Nonlinear Equations - UFRJ

[SEC. 10.1: HOMOTOPY ALGORITHM 139
As this is expository material, we will make suppositions about
intermediate quantities that need to be computed. Namely, the following
operations are assumed to be performed exactly and at unit
cost: Sum, subtraction, multiplication, division, deciding x > 0, and
square root.
In particular, Newton iteration N(F, X) = X−DF(X) † F(X) can
be computed in O(n dim(H d )) operations.
It would be less realistic to assume that we can compute condition
numbers (that have an operator norm). Operator norms can be
approximated (up to a factor of √ n) by the Frobenius norm, which
is easy to compute. Therefore, let
µ F (F, X) =
= ‖F‖
DF(X) −1
∥
⎡
⎢
|X ⎣ ⊥
‖X‖ d1−1√ ⎤
d 1 . ..
⎥
⎦
‖X‖ dn−1√ ∥
d n
be the ‘Frobenius’ condition number. It is invariant by scaling, and
µ(f, x) ≤ µ F (f, x) ≤ √ n µ(f, x).
Also, we need to define the following quantity:
Φ t,σ (X) = ∥ ∥DF t (X) † (F σ (X) − F t (X)) ∥ ∥ .
∥
F
The algorithm will depend on constants a 0 , α, ɛ 1 , ɛ 2 . The constant
a 0 is fixed so that
a 0 + ɛ 2
= α. (10.3)
(1 − ɛ 1 )
2
The value of the other constants was computed numerically (see
remark 10.14 below). The constant C will appear as a complexity
bound, and depends on the other constants. There is no claim of
optimality in the values below:
Constant Value
α 7.110 × 10 −2
ɛ 1 5.596 × 10 −2
ɛ 2 5.656 × 10 −2
a 0 6.805, 139, 185, 76 × 10 −3
C 16.26 (upper bound).