Quadratic Pade approximation : numerical aspects and applications
Date
2019
Journal Title
Journal ISSN
Volume Title
Publisher
Steklov Mathematical Institute
Abstract
Pad´e approximation is a useful tool for extracting singularity information from a power series. A linear
Pad´e approximant is a rational function and can provide estimates of pole and zero locations in the complex
plane. A quadratic Pad´e approximant has square root singularities and can, therefore, provide additional
information such as estimates of branch point locations. In this paper, we discuss numerical aspects of computing
quadratic Pad´e approximants as well as some applications. Two algorithms for computing the coefficients in
the approximant are discussed: a direct method involving the solution of a linear system (well-known in the
mathematics community) and a recursive method (well-known in the physics community). We compare the
accuracy of these two methods when implemented in floating-point arithmetic and discuss their pros and cons.
In addition, we extend Luke’s perturbation analysis of linear Pad´e approximation to the quadratic case and
identify the problem of spurious branch points in the quadratic approximant, which can cause a significant loss
of accuracy. A possible remedy for this problem is suggested by noting that these troublesome points can be
identified by the recursive method mentioned above. Another complication with the quadratic approximant arises
in choosing the appropriate branch. One possibility, which is to base this choice on the linear approximant, is
discussed in connection with an example due to Stahl. It is also known that the quadratic method is capable of
providing reasonable approximations on secondary sheets of the Riemann surface, a fact we illustrate here by
means of an example. Two concluding applications show the superiority of the quadratic approximant over its
linear counterpart: one involving a special function (the Lambert W-function) and the other a nonlinear PDE
(the continuation of a solution of the inviscid Burgers equation into the complex plane).
Description
CITATION: Fasondini, M. et al. 2019. Quadratic Pade approximation : numerical aspects and applications. Computer Research and Modeling, 11(6):1017–1031, doi:10.20537/2076-7633-2019-11-6-1017-1031.
The original publication is available at http://www.mathnet.ru
The original publication is available at http://www.mathnet.ru
Keywords
Pade approximant, Approximation theory, Power series, Singularities (Mathematics), Floating-point arithmetic, Equations, Quadratic
Citation
Fasondini, M. et al. 2019. Quadratic Pade approximation : numerical aspects and applications. Computer Research and Modeling, 11(6):1017–1031, doi:10.20537/2076-7633-2019-11-6-1017-1031.