The Newton iteration from the point of view of Thurston theory and root finding

We describe the dynamical system generated by Newton’s root-finding method from the point of view of Thurston’s characterization theory of rational maps, leading to a complete classification of all postcritically finite Newton maps, and outline an extension to a classification of all Newton maps (postcritically finite or not). We also discuss Newton’s method from the point of view of root finding; it has successfully found all roots of polynomials of degrees greater than 10^{^9}.