Iterative schemes for linear and non-linear problems with their applications

Abstract

Computational methods have been successfully applied to study practical problems such as electrical circuits, chemical reactions, modal analysis, analysis of statically determinant trusses, physical law, biological phenomena, and computational economics with the advent of the modern high-speed electronic digital computers. The art and science of preparing and solving scientific and engineering problems have undergone considerable changes. This is due to the complex structure interpreted in the mathematical modeling of many real-life phenomena. At the same time, numerical answers to problems are never needed exactly. Rather, an approximation to the answer is required, which is accurate ``to a certain number of decimal places" or accurate within a given tolerance. So, many numerical methods for finding the solution of a given problem merely produce a sequence of approximations which is shown to converge to the desired answer. The main objectives of this thesis are to address the two most frequently occurring problems in scientific work. One of the major concerns is determining the best approximate solution for scalar non-linear equations that include polynomials of higher-degree, non-polynomial expressions, and non-smooth functions. Secondly, it deals with the computation of different generalized inverses that are further used for finding solutions of given linear systems. As the name indicates, the generalized inverse is the axiomatic study of the intuitive notions of non-singular matrix inverse and pseudoinverse of rectangular or rank-deficient square matrices, which are the central themes of matrix theory. We shall be interested in constructing iterative methods to solve the above-mentioned problems. A systematic proof of the convergence of the developed methods to the desired solution is discussed. Also, an attempt has been made to follow the essential standard of numerical iterative procedure that can be gained through computational efficiency. For computing the matrix inverse using iterative techniques, some vital factors are concerned in order to measure the efficacy of a method under the fact of asymptotically stable Schulz-type methods. Some factors are efficiency index, convergence order, count of matrix multiplications with matrix, termination criteria, count of iterations, etc. Keeping these points in mind, we will try to answer: ``Is the computational complexity of the new iterative algorithm reasonable?"