Abstract:

This paper introduces two geometrically motivated iterative algorithms for computing real roots of differentiable functions: the Tangents Method and the SecTan Method. Both methods exploit the relationship between secant and tangent lines to construct convergence-preserving updates that do not require initial guesses near the true root. The Tangents Method is based on the intersection of tangent lines drawn at the endpoints of an interval containing a root, while the SecTan Method identifies points where the tangent to the function is parallel to the secant line across the interval. We present rigorous convergence analyses for both methods, establish sufficient conditions for linear convergence, and compare their performance with classical iterative schemes such as the Newton and Bisection methods. Numerical examples demonstrate that the proposed algorithms combine robustness with competitive accuracy. The geometric construction underlying both approaches suggests natural extensions to higher-dimensional root-finding problems.