Nonlinear Equations - UFRJ

158 [CH. A: OPEN PROBLEMS
of modern scientific computation, where most of the algorithms ...
are real number algorithms.”
Then the authors develop a model of computation on the real
numbers known today as the BSS model following the lines of a seminal
paper [19]. This model is well adapted to study the complexity
of numerical algorithms.
However this ideal picture suffers from an important defect. Numerical
analysts do not use the exact arithmetic of real numbers but
floating-point numbers and a finite precision arithmetic. The cited
authors remark on the ultimate need to take input and round-off error
into account in their theory. But now about twenty years later
there is scant progress in this direction. For this reason we feel important
to develop a model of computation based on floating-point
arithmetic and to study, in this model, the concepts of stability and
complexity of numerical computations.
A.2 A deterministic solution to Smale’s
17th problem
Smale’s 17th problem asks
“Can a zero of n complex polynomial equations in n unknowns
be found approximately, on the average, in polynomial time with a
uniform algorithm?”
The foundations to the study of this problem where set in the
so–called “Bezout Series”, that is [70–74]. The reader may see [79]
for a description of this problem.
After the publication of [79] there has been much progress in the
understanding of systems of polynomial equations. An Average Las
Vegas algorithm (i.e. an algorithm which starts by choosing some
points at random, with average polynomial running time) to solve this
problem was described in [11,12]. This algorithm is based on the idea
of homotopy methods, as in the Bezout Series. Next, [69] showed that
the complexity of following a homotopy path could actually be done