Nonlinear Equations - UFRJ

Chapter 7
Newton Iteration and
Alpha theory
Let f be a mapping between Banach spaces. Newton
Iteration is defined by
N(f, x) = x − Df(x) −1 f(x)
wherever Df(x) exists and is bounded. Its only possible fixed points
are those satisfying f(x) = 0. When f(x) = 0 and Df(x) is invertible,
we say that x is a nondegenerate zero of f.
It is well-known that Newton iteration is quadratically convergent
in a neighborhood of a nondegenerate zero ζ. Indeed, N(f, x) − ζ =
D 2 f(ζ)(x − ζ) 2 + · · · .
There are two main approaches to quantify how fast is quadratic
convergence. One of them, pioneered by Kantorovich [48] assumes
that the mapping f has a bounded second derivative, and that this
bound is known.
Gregorio Malajovich, Nonlinear equations.
28 o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 2011.
Copyright c○ Gregorio Malajovich, 2011.
82