Nonlinear Equations - UFRJ

[SEC. 10.3: AVERAGE COMPLEXITY OF RANDOMIZED ALGORITHMS 149
Thus we obtain a pair (f 0 , z 0 ) in the solution variety V ⊂ P(H d )×
P n . This pair is a random variable, and hence has a certain probability
distribution.
Proposition 10.15 (Beltrán-Pardo). The procedure described above
provides a random pair (f 0 , z 0 ) in V, with probability distribution
1
B π∗ 1 dH d ,
where B = ∏ d i is the Bézout bound and dH d is the Gaussian probability
volume in H d . Thus π ∗ 1 dH d denotes its pull-back through the
canonical projection π 1 onto the first coordinate.
Proof. For any integrable function h : V → R,
∫
1
h(v)π
B
1dH ∗ d (v) =
V
= 1 ∫ ∫
dV (z) h(F, z) det |Df(z)Df(z)∗ |
∏ dH d ) z
B P n (H d ) z Ki (z, z)
∫ ∫
= dV (z) h(F, z) det |L z(f)L z (f) ∗ |
P n (H d ) z
(1 + ‖z‖ 2 ) n dH d ) z
=
∫H 1
∫
R M
h(M + F, z)dH 1
We need to quote from their paper [13, Theorem 20] the following
estimate:
Theorem 10.16. Let M be a random complex matrix of dimension
(n + 1) × n picked with Gaussian probability distribution of mean 0
and variance 1. Then,
E ( ‖M † ‖ 2) ≤ n (
1 + 1 ) n+1
− n − 1 2 n
2
Assuming n ≥ 2, the right-hand-side is immediately bounded
above by ( e3/2
2
− 1)n < 1.241n. In exercise 10.1, the reader will
show that when the variance is σ 2 , then
E ( ‖M † ‖ 2) ( ) e
3/2
≤
2 − 1 nσ −2 . (10.19)