  Finding roots of equations is at the heart of most computational science. A
well-known and widely used iterative algorithm is the Newton's method. However,
its convergence depends heavily on the initial guess, with poor choices often
leading to slow convergence or even divergence. In this paper, we present a new
class of methods that improve upon the classical Newton's method. The key idea
behind the new approach is to develop a relatively simple multiplicative
transformation of the original equations, which leads to a significant
reduction in nonlinearities, thereby alleviating the limitations of the
Newton's method. Based on this idea, we propose two novel classes of methods
and present their application to several mathematical functions (real, complex,
and vector). Across all examples, our numerical experiments suggest that the
new methods converge for a significantly wider range of initial guesses with
minimal increase in computational cost. Given the ubiquity of Newton's method,
an improvement in its applicability and convergence is a significant step
forward, and will reduce computation times several-folds across many
disciplines. Additionally, this multiplicative transformation may improve other
techniques where a linear approximation is used.
