A Study of Numerical Methods for Nonlinear Problems and Their Applications

Abstract

Computational methods have been effectively used to tackle real-world problems like global positioning systems, fluid flow, control systems, chemical reactions, computational economics, and biological and physical phenomena with the advent of modern high-speed digital electronic computers. These applications demonstrate the growing dependence on computational approaches in solving complex practical problems. Because of its practical aspect, many look for solutions that are “good enough,” but defining what “good enough” means is more difficult. Determining the required accuracy therefore becomes a central topic in computational problem solving. The methodology and techniques for addressing scientific and engineering challenges have experienced significant transformations. The increasing complexity of the nonlinear problems under study largely drives these changes. This complexity arises from the complex structures analyzed in the mathematical modeling of various real-world instances. As a result, exact analytical solutions are often difficult or impossible to obtain. Simultaneously, approximate numerical solutions to problems are often sufficient. In many cases, such approximations provide results that are both accurate and computationally efficient. An approximation of the solution is necessary, precise to a specified number of decimal places or within a defined tolerance. Various numerical techniques for solving a specific problem generate a sequence of approximations that converge to the desired solution. The quality of these approximations must be evaluated in relation to the problem’s requirements. But “good enough” depends on the application area. The primary objectives include finding the best possible approximate solutions for scalar or systems of nonlinear equations, along with higher-degree polynomials, transcendental functions, extensive multidimensional systems, and both ordinary and partial differential equations. Secondly, it addresses the calculation of the matrix sign function, which is subsequently utilized to compute solutions for nonlinear matrix equations. This function plays an important role in extending numerical techniques to matrix-based problems. The matrix sign function is an axiomatic study of the intuitive concepts that govern the matrix function of non-singular square matrices, which are fundamental concepts in matrix theory. Therefore, performing these computations requires strong theoretical knowledge. The present study requires the knowledge of not only the original problem but also the derivation, error analysis, and performance limits of the numerical methods used to solve it. This research develops new optimized iterative methods for solving nonlinear scalar, vectorial, and matrix equations, aiming for high convergence orders with minimal computational cost and adhering to theoretical rigor to avoid common numerical pitfalls. This emphasis is particularly important when numerical methods are applied to practical problems. As a fact, some algorithms do not work for real-world problems. Although numerical techniques have improved greatly, some limitations still exist. Comprehensive numerical experiments consistently validate the theoretical developments. This computational experience is useful, as it connects exact arithmetic theory with finite-precision computation. At the same time, studying dynamical behavior through basins of attraction proves that the proposed schemes are better, efficient, and useful in real-world computing instances. To present these results in a systematic manner, the thesis is organized as follows: Chapter 1 provides an introduction and developments in the theory related to nonlinear scalar, vector, and matrix equations. We present a thorough literature review of algorithms for nonlinear problems. The applications of these algorithms for computing the matrix sign function are studied as well. Furthermore, we review basic mathematical concepts that drive the study of nonlinear solvers and the existence of the matrix sign function. Chapter 2 deals with developing a higher-order optimal family of Chebyshev–Halley type methods to solve a univariate nonlinear equation with multiple roots. The proposed scheme considers weight functions that are selected adequately to optimize the convergence order and demands only four functional evaluations at each iteration. An extensive convergence analysis is also provided that demonstrates the establishment of eighth-order convergence for the developed scheme. As a result, the efficiency index of the proposed scheme is optimal according to the Kung-Traub conjecture. Finally, the theoretical findings are verified through numerical experiments that include real-life and nonlinear academic problems. In addition, the dynamical study of iterative schemes reflects a comprehensive overview of their stability, convergence properties, and graphical aspects by drawing attraction basins in the complex plane. Chapter 3 aims to develop and analyze a new derivative-free class of higher-order iterative methods for locating multiple roots numerically. The scheme is generated by using King’s type it erative method. By employing the Traub-Steffensen technique, a derivative-free family is proposed, which requires three functional evaluations to achieve optimal fourth-order convergence. Moreover, it can be observed that the theoretical convergence results of the family are symmetrical for different multiplicities of roots. Finally, a wide variety of nonlinear problems are included to confirm the applicability and effectiveness of the proposed methods. Chapter 4 deals with the development of multi-step vectorial iterative schemes for solving non linear systems, achieving fourth and sixth-order convergence. The proposed methods are designed to minimize computational costs by employing a single inverse operator and reducing the number of functional evaluations per iteration. Furthermore, the proposed three-step scheme is generalized into a (q +1)-step family, increasing the convergence order to 2q +2. While standard local convergence analysis based on Taylor series expansion is common, but it has limitations, as it requires the use of higher-order derivatives. To overcome this limitation, the theoretical analysis in a Banach space setting is conducted that relies solely on first-order derivatives. The existence of a unique solution is guaranteed within a specific domain, whose radius of convergence is formally obtained using Lipschitz constants. Furthermore, the theoretical results are tested on several numerical examples to check the performance and stability of the proposed methods in comparison to existing counterparts. In chapter 5, some new iterative families that utilizes a derivative-free approach, requiring two frozen divided differences and one matrix inversion per iteration are proposed. Additionally, a general multi-point family using a fifth-order scheme as a predictor is derived. The proposed families are analyzed both theoretically and numerically, and experimentation is conducted using a range of numerical problems to confirm the theoretical results on convergence order and computational efficiency. The dynamical properties of each iterative method are studied by means of basins of attraction. In chapter 6, a novel matrix iteration for numerically computing the matrix sign function is proposed. An extensive convergence analysis is carried out in the matrix case to achieve fourth order convergence. The basins of attraction are illustrated to demonstrate the global convergence behavior of the proposed method. Analytically, it is shown that the presented scheme is asymptotically stable. Furthermore, theoretical developments are numerically verified by discussing the clustering of eigenvalues around ±1. Moreover, a class of numerical examples for matrices of various dimensions is assessed to exhibit the efficacy of the proposed method. In chapter 7, the primary objective is to develop two novel iterative schemes for computing the sign of a matrix that has no pure imaginary eigenvalues. Detailed convergence analysis and asymptotic stability of the proposed methods are discussed. It is shown that the new schemes converge globally by drawing attraction basins. In addition, the obtained results are extended to compute non-trivial solutions of the Yang-Baxter-like equation, provided the given matrix has no eigenvalues on the imaginary axis. To illustrate the effectiveness of theoretical developments, a class of numerical examples for matrices of various dimensions, including eigenvalue clustering, is worked out. Chapter 8 discusses the summary and future aspects of the presented work.