Nonlinear Equations - UFRJ

[SEC. A.5: EXTENSION OF THE ALGORITHMS... 161
logarithmic in the condition number of the extremes, see [14]. Their
computation is however a difficult task. A simple question that one
may ask is the following: let (f t , ζ t ), 0 ≤ t ≤ 1 be a geodesic for the
condition metric. Is it true that max{µ(f t , ζ t : 0 ≤ t ≤ 1} is reached
at the extremes t = 0, 1? More generally, one can ask for convexity
of µ along these geodesics, or even convexity of log µ (which implies
convexity of µ).
Following [8,9,21], let us put the question in a general setting. Let
M be a Riemannian manifold and let κ : M → (0, ∞) be a Lipschitz
function. We call that conformal metric in M obtained by pointwise
multiplying the original one by κ the condition metric. We say that
a curve γ(t) in M is a minimizing geodesic (in the condition metric)
if it has minimal (condition) length among all curves with the same
extremes. A geodesic in the condition metric is then by definition any
curve that is locally a minimizing geodesic. Then, we say that κ is
self–convex if the function
t → log(κ(γ(t)))
is convex for any geodesic γ(t) in M. The question is then: Is µ
self–convex in W ?
It is interesting to point out that the usual unnormalized condition
number of linear algebra (that is, κ(A) = ‖A −1 ‖) is a self–convex
function in the set of maximal rank matrices, see [8,9] In [8] it is also
proved that functions given by the inverse of the distance to a (sufficiently
regular) submanifold of R n is log–convex when restricted to
an open set. Another interesting question is if that result can be extended
to arbitrary submanifolds of arbitrary Riemannian manifolds.
A.5 Extension of the algorithms for Smale’s
17th problem to other subspaces
The algorithms described above are all designed to solve polynomial
systems which are assumed to be in dense representation. In particular,
the “average” running time is for dense polynomial systems.
As any affine subspace of H d has zero–measure in H d , one cannot
conclude that the average running time of any of these algorithms