Nonlinear Equations - UFRJ

122 [CH. 9: THE PSEUDO-NEWTON OPERATOR
depends on the connection.
Luckily, it turns out that of the two conditions defining the geodesic,
only one is actually relevant for the purpose of Newton iteration: the
condition at t = 0 should be ẋ.
A more general procedure is to replace the exponential map by a
retraction map R : T M → M with
∂
R(x, tẋ)ẋ.
∂t |t=0
This is discussed in [1]. A previous example, studied in the literature,
is projective Newton [20, 68, 70].
Through this chapter and the next, we adopt the following notations.
Given a point x ∈ P n or in a quotient manifold M/H, X
denotes a representative of it in C n+1 (or in M). The class of equivalence
of X may be denoted by x or by [X]. With this convention,
projective Newton is
N proj (f, x) = [X − Df(X) −1
X ⊥ f(X)].
This iteration has advantages and disadvantages. The main disadvantage
is that its alpha-theory is much harder than the usual Newton
iteration.
In this book, we will follow a different approach. The following
operator was suggested by [2]:
N pseu (f, X) = X − Df(X) −1
| ker Df(X) ⊥ f(X).
This holds in general for manifolds that are quotient of a linear
space (or an adequate subset of it) by a group. For instance, P n as
quotient of C n+1 \ 0 by C × . In this case, results of convergence and
robustness are not harder than in the classical setting [56].
This whole approach was extended to the multi-projective setting
in [33]. More precisely, let n = n 1 + · · · + n s − s and consider multihomogeneous
polynomials in X = (X 1 , . . . , X s ). Let Ω be the set of
X ∈ C n+s such that at least one of the X i vanishes. Then we set M =
C n+s \ Ω and H = (C × ) s , acting on M by hX = (h 1 X 1 , . . . , h s X s ).
Through this chapter, F 1 , . . . , F n will denote spaces of multihomogeneous
polynomials, such that elements of F i have degree d ij