Nonlinear Equations - UFRJ

110 [CH. 8: CONDITION NUMBER THEORY
Exercise 8.2. Prove Theorem 8.3.
Exercise 8.3. Assume furthermore that m < n. Show that the same
interpretation for the condition number still holds, namely the norm
of the perturbation of some solution is bounded by the condition
number, times the perturbation of the input.
8.2 The linear term
As in Chapter 5, let M be an analytic manifold and let F be a
non-degenerate fewspace of holomorphic functions from M to C. A
possibly trivial homogenization group H acts on M, and f(hx) =
χ(h)f(x) for all f ∈ F, x ∈ M, where χ(h) is a multiplicative character.
Furthermore, we assume that M/H is an n-dimensional manifold.
Given x ∈ M , F x denotes the space of functions f ∈ F vanishing
at x. Using the kernel notation, F x = K(·, x) ⊥ . The later is non-zero
by Definition 5.2(2).
Let x ∈ M and f ∈ F x . The derivative of f at x is
Df(x)u ↦→ 〈f(·), D¯x K(·, x)u〉 F = 〈f(·), P x D¯x K(·, x)u〉 Fx
where P x : F → F x is the orthogonal projection operator (Lemma 5.10).
Note that since F is a linear space, D¯x K(·, x) and P x D¯x K(·, x) are
also elements of F.
Lemma 8.4. Let L = L x : F → T x M ∗ be defined by
〈
〉
1
L x (f) : u ↦→ f(·), √ P x D¯x K(·, x)ū
K(x, x)
Then L is onto, and L | ker L ⊥
is an isometry.
Proof. Recall that the metric in M is the pull-back of the Fubini-
Study metric in F by x ↦→ K(·, x). The adjoint of L = L x is
L ∗ : T x M → F ∗ ,
u
↦→
(
f ↦→
〈
〉 )
1
f(·), √ P x D¯x K(·, x)ū .
K(x,x)
F
F
.