Nonlinear Equations - UFRJ

140 [CH. 10: HOMOTOPY
We will need routines to compute the following quantities:
• S 1 (X, t) is the minimal value of s > t with
‖F s − F t ‖ =
ɛ 1
µ F (F t , X) .
This can be computed by computing easily with elementary
operations and exactly one square root.
• S 2 (X, t) is the maximal value of s > t such that, for all t < σ <
s,
2ɛ 2
Φ t,σ (X) ≤
D 3/2 µ F (F t , X)
In particular, when S 2 (t) is finite,
Φ t,S2(t)(X) =
2ɛ 2
D 3/2 µ F (F t , X)
Again, S 2 may be computed by elementary operations, and then
solving one degree two polynomial (that is, one square root).
Algorithm Homotopy.
Input: F 0 , F 1 ∈ H d \ {0}, X 0 ∈ C n+1 \ {0}.
i ← 0, t 0 ← 0, X 0 ← 1
‖X 0‖ X 0.
Repeat
t i+1 ← min
(
)
S 1 (X i , t i ), S 2 (X i , t i ), 1 .
X i+1 ←
1
‖N(F ti+1 ,X i)‖ N(F t i+1
, X i ).
i ← i + 1.
Until t i = 1.
Return X ← X i
Theorem 10.5 (Dedieu-Malajovich-Shub). Let n, D = max d i ≥ 2.
Assume that F 0 and F 1 satisfy (10.1), (10.2) and moreover X 0 is a
(β, µ, a 0 ) certified approximate zero for F 0 .