Nonlinear Equations - UFRJ

[SEC. 9.3: APPROXIMATE ZEROS 127
Here is another useful estimate, that we state for homogeneous
systems only:
Lemma 9.5. Let X ∈ C n+1 and f, g ∈ H d . Assume that v =
µ(f, X) < 1. Then, for all Y ⊥ ker Df(X),
‖f−g‖
‖f‖
‖Y‖
√
1 − v
2
1 + v
≤ ‖Dg(X) † Df(X)Y‖ ≤ ‖Y‖
1 − v .
The rightmost inequality holds unconditionally.
Proof. By Lemma 8.9,
∥ ∥ ∥ ( )∥
Df(X)
† ∥∥∥ g − f ∥∥∥
‖Dg(X) − Df(X)‖ ≤ µ(f, X) L x ≤ v
‖f‖
In particular
∥ Df(X) † Dg(X) ker Df(X) ⊥ − I ker Df(X)
∥
⊥ ≤ v.
By Lemmas 9.2 and 7.8,
∥ Dg(X) † Df(X)Y ∥ ∥ ∥∥Dg(X) ≤
−1
Df(X)Y
ker Df(X) ⊥
The lower bound follows from Lemma 9.3:
∥ Dg(X) † Df(X)Y ∥ ∥ ≥
‖Y ‖ √ 1 − v 2
1 + v
∥ ≤ ‖Y ‖
1 − v
9.3 Approximate zeros
The projective distance is defined in C n+1 by
‖X − λY‖
d proj (X, Y) = inf
.
λ∈C × ‖X‖
Since it is scaling invariant, is defines a metric in projective space
that is related to the Riemannian distance by
d proj (x, y) = sin(d Riem (x, y)) ≤ d Riem (x, y)