Computing Topological Degree and Equation Solving
This is the title of my PhD thesis, which I am in the process of writing
up. I am/was a research student at the
University of Bath,
under the supervision of
Dan Richardson.
Topological Degree
Given a system of (non-linear) equations F = 0 where
F : Rn -> Rn
and each fi : Rn -> R is continuous, the degree of F at 0 relative to
B, deg(F,B,0), where B is a box, is defined to be
an integer with the following properties:
- It has a root-counting property, being the sum of the signs of
the Jacobian determinants at all the solutions (assuming no singular
solutions).
- The degree is continuous in F.
- The degree is additive in B, so that if B = B1 U B2, where B1 and B2 are disjoint, then:
deg(F,B,0) = deg(F,B1,0) + deg(F,B2,0)
The basis of the original method of Oliver Aberth is to project the boundary of B under the
mapping F, and determine the number of intersections with an
arbitrary ray emanating from the origin.
The degree is determined only by the behaviour of F on the boundary
of B, and the computation can be expressed as a (complicated)
sequence of interval evaluations of the
fi over the boundary.
I have implemented and tested a number of variants of this algorithm, and
have undertaken some abstract complexity analysis.
Equation Solving
Calculating topological degree would seem to be useful for equation
solving. I have developed a prototypical solver, based upon repeated
degree computation and box subdivision. The solver will isolate and
find to reasonable accuracy all solutions (even singular ones) within
a specified region of interest.
So far, the solver has been used to find all the common points of
intersection of 3 arbitrary tori (quartics) in R3. I tentatively believe there can be a maximum of 8 such points...
(I will have my thesis available here in due course.)