Optimize Puiseux-series sample patterns for endgame efficiency and accuracy

Optimize the sample pattern used to incorporate additional accurately corrected points into a truncated Puiseux-series approximation so as to improve computational efficiency and endpoint accuracy while verifying that the convergence radius and winding number are correct.

Background

The spiral, Cauchy, and power-series endgames approximate singular homotopy-path endpoints by interpolating samples with a truncated Puiseux series. Once the samples lie inside the convergence disk and the winding number is known, the truncated series can predict additional points in the transformed s-plane; Newton correction can then produce accurate samples for a refined approximation.

The paper identifies an unresolved design problem: selecting where to sample and how to use corrected samples may affect both the computational cost and the accuracy of endpoint estimation. Any such optimization must also ensure that the assumed convergence radius and winding number are valid, since the reliability of the Puiseux approximation depends on those prerequisites.