was coined. Our review associate determinant with the matrix as part of linear systems but not with polynomials. Thus, the paper first gives the background on matrix with vast applications in all fields of study and then reviews the history of determinants which is based on its major contributors in chronological order from the sixteenth century to the twenty-first century
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