Feb 13 2008

MAIN POINTS
The first section defines the rate of convergence, particularly focusing on linear and quadratic convergence. This is defined by a limit as a fraction of successive terms goes to infinity. A table demonstrates the dramatic difference between linear convergence and quadratic convergence. Linear convergence doesn't seem to even be much of an improvement upon the bisection method, for instance. I'm assuming that the lambda term is a goal of accuracy decided upon beforehand.
The second section defines zero multiplicity as.. sort of the degree of the polynomial around the zero. A zero of multiplicity one is 'simple.' Thm 2.10 says there is a simple zero at p only if f(p)=0 and f'(p) != 0. This is extended to zeroes of any multiplicity by making further and further claims about deeper derivatives. Examples are given that show how to deal with zeroes of various multiplicity.

CHALLENGES
In the first section, the relationship between the lambda term and the alpha term confuses me— are they determined by the equation in question, or by the person doing the calculation. In the second part, I don't really understand the significance of zeroes of various multiplicities— does this directly relate to how the method converges, and if so, how?

REFLECTION
The quantification of convergence sheds more light on when Newton's method is inefficient, which was not obvious before. This seems to be furthered by the investigation into zeros of various multiplicity which affect the type of convergence— leading to a clear classification of the behavior of the method.