The Newton method applied to for finding is sometimes called Smith iteration.

For generalisations to matrix roots and other functions as well as analysis, the work of Nick Higham may be of interest [1,2,3].

[1] http://eprints.ma.man.ac.uk/637/

[2] http://eprints.ma.man.ac.uk/310/

[3] http://www.maths.manchester.ac.uk/~higham/fm/index.php

Specifically, pages 27 and 35. But the whole paper in general.

You will find the whole explanation for the definition of the two series, and all proofs & conditions of convergence.

(Actually, this whole codebase is based in this paper.) ]]>