Comparison of Some Numerical Methods for Solving Real-Life Nonlinear Equations by Using Python Programming

Abstract

 This research evaluates the efficiency and accuracy of some iterative methods for solving scalar nonlinear equations. The study focuses on four types of iterative methods, such as Newton-Raphson, Bisection, Secant, and Algorithm 1, a novel fourth-order and derivative-free root-finding algorithm exhibiting different convergence orders. The accuracy of these methods is tested numerically on some real-life nonlinear equations such as the fluid permeability problem, the blood rheology model, Van Der Wall’s and Planck’s radiation laws, and lastly, the beam design problem. Numerical experiments are conducted using the Python programming language with a tolerance of 10-14 and 10-15. The results indicate that the Secant method requires fewer iterations and CPU time if compared with Newton-Raphson, Bisection and Algorithm 1 in solving fluid permeability problems, blood rheology models, and Van Der Wall’s problems. The bisection method converges quickly compared with other methods for solving Planck’s radiation law, while Algorithm 1 converges quicker than other methods for solving the beam design problem. In terms of accuracy, the Secant method gives greater accuracy in solving fluid permeability and Van Der Wall’s problems. Meanwhile, for the blood rheology model, Newton Raphson's methods overcome other methods. On the other hand, the Bisection method gives greater accuracy for Planck’s radiation law and beam design problems. Algorithm 1 performance showed effective convergence to the root, but, in many cases, it encounters a division by zero issue. The study suggests extending investigations to Algorithm 1 refinement, comparative studies on various equation types, exploration of hybrid methods, real-world application and validation, and user-friendly implementation.