From Halley to Secant: Redefining root finding with memory-based methods including convergence and stability

Root-finding methods solve equations and identify unknowns in physics, engineering, and computer science. Memory-based root-seeking algorithms may look back to expedite convergence and enhance computational efficiency. Real-time systems, complicated simulations, and high-performance computing demand frequent, large-scale calculations. This article proposes two unique root-finding methods that increase the convergence order of the classical Newton-Raphson (NR) approach without increasing evaluation time. Taylor's expansion uses the classical Halley method to create two memory-based methods with an order of 2.4142 and an efficiency index of 1.5538. We designed a two-step memory-based method with the help of Secant and NR algorithms using a backward difference quotient. We demonstrate memory-based approaches' robustness and stability using visual analysis via polynomiography. Local and semilocal convergence are thoroughly examined. Finally, proposed memory-based approaches outperform several existing memory-based methods when applied to models including a thermistor, path traversed by an electron, sheet-pile wall, adiabatic flame temperature, and blood rheology nonlinear equation.