Simple rational approximation: Difference between revisions

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'''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 [[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 [[Rootroot (mathematics)of a function|zeros]] of [[secular function|secular functions]]. 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==