What iterative method did Isaac Newton develop for approximating roots?

The iterative method Isaac Newton developed for approximating roots is Newton’s method, also called the Newton–Raphson method.

The procedure begins with an initial estimate for a root of f(x)=0. It replaces the curve near that estimate with its tangent line, finds where the tangent crosses the x-axis, and uses that intercept as the next estimate. Repeating the calculation gives the standard iteration xₙ₊₁ = xₙ − f(xₙ)/f′(xₙ). When the starting value is sufficiently close and the root is well behaved, convergence is often quadratic, so the number of correct digits can roughly double at each step.

Newton described an early form of the technique in work written in the late 1660s and early 1670s, but his original presentation focused mainly on polynomial equations and did not use the modern general derivative formula. Joseph Raphson later published a more explicit iterative version, which is why both names appear in textbooks.

Newton’s method is not guaranteed to converge from every starting point. A poor guess, a zero derivative, or complicated function behavior can cause slow convergence, divergence, or cycling. Secant and bisection methods are different root-finding algorithms.