  There are thousands of papers on rootfinding for nonlinear scalar equations.
Here is one more, to
  talk about an apparently new method, which I call ``Inverse Cubic Iteration''
(ICI) in analogy to the Inverse Quadratic Iteration in Richard Brent's zeroin
method. The possibly new method is based on a cubic blend of tangent-line
approximations for the inverse function. We rewrite this iteration for
numerical stability as an average of two Newton steps and a secant step: only
one new function evaluation and derivative evaluation is needed for each step.
The total cost of the method is therefore only trivially more than Newton's
method, and we will see that it has order $1+\sqrt{3} = 2.732...$, thus
ensuring that to achieve a given accuracy it usually takes fewer steps than
Newton's method while using essentially the same effort per step.
