Nonlinear Equations - UFRJ

160 [CH. A: OPEN PROBLEMS
ζ 1 of f 1 which is reached by the homotopy starting at f 0 ◦ U ∗ will
be different for different choices of U. The question is then, assuming
that all the roots of f 1 are non–singular, what is the probability
(of the set of unitary matrices with Haar measure) of finding each
root? Some experiments [10] seem to show that all roots are equally
probable, at least in the case of quadratic systems. But, there is no
theoretical proof of this fact yet.
A.4 Log–Convexity
Let H d be the projective space of systems of n homogeneous polynomials
of fixed degrees (d) = (d 1 , . . . , d n ) and n + 1 unknowns. In
[69], it is proved that following a homotopy path (f t , ζ t ) (where f t is
any C 1 curve in P(H d ), and ζ t is defined by continuation) requires
at most
∫ 1
L κ (f t , ζ t ) = CD 3/2 µ(f t , ζ t )‖( f ˙ t , ˙ζ t )‖ dt (A.1)
0
homotopy steps (see [7,10,25,31] for practical algorithms and implementation,
and see [55, 56] for different approaches to practical implementation
of Newton’s method). Here, C is a universal constant,
D is the max of the d i and µ is the normalized contition number,
sometimes denoted µ norm , and defined by
(
∥
µ(f, z) = ‖f‖ ∥(Df(z) | z ⊥) −1 Diag
∀ f ∈ P(H d ), z ∈ P(C n+1 ).
‖z‖ di−1 d 1/2
i
)∥ ∥∥ ,
Note that µ(f, z) is essentially the operator norm of the inverse of the
matrix Df(z) restricted to the orthogonal complement of z. Then,
(A.1) is the length of the curve (f t , ζ t ) in the so–called condition
metric, that is the metric in
W = {(f, z) ∈ P(H d ) × P n : µ(f, z) < +∞}
defined by pointwise multiplying the usual product structure by the
condition number.
Thus, paths (f t , ζ t ) which are, in some sense, optimal for the homotopy
method, are those defined as shortest geodesics in the condition
metric. They are known to exist and to have length which is