Locating the Closest Singularity in a Polynomial Homotopy

05/15/2022
by   Jan Verschelde, et al.
0

A polynomial homotopy is a family of polynomial systems, where the systems in the family depend on one parameter. If for one value of the parameter we know a regular solution, then what is the nearest value of the parameter for which the solution in the polynomial homotopy is singular? For this problem we apply the ratio theorem of Fabry. Richardson extrapolation is effective to accelerate the convergence of the ratios of the coefficients of the series expansions of the solution paths defined by the homotopy. For numerical stability, we recondition the homotopy. To compute the coefficients of the series we propose the quaternion Fourier transform. We locate the closest singularity computing at a regular solution, avoiding numerical difficulties near a singularity.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
12/15/2016

The Method of Gauss-Newton to Compute Power Series Solutions of Polynomial Homotopies

We consider the extension of the method of Gauss-Newton from complex flo...
research
08/11/2020

A Neumann series of Bessel functions representation for solutions of the radial Dirac system

A new representation for a regular solution of the radial Dirac system o...
research
02/21/2020

Robust Numerical Tracking of One Path of a Polynomial Homotopy on Parallel Shared Memory Computers

We consider the problem of tracking one solution path defined by a polyn...
research
10/10/2021

Polynomial and trigonometric splines

Classes of simple polynomial and simple trigonometric splines given by F...
research
05/21/2020

An improved Neumann series of Bessel functions representation for solutions of perturbed Bessel equations

New representations for an integral kernel of the transmutation operator...
research
09/28/2016

Solving polynomial systems via homotopy continuation and monodromy

We study methods for finding the solution set of a generic system in a f...
research
10/04/2019

A Polyhedral Homotopy Algorithm For Real Zeros

We design a homotopy continuation algorithm for finding real zeros of sp...

Please sign up or login with your details

Forgot password? Click here to reset