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 zeros of 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 secular fuctions in order to find its roots. 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.

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.