Introduction

In numerical analysis, the solution of polynomial and transcendental equations is a fundamental topic with widespread applications in various fields of science and engineering. These equations, which are often too complex to solve analytically, require iterative numerical methods to find their roots.

A polynomial of degree $n$ is of the form \[ f(x) = a_0 + a_1 x + a_2 x^2 + \dots + a_n x^n, \quad \text{where } a_n \neq 0. \] where \( a_n, a_{n-1}, \dots, a_0 \) are coefficients, and \( x \) is the unknown variable. The polynomial $f(x) = 0$ is called an algebraic equation of degree $n$. On the other hand, transcendental equations involve transcendental functions such as exponential, logarithmic, trigonometric, or hyperbolic functions. An example of a transcendental equation is: \[ f(x) = e^x - x^2 = 0 \] Due to their non-algebraic nature, transcendental equations do not have general solutions and must be solved using numerical methods.

If there exists a real number $\alpha$ such that $f(\alpha) = 0$, then we say $\alpha$ is a real root (or a zero) of the equation $f(x) = 0$. The main goal of this section is to find a real root of the equation $f(x) = 0$.

The root of an equation can be found either analytically or numerically. Not all equations can be solved analytically. For example, the zeros of the equation \[ x^2 - 3x + 2 = 0 \] can be found using the quadratic formula. Similarly, the roots of cubic and quartic polynomials can be determined analytically. However, for more complex equations, such as \[ \sin x - e^{x} = 0, \] we must rely on iterative or numerical methods. In iterative methods, the root is approximated by starting with an initial guess (or initial approximation) and gradually improving this guess to achieve a more accurate approximation of the root.

In this section, we will explore various techniques for solving these types of equations, including:

• Bisection Method: A simple and robust method that repeatedly bisects an interval and selects the subinterval in which the root lies.
• Regula Falsi Method (False Position Method): A method that improves upon the bisection method by interpolating a straight line between two points on the function curve.
• Newton-Raphson Method: A powerful and fast method that uses the derivative of the function to iteratively approximate the root.

Each method has its own advantages, limitations, and applicability depending on the nature of the equation and the desired accuracy of the solution.