Simple rational approximation

Simple rational approximation (SRA) is a subset of interpolating methods using rational functions. Especially, SRA interpolates a given function with a specific rational function whose poles and 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 of secular functions. A divide-and-conquer algorithm to find the eigenvalues and eigenvectors for symmetric matrices is well-known in numerical analysis. In a strict sense, SRA imples a specific interpolation using simple rational functions as a part of the devide-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 secular fuctions in order to find its roots. Moreover, based on previous researches, a simple root that lies between adjacent poles can be considerably well interpolated by using a two-dominiant-pole rational function as an approximating function.

References

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