Numerical Methods for Solving Nonlinear Equations

Abstract

AbstractThe nonlinear description has continuously been crucial in a wide range of disciplines to provide an accurate prediction of a natural phenomenon. Thus, finding a reliable solution method for these nonlinear models is of significant importance since, in most real-life applications, direct solution methods are not feasible, even in linear cases. Moreover, an inefficient method is likely to take additional computational cost and effort. This chapter attempts to provide a fundamental description of various iterative methods for solving nonlinear discretized equations. In the first part, a theoretical account of nonlinear systems with different types of iterative methods are depicted. The second part deals with both one-point and multi-point iterative methods; this includes a description of the method, mathematical formulations, and the weak and strong points. Different iterative methods to solve a system of nonlinear equations are then described. Some discussed methods include the family of conjugate gradient, multi-step, and Newton-like. This part also identifies intricacies regarding a system of nonlinear equations, offering different remedies to solve these issues. Finally, a comparative study of the discussed methods and their applications in solving conventional equations are outlined in brief. The iterative methods mentioned in this chapter can be useful not only in solving nonlinear problems but also in linear problems and optimization.KeywordNonlinear equationsIterative methodsRoot-findingComputational efficiencyJacobian matrix