Simple rational approximation: Difference between revisions

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'''Simple rational approximation (SRA)''' is a subset of [[Interpolation|interpolating]] methods using [[rational functionsfunction]]s. Especially, SRA interpolates a given function with a specific [[rational function]] whose poles[[pole (complex analysis)|pole]]s and [[root of a function|zeros]] are [[simple]], which means that there is no multiplicity in poles and zeros. Sometimes, it only implies simple poles.

The main application of SRA lies in finding the [[root of a function|zeros]] of [[secular function]]s. 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 method|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>

We can determine a, b, and c so that

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:<math>x_{n+1}=x_{n}-\frac{f(x_n)}{f'(x_n)} \left({\frac{1}{1-\frac{f(x_n)f''(x_n)}{2(f'(x_n))^2}}}\right).</math>

This is referred to as Halley's formula.

This ''geometrical interpretation'' <math>h(z)</math> was derived by Gander (1978), where the equivalent iteration also was derived by applying Newton's method to

:<math>g(x)=\frac{f(x)}{\sqrt{f'(x)}}=0.</math>

We call this ''algebraic interpretation'' <math>g(x)</math> of Halley's formula.

There are no known explanation for how one is supposed to calculate a, b or c out of these equations.

==One-point second-order iterative method: Simple rational approximation==

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The algebraic interpretation of this iteration is obtained by solving

:<math>g(x)=1-\frac{\alpha}{{f(x)}}=0.</math>

This one-point second-order method is known to show a locally quadratic convergence if the root of the equation is simple.

SRA strictly implies this one-point 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 analyzing the rate of convergence. See Gander (1978).

== References ==

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| title = The spectrum of a modified linear pencil

| volume = 46

| year = 2003}}.| doi-access = free

}}.

*{{citation

| last1 = Gu | first1 = Ming

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| title = A divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem

| volume = 16

| year = 1995}}| url = https://zenodo.org/record/1236142

}}.

*{{citation

| last = Gander | first = Walter