University of Illinois, Chicago
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 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. We conclude with a few illustrative examples, with an eye toward homotopy continuation.