Inverse quadratic interpolation: Difference between revisions

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{{Short description|Method of solving equations}}

In [[numerical analysis]], '''inverse quadratic interpolation''' is a [[root-finding algorithm]], meaning that it is an algorithm for solving equations of the form ''f''(''x'') = 0. The idea is to use [[polynomial interpolation|quadratic interpolation]] to approximate the [[inverse function|inverse]] of ''f''. This algorithm is rarely used on its own, but it is important because it forms part of the popular [[Brent's method]].

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:<math> f^{-1}(y) = \frac{(y-f_{n-1})(y-f_n)}{(f_{n-2}-f_{n-1})(f_{n-2}-f_n)} x_{n-2} + \frac{(y-f_{n-2})(y-f_n)}{(f_{n-1}-f_{n-2})(f_{n-1}-f_n)} x_{n-1} </math>

:::::<math> {}\qquad + \frac{(y-f_{n-2})(y-f_{n-1})}{(f_n-f_{n-2})(f_n-f_{n-1})} x_n. </math>

We are looking for a root of ''f'', so we substitute ''y'' = ''f''(''yx'') = 0 in the above equation, and this results in the above recursion formula.

==Behaviour==

The asymptotic behaviour is very good: generally, the iterates ''x''_''n'' converge fast to the root once they get close. However, performance is often quite poor if youthe doinitial notvalues startare verynot close to the actual root. For instance, if by any chance two of the function values ''f''_{''n''&minus;2}, ''f''_{''n''&minus;1} and ''f''_''n'' coincide, the algorithm fails completely. Thus, inverse quadratic interpolation is seldom used as a stand-alone algorithm.

The order of this convergence is approximately 1.84 as can be proved by [[secant method]] analysis.

==Comparison with other root-finding methods==

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Inverse quadratic interpolation is also closely related to some other root-finding methods.

Using [[linear interpolation]] instead of quadratic interpolation gives the [[secant method]]. Interpolating ''f'' instead of the inverse of ''f'' gives [[Muller's method]].

==See also==

* [[Successive parabolic interpolation]] is a related method that uses parabolas to find extrema rather than roots.

==References==

*[[James F. Epperson]], [https://books.google.com/books?id=Mp8-z5mHptcC&pg=PA182 An introduction to numerical methods and analysis], pages 182–185, Wiley-Interscience, 2007. {{isbn|978-0-470-04963-1}}

[[Cleve Moler]], [http://www.mathworks.com/moler/chapters.html Numerical Computing with MATLAB], Section 4.5, Society for Industrial and Applied Mathematics (SIAM), 2004. ISBN 0-89871-560-1.

{{root-finding algorithms}}

[[Category:Root-finding algorithms]]