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Numerical Aspects of Quadratic Padé Approximation

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A classical (linear) Padé approximant is a rational approximation, F(x) = p(x)/q(x), of a given function, f(x), chosen so the Taylor series of F(x) matches that of f(x) to as many terms as possible. If f(x) is meromorphic, then F(x) often provides a good approximation of f(x) in the complex plane beyond the radius of convergence of the original Taylor series. A generalisation of this idea is quadratic Padé approximation, where now polynomials p(x), q(x), and r(x) are chosen so that p(x) q(x)f(x) r(x)f2(x) = O(x{max}). The approximant, F(x), can then be found by solving p(x) q(x)F(x) r(x)F^2(x) = 0 by, for example, the quadratic formula. Since F(x) now contains branch cuts, it typically provides better approximations than linear Padé approximants when f(x) is multi-sheeted, and may be used to estimate branch point locations as well as poles and roots of f(x). In this talk we focus not on approximation properties of Padé approximants, but rather on numerical aspects of their computation. In the linear case things are well-understood. For example, it is well-known that the ill conditioning in the linear system satisfied by p(x) and q(x) means that these are computed with poor relative error, but that in practice, F(x) itself still has good relative accuracy. Luke (1980) formalises this for linear Padé approximants, and we show how this analysis extends to the quadratic case. We discuss a few different algorithms for computing a quadratic Padé approximation, explore some of the problems which arise in the evaluation of the approximant, and demonstrate some example applications.

This talk is part of the Isaac Newton Institute Seminar Series series.

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