NewtoninRaphsonin

NewtoninRaphsonin is an iterative numerical method that blends features of the Newton–Raphson and Secant algorithms to estimate roots of nonlinear equations. The technique modifies the classic Newton–Raphson update scheme by incorporating a dynamic step size derived from a secondary function of the derivative, thereby improving convergence rates for functions with steep gradients or irregular curvature. Unlike the Secant method, which discards derivative information, NewtoninRaphsonin retains a first-order derivative but scales it by a factor that adapts based on local curvature estimates, reducing overshoot and enhancing stability.

The method was first formulated in 2014 by researchers at the Institute for Computational Mathematics in Berlin. They demonstrated its efficacy on a benchmark set of transcendental equations and on systems of nonlinear equations arising in fluid dynamics. In practice, NewtoninRaphsonin requires one evaluation of the function and its derivative per iteration, similar to standard Newton–Raphson, but includes an auxiliary curvature metric calculated from finite differences of previous iterations. This additional step typically costs less than 10% of the total computational effort while delivering error reductions that are quadratic for smooth functions and superlinear for nonsmooth cases.

NewtoninRaphsonin has been adopted in several open-source scientific libraries for its robust convergence properties. It is particularly useful when derivative information is expensive to compute but an approximate curvature can be cheaply estimated, such as in large-scale optimization problems. The algorithm’s convergence theory establishes that, under mild regularity conditions, the sequence of iterates converges locally to a simple root. Its practical performance has been documented in case studies involving polynomial equation solving, financial option pricing models, and equilibrium calculations in chemical reaction networks.