Nonlinear Equations - UFRJ

[SEC. 9.1: THE PSEUDO-INVERSE 123
in X j . An alternative definition of Ω is: the set of points X at C n+s
where axiom 5.2.2 fails, namely the evaluation map at X is the zero
map for some F i .
In order to define the Newton iteration on multiprojective space
P n1 × · · · × P ns , Dedieu and Shub [33] endow M = C n+s \ Ω with a
metric that is H-invariant. Their construction amounts to scaling X
by h such that ‖h 1 X 1 ‖ = · · · = ‖h s X s ‖ = 1 and then
N pseu (f, x) = [hX − Df(hX) −1
ker Df(hX) ⊥ f(hX)].
In this book, we are following a different philosophy. While condition
numbers are geometric invariants that live in quotient space
(or on manifolds), Newton iteration operates only on linear spaces.
Hence we will define
N(f, X) = X − Df(X) −1
ker Df(X) ⊥ f(X)
as a mapping from M into itself. It may be undefined for certain
values of X. While it coincides with N pseu for values of X scaled such
that ‖X 1 ‖ = · · · = ‖X s ‖, it is not in general a mapping in quotient
space. This will allow for iteration of N, without rescaling. In chapter
10 we will take care of rescaling the vector X when convenient,
and will say that explicitly.
9.1 The pseudo-inverse
The iteration N pseu is usually expressed in terms of a generalization
of the inverse of a matrix:
Definition 9.1. Let A be a matrix, with svd decomposition A =
UΣV ∗ (see Th. 8.1). Its pseudo-inverse A † is
where (Σ † ) ii = Σ −1
ii
A † = V Σ † U ∗
when Σ ii ≠ 0, or zero otherwise.
Note that if A is a rank m, m×n matrix with m ≤ n, then AA † =
I m and A † A is the orthogonal projection onto ker A ⊥ . Moreover,
A † = (AA ∗ ) −1 A.