## Graduate Computational Algebraic Geometry Seminar

Nathan Bliss

UIC

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

**Abstract:**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 instead a series with matrix coefficients. We show that in the regular case, the solution of the linear system satisfies the conditions of the Hermite interpolation problem. In general, we solve a Hermite-Laurent interpolation problem, via a lower triangular echelon form on the coefficient matrix. To find the matrix series without symbolic evaluation, we apply techniques of algorithmic differentiation. With a couple of illustrative examples, we demonstrate the application to polynomial homotopy continuation.

Thursday February 9, 2017 at 3:00 PM in SEO 1227