Nonlinear Equations - UFRJ

152 [CH. 10: HOMOTOPY
by the geometric Lemma, setting U = F t , A = F 0 , B = F 1 and
scaling. Replacing ‖F 0 ‖, F 1 by √ 2N and passing to expectations,
(∫ 1
µ 2 )
E (M(f t ; 0, 1)) ≤ 2NE
2(F t )
0 ‖F t ‖ 2 dt
∫ 1
( ) µ
2
≤ 2N E 2 (F t )
‖F t ‖ 2 dt .
Now, in the rightmost integral, F 0 and F 1 are sampled from the
probability space
(B(0, √ 2N), κ −1 dH d
)
.
The integrand is positive, so we can bound the integral by
E (M(f t ; 0, 1)) ≤ κ −2 ∫ 1
0
0
( ) µ
2
E 2 (F t )
‖F t ‖ 2 dt
where now F 0 and F 1 are Gaussian random variables. Using that
κ ≥ 1/2,
∫ 1
( ) µ
2
E (M(f t ; 0, 1)) ≤ 8N E 2 (F t )
‖F t ‖ 2 dt .
Let N(¯F, σ 2 I) denote the Gaussian normal distribution with mean
¯F and covariance σ 2 I (a rescaling of what we called dH d ).
From Corollary 10.17,
( ) e
3/2
∫ 1
E (M(f t ; 0, 1)) ≤ 8
2 − 1 n
N
0 t 2 dt = 4(e3/2
+ (1 − t)
2
2 −1)πNn.
This establishes:
Proposition 10.20. The expected number of homotopy steps of the
algorithm of Theorem 10.5 with F 0 , z 0 sampled by the Beltrán-Pardo
method, is bounded above by
( ) e
3/2
1 + 4
2 − 1 πCNn 3/2 D 3/2
0