Roots of Nonlinear Equations

When we encounter zeros or roots of a function f(x), it indicates that we are referring to finding the solutions of a scalar equation f(x) = 0. In this chapter, we explore several widely used methods in mathematics to find the real roots of nonlinear equations, including the bisection method, the secant method, the method of successive substitution, Newton’s iteration method, and the graphical method. The bisection method involves repeatedly dividing an interval until a root is found, and the secant method approximates the root using linear interpolation between two points. In the successive substitution method, an initial guess is iteratively refined until the equation is satisfied. This method can be particularly useful when dealing with equations that do not have a closed-form solution. Newton’s iteration method is widely used for finding the roots of nonlinear equations. It involves iteratively improving an initial guess by using the tangent line of the function at that point. The graphical method plots the process and visually identifies the points where it intersects the x-axis, indicating the roots. While MATLAB provides the roots command for finding the roots of polynomials, the additional methods discussed in this chapter offer more flexibility and precision when dealing with nonlinear equations. Understanding and applying these methods to find the roots of equations can greatly enhance problem-solving capabilities in various mathematics, engineering, and scientific research fields. Section 7.1, a short introduction to the chapter, is given. Section 7.2 describes the essential elements of the graphical method. Section 7.3 presents the implementation of the bisection method. Additionally, Sect. 7.4 reviews the most important features of Newton’s iteration method. Section 7.5 gives an overview of the secant method, and Sect. 7.6 addresses the method of successive substitutions. Finally, in Sect. 7.7, nonlinear simultaneous equations are discussed.