Nonlinear Equations - UFRJ

144 [CH. 10: HOMOTOPY
Let d Riem denote the Riemannian distance between x ti
iteration [N(F s , X ti )].
and Newton
sin(d Riem ) = d proj (X ti , N(F s , X ti )) ≤ β.
Because projective space has radius π/2, we may always bound
d Riem (x, y) ≤ π 2 d proj(x, y)
so that we should set u = Dπ
2
µβ in order to apply Theorem 8.23. We
obtain
µ
µ(f s , [N(F s , X i )]) ≤
1 − ɛ 1 − πa 0 / √ D
The estimate on (10.13) follows from (9.6). Using (10.11),
The estimate
β(F s , N(F s , X i )) ≤
α(1 − α)
β(F s , X i )
ψ(α)
(1 − ɛ 1 )(1 − α)
(1 − (1 − ɛ 1 )α/2)(1 − ɛ 1 − πa 0 / √ 2) ψ(α) ≤ a 0 (10.15)
was obtained numerically. It implies (10.14)
Remark 10.10. (10.15) seems to be the main ‘active’ constraint for
the choice of α, ɛ 1 , ɛ 2 .
α 2
Lemma 10.11. Under the conditions of Lemma 10.8,
µ
µ(f s , z s ) ≥
1 + ɛ 1 + π(1 − ɛ 1 )αr 0 (α)/ √ D
(10.16)
where r 0 = r 0 (α) is defined in Theorem 9.9.
Proof. From Theorem 9.9 applied to F s and X i , the projective distance
from X i to z s is bounded above by r 0 (α)β(F s , X i ). Therefore,
we set
u = π(1 − ɛ 1 )r 0 (α)α/ √ D v = ɛ 1
and apply Theorem 8.23.