Nonlinear Equations - UFRJ

114 [CH. 8: CONDITION NUMBER THEORY
Lemma 8.9. 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 .
When n = s, this is precisely the Shub-Smale condition number:
∥ ⎡√ ∥∥∥∥∥∥ d1 ‖x‖ d1−1
⎤
2
µ(f, x) = ‖f‖ Hd (Df(x) |x ⊥) −1 ⎢
⎣
. ..
⎥
⎦
.
√ dn ‖x‖ dn−1 ∥
2 2
(8.1)
Theorem 8.10 (Condition number theorem, homogeneous). Let f ∈
F x = (H d1 × · · · × H ds ) x
. Let r = min(n, s). Then
µ(f, x) −1 =
‖g‖
min
g∈Fx ‖f‖ .
rank(D(f+g)(x))