A comparative study of the Regula Falsi method, Newton's method, and the steepest descent method for solving nonlinear equations

A numerical method has been introduced to help mathematicians to solve the functions. Numerical methods are powerful tools used to approximate solutions to equations that may not have exact solutions or are difficult to solve analytically. This study presents a comparative analysis of numerical methods that is the Regula Falsi method, Newton's Method, and Steepest Descent method. These methods are employed for solving nonlinear equations. All these methods will be compared and tested with eight different types of test functions including polynomials, exponentials, cubic, and trigonometric functions and also with different tolerance and initial guess. The performance of these methods was evaluated using performance profiles based on the number of iterations and CPU time. The results demonstrate that Newton's Method outperforms the other approaches, exhibiting the fastest convergence, and the least computational cost. Regula Falsi shows moderate performance, while Steepest Descent lags in efficiency due to its higher iteration count and CPU usage. The findings underscore the significance of selecting appropriate numerical techniques to optimize computational efficiency, with potential applications across diverse scientific and engineering disciplines.