The Method of Gauss-Newton to Compute Power Series Solutions of Polynomial Homotopies
This work provides a theoretical extension of Gauss-Newton to power series for polynomial homotopy continuation, but the results are incremental and primarily of interest to specialists in numerical algebraic geometry.
The paper extends the Gauss-Newton method to truncated power series for solving polynomial homotopies, showing that in the regular case the solution cost is cubic in problem size, and at singular points it uses tropical algebraic geometry to compute Puiseux series.
We consider the extension of the method of Gauss-Newton from complex floating-point arithmetic to the field of truncated power series with complex floating-point coefficients. With linearization we formulate a linear system where the coefficient matrix is a series with matrix coefficients, and provide a characterization for when the matrix series is regular based on the algebraic variety of an augmented system. The structure of the linear system leads to a block triangular system. In the regular case, solving the linear system is equivalent to solving a Hermite interpolation problem. We show that this solution has cost cubic in the problem size. In general, at singular points, we rely on methods of tropical algebraic geometry to compute Puiseux series. With a few illustrative examples, we demonstrate the application to polynomial homotopy continuation.