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 roots[[root of [[seculara function|zeros]] of [[secular functionsfunction]]s. A [[divide-and-conquer algorithm]] to find the [[eigenvalues]] and [[eigenvectors]] for symmetricvarious kinds of [[Matrix (mathematics)|matrices]] is well- known in [[numerical analysis]]. In a strict sense, SRA implesimplies a specific [[interpolation]] using simple rational functions as a part of the devidedivide-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 secularzeros fuctionsof inthe ordersecular to find its rootsfunction. Moreover, based on previous researches, a simple rootzero that lies between two adjacent poles can be considerably well interpolated by using a two-dominiantdominant-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

:<math>h^{(i)}(x)=f^{(i)}(x), \qquad i=0,1,2.</math>

Then solving <math>\,h(z)=0</math> yields the iteration

:<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.

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

Similarly, we can derive a variation of Halley's formula based on a one-point ''second-order'' iterative method to solve <math>\,f(x)=\alpha(\neq 0)</math> using simple rational approximation by

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

Then we need to evaluate

:<math>h^{(i)}(x)=f^{(i)}(x), \qquad i=0,1.</math>

Thus we have

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

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 ==

*{{citation

* S. Elhay, G. H. Golub and Y.M. Ram, "The spectrum of a modified linear pencil", ''Computers and Mathematics with Applications'', vol. 46, pp. 1413-1426, 2003.

| last = Demmel | first = James W. | authorlink = James Demmel

* M. Gu and S. Eisenstat, "A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem," ''SIMAX'', vol. 16, no. 1, pp. 172-191, 1995.

| mr = 1463942

| isbn = 0-89871-389-7

| ___location = Philadelphia, PA

| publisher = [[Society for Industrial and Applied Mathematics]]

| title = Applied Numerical Linear Algebra

| year = 1997}}.

*{{citation

| last1 = Elhay | first1 = S.

| last2 = Golub | first2 = G. H. | author2-link = Gene H. Golub

| last3 = Ram | first3 = Y. M.

| doi = 10.1016/S0898-1221(03)90229-X

| mr = 2020255

| issue = 8–9

| journal = Computers & Mathematics with Applications

| pages = 1413–1426

| title = The spectrum of a modified linear pencil

| volume = 46

| year = 2003| doi-access = free

}}.

*{{citation

| last1 = Gu | first1 = Ming

| last2 = Eisenstat | first2 = Stanley C.

| doi = 10.1137/S0895479892241287

| mr = 1311425

| issue = 1

| journal = SIAM Journal on Matrix Analysis and Applications

| pages = 172–191

| 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

| publisher = [[Stanford University]], School of Humanities and Sciences, Computer Science Dept.

| title = On the linear least squares problem with a quadratic constraint

| year = 1978}}.

[[Category:interpolationInterpolation]]