Inverse Cubic Iteration
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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+√(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.
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