Simple rational approximation: Difference between revisions

Revision as of 21:24, 3 May 2006

Revision as of 22:06, 3 May 2006

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The main application of SRA lies in finding the [[Root (mathematics)|zeros]] of [[secular function|secular functions]]. A divide-and-conquer algorithm to find the [[eigenvalues]] and [[eigenvectors]] for various kinds of [[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.

==Helley's formular==

The origin of the interpolation with rational functions can be found in the previous work done by [[Edmond Halley]]. The Helley's formular 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>

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SRA strictly implies this one-opint second-order interpolation by a simple rational function.

We can notice that even third order method is a variation of Newton's method. We see the Newton's steps are multiplied by some factors. These factors are called the convergence factors of the variations, which are useful for analizinganalyzing the rate of convergence. See Gander(1978).

== References ==