How does the inverse quadric interpolation help us find the root?

Question

I can't really understand how taking inverse of a parabola helps us finding a root. I understand that it intersects x-axis close to the root but why does that happen?

Answer 1

In the case that the segment of the function is clearly dominated by the lower order derivatives, that is, strictly monotonously increasing or decreasing and strictly convex or concave, any non-linear approximation of the function (with more than 2 function values as data) will provide a closer approximation of the function than the secant approximation and thus produce a closer root approximation. Or formulated the other way around, for a given accuracy of the root approximation, a quadratic or similar approximation will give such a root approximation over a larger interval than a secant root.

Given such a standard situation, it does not make a big difference in terms of accuracy if the function itself or the inverse function is approximated. The formula for the root location and any subsequent convergence analysis is structurally simpler for the inverse function.

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In the middle between the quadratic and inverse quadratic approximation is the hyperbolic approximation $f(x)\approx\frac{c-x}{d_1x+d_0}$, as its inverse has an analogous form.

Another explored but not as successful approximation uses exponential terms to generate non-linearity, like $f(x)=(c_0+c_1x)e^{dx}$ or similar, see Why does Ridders' method work as well as it does?