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

*Nathan Bliss* and *Jan Verschelde*

#### 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 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.
In general, we solve a Hermite-Laurent interpolation problem,
via a lower triangular echelon form on the coefficient matrix.
We show that this solution has cost cubic in the problem size.
With a few illustrative examples, we demonstrate the
application to polynomial homotopy continuation.