Nonlinear Equations - UFRJ

[SEC. 10.2: PROOF OF THEOREM 10.5 141
1. If the algorithm terminates, then X is a (β, µ, a 0 ) certified approximate
zero for F 1 .
2. If the algorithm terminates, and z 0 denotes the zero of F 0 associated
to X 0 , then z 1 is the zero of F 1 associated to X where
f t (z t ) ≡ 0 is a continuous path.
3. There is a constant C < 16.26 such that if the condition length
L(f t , z t ; 0, 1) is finite, then the algorithm always terminates after
at most
1 + Cn 1/2 D 3/2 L(f t , z t ; 0, 1) (10.4)
steps.
The actual theorem in [31] is stronger, because the algorithm
thereby allows for approximations instead of exact calculations. It
is more general, as the path does not need to be linear. Also, it is
worded in terms of the projective Newton operator N proj . This is
why the constants are different. But the important feature of the
theorem is an explicit step bound in terms of the condition length,
and this is reproduced here.
Remark 10.6. We can easily bound
L(f t , z t ; 0, 1) ≤
∫ 1
0
‖ḟt‖ ft µ(f t , z t ) 2 dt
and recover the complexity analysis of previously known algorithms.
Remark 10.7. The factor on √ n in the complexity bound comes from
the approximation of µ by µ F . It can be removed at some cost. The
price to pay is a more complicated subroutine for norm estimation,
and a harder complexity analysis.
10.2 Proof of Theorem 10.5
Towards the proof of Theorem 10.5, we need five technical Lemmas.
For the geometric insight, see figure 10.1.