Nonlinear Equations - UFRJ

[SEC. 8.3: THE CONDITION NUMBER FOR UNMIXED SYSTEMS 111
Thus, for all f, g ∈ F,
〈L ∗ f(·), L ∗ g(·)〉 F ∗ = 〈L ∗ f(·) ∗ , L ∗ g(·) ∗ 〉 F = 〈f(·), g(·)〉 x .
This says that L ∗ is unitary, hence it has zero kernel and is an isometry
onto its image. Thus (Theorem 8.1) L | ker L ⊥ is an isometry.
8.3 The condition number for unmixed
systems
Let f = (f 1 , . . . , f s ) ∈ F s . Let K(·, ·) and L = L x be as above. We
define now
L = L x : F s → L(T x M, C s ),
⎡ ⎤
L x (f 1 )
(f 1 , . . . , f s ) ↦→
⎢
⎣ .
L x (f s )
⎥
⎦ .
The space L(T x M, C s ) is endowed with ‘Frobenius norm’,
⎡ ⎤
θ 1
⎢ ⎥
⎣
. ⎦
∥ θ s
∥
2
F
s∑
= ‖θ i ‖ 2 x
each θ i interpreted as a 1-form, that is an element of T x M ∗ .
immediate consequence of Lemma 8.4 is
i=1
An
Lemma 8.5. L x is onto, and L | ker L ⊥
is an isometry.
The condition number of f at x is defined by
µ(f, x) = ‖f‖ σ min(n,s) (L x (f)) −1 .
We will see in the next section that when F = H d,d,··· ,d and n = s,
this is exactly the Shub-Smale condition number of [70], known as the
normalized condition number µ norm in [20].