Simple rational approximation: Difference between revisions

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Revision as of 15:49, 19 October 2010

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The main application of SRA lies in finding the [[root of a function|zeros]] of [[secular function|secular functions]]. A divide-and-conquer algorithm to find the [[eigenvalues]] and [[eigenvectors]] for various kinds of [[Matrix (mathematics)|matrices]] is well-known in [[numerical analysis]]. In a strict sense, SRA implies a specific [[interpolation]] using simple rational functions as a part of the divide-and-conquer algorithm. Since such secular functions consist of a series of rational functions with simple poles, SRA is the best candidate to interpolate the zeros of the secular function. Moreover, based on previous researches, a simple zero that lies between two adjacent poles can be considerably well interpolated by using a two-dominant-pole rational function as an approximating function.

== One-point third-order iterative method: Halley's formula ==

The origin of the interpolation with rational functions can be found in the previous work done by [[Edmond Halley]]. Halley's formula is known as one-point third-order iterative method to solve <math>\,f(x)=0</math> by means of approximating a rational function defined by

:<math>h(z)=\frac{a}{z+b}+c.</math>