Comparison of the Newton–Raphson and Secant Methods in a Simple Pendulum Model

Naufal Afi Adani (Department of Mathematics Education, Faculty of Tarbiyah, Universitas Islam Negeri Raden Mas Said Surakarta, Indonesia)
Anindhita Maheswari (Department of Mathematics Education, Faculty of Tarbiyah, Universitas Islam Negeri Raden Mas Said Surakarta, Indonesia)
Ari Wibowo (Department of Mathematics Education, Faculty of Tarbiyah, Universitas Islam Negeri Raden Mas Said Surakarta, Indonesia)

Sainmatika: Jurnal Ilmiah Matematika dan Ilmu Pengetahuan Alam
Vol. 23 No. 1 (2026): Sainmatika : Jurnal Ilmiah Matematika dan Ilmu Pengetahuan Alam

Various problems in mathematics and physics, including the nonlinear pendulum model, cannot be solved analytically, so numerical methods are used to obtain approximate solutions with a certain error tolerance. This study compares the Newton–Raphson and Secant methods in solving nonlinear equations in a pendulum system based on iteration count, error, and convergence stability using a comparative quantitative approach.  The results show that neither method is absolutely superior, as both successfully produced approximate solutions. The value of θ (angular displacement) decreases as the pendulum length (L) increases due to the proportional relationship involving potential energy and the factor mgL. The Newton–Raphson method reached the solution in 4 iterations, while the Secant method required 4–6 iterations. The average order of convergence for Newton–Raphson approaches p ≈ 2 (quadratic), whereas the Secant method approaches p ≈ 1.62 (superlinear). The differences between the two methods are more influenced by the choice of initial guesses and the respective mechanisms of each method.