Nonlinear Equations - UFRJ

Chapter 9
The pseudo-Newton
operator
Newton iteration was originally defined on linear spaces,
where it makes sense to add a vector to a point. Manifolds in general
lack this operation. A standard procedure in geometry is to replace
the sum by the exponential map
exp : T M → M,
(x, ẋ) ↦→ exp x (ẋ),
that is the map such that exp x (tẋ/‖ẋ‖) is a geodesic with speed ẋ
at zero. This approach was developed by many authors, such as [82]
or [40]. The alpha-theory for the Riemannian Newton operator
N Riem (f, x) = exp x −Df(x) −1 f(x)
appeared in [32]. This approach can be algorithmically cumbersome,
as it requires the computation of the exponential map, which in turn
Gregorio Malajovich, Nonlinear equations.
28 o Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, 2011.
Copyright c○ Gregorio Malajovich, 2011.
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