Theoretical Optimal Strategy

Suppose we have an-dimensional box X and a continuous function F :X →Rⁿ given by (7.1). Given an m-dimensional face which arises in a topological degree computation over X, we are required to evaluate some or all of the functions fn−m, . . . , fn over the face. A subdivision is required precisely when all of the resultant intervals contain zero; in this case the face is partitioned into sub-faces. A sub-face will itself require further subdivision if all of the interval evaluations over it contain zero. Ideally, therefore, we would partition the original face into the smallest number of sub-faces that do not require further subdivision, so that the total number of faces to be processed is minimised.

Abstracted Subdivision Problem

As before, we can abstract the notion of face subdivision away from the broader context of computing the topological degree. Here we will consider the first iteration of the algorithm, without loss of generality. Disregarding the assigned orientation, a facescan be considered to be just ann−1-dimensional box inRⁿ. We havencontinuous functionsf1, . . . , fn:s→R, not identically zero, with no zero inscommon to all functions. We wish to partitionsinto sub-boxess₁, . . . , s_k such that for eachs_i,i= 1, . . . , k, there exists some j_i ∈ {1, . . . , n} for which 06∈fji(si), where the latter denotes the interval extension used forfji.

Definition 7.1(Non-terminal). A face sisnon-terminal, with respect to a set of (interval extensions for) functions f1, . . . ,fn, if ∀i= 1, . . . , n, 0∈fi(s).

The non-terminal faces are precisely those which require subdivision in the degree com-putation algorithm.

Definition 7.2 (Subdivision point). Asubdivision point is a point chosen inside ann− 1-dimensional face as the basis for a partitioning of that face into2ⁿ⁻¹ sub-faces, by dividing all component intervals about that point. More generally, we may choose not to divide all component intervals of a face s; in this case we just say that it is partitioned into sub-faces s1, . . . , sk. The term subdivision will either refer to the partition itself (as a set of sub-faces) or to the n−2-dimensional box which forms the common boundary of any two

7 Computation of Topological Degree

Optimal Face Partitioning

Definition 7.3(Optimal partitioning). A partitioning ofsinto terminal sub-facess1, . . . , sk

isoptimal, with respect to a set of (interval extensions for) functions f₁, . . . ,f_n, if, for any other partitioning into terminal sub-faces s⁰₁, . . . s⁰_l, k≤l, i.e. k is minimal.

Let us consider the solution set for each of f₁(x) = 0, . . . , f_n(x) = 0 over s. Under the starting assumptions, each solution set must consist of one or more closed subsets of s of dimension n−2 or less: curves in R², surfaces in R³, hypersurfaces in R⁴⁺. Under the terminology of cylindrical decomposition (cf. Subsection 5.3.4), sign conditions on all the f_i will, collectively, form a stratification ofs into manifolds.

Typical Case

Let us assume that the solution set of f_i(x) = 0 for at least one i ∈ {1, . . . , n} does not project fully onto all edges (child faces) of the face, viz. the box enclosure for the intersection of the solution set of f_i(x) = 0 with the face sis smaller thansitself.

The following abstract mechanism for determining the minimum number of subdivisions is then proposed: We can arbitrarily select ann−2-dimensional cross-section of the face which does not intersect at least one of these solution sets, by restricting one of the interval fields to a point. This subset of the face can then be expanded (back into ann−1-dimensional box) by gradually extending the width of the interval field that was restricted, only until all of the solution sets intersect — this now constitutes a non-terminal subface. We are now guaranteed that at least one subdivision must occur within this designated region. Since the last of the solution sets (continuous curves) to be included only intersects minutely, we can be sure that only one subdivision within this region will suffice.

The remainder of the face can be partitioned by repetition of this method, assuming we always remain in the typical case. We then have a partition into r1+r0 (box) regions, of whichr₁ should contain exactly one (distinct) subdivision, andr₀ of which (possiblyr₀= 0) do not require a subdivision. Therefore the minimum possible number of subdivisions is preciselyr1, which results in an optimum partitioning into r1 + 1 sub-faces. With such a partitioning, the maximum number of interval function evaluations over this face is thus n(r₁+ 1).

Example 7.5. Consider a 2-dimensional face s from a box in R³, where the solution sets of f₁(x) = 0, f₂(x) = 0, f₃(x) = 0 are as depicted in Figure 7.12.

• Starting from the left-hand edge of the face, we must have a sub-face which intersects the f₂(x) = 0 curve. Gradually extending this rightward, we can incorporate part of f₃(x) = 0 and finally we can just include the smallest part off₁ = 0. This is the first region,R1, which must contain a subdivision.

• Continuing on in the same fashion, we can designate a region R₂ which intersects f1(x) = 0, f2(x) = 0, and the smallest part of f3(x) = 0.

7 Computation of Topological Degree

f (x)=0 ₁ f (x)=0 2 f (x)=0 3

Figure 7.12: Solution sets of fi(x) = 0,i= 1,2,3, over a face sin Example 7.5.

• The left-hand edge of the remainder of the face (s\(R₁∪R₂)) is intersected by all three solution sets, so we can instead expand a new region from the top. This can grow down until it finally includesf1(x) = 0 in its corner. This region is designated R3.

• The remaining region, R₄, is not intersected by f₂(x) = 0, so it need not contain a subdivision.

We thus have a partition into four regions; R1, . . . , R4, three of which (R1, R2, and R3) need a subdivision, as depicted in Figure 7.13. An optimal subdivision using precisely three subdivisions, i.e. into four subfaces (s1, . . . , s4), is therefore possible, as illustrated in Figure 7.14.

The reader’s attention is drawn to the difference between the partition of the face into the regions Ri, each of which must contain a subdivision, and the (example) partition into the actual sub-facessi.

The analysis here is predicated upon the assumption of exact interval arithmetic (i.e.

ignoring the dependency problem), which is in general not delivered in practice. To cater for interval extensions of functions, the individual solution sets may need to be expanded;

also their extent (and the extent of overestimation) depends on the size of the (sub-)face in question. However, the approach is still fundamentally applicable.

Linear Case

In the case where all of thef_i are linear, the solution sets of allf_i(x) = 0can be determined exactly, and it is feasible to actually implement some variant of an optimal subdivision strategy. This is of limited merit in itself, since the computation of topological degree for linear systems is easy in any case. However, if a sufficiently good approximation can be found for each solution set over a face, a method based on this optimal strategy may also be applicable for nonlinear systems.

Maximal Projection Case

In the (presumbly rare) case where the solution sets for all of the fi(x) = 0 project onto

7 Computation of Topological Degree

R R

R

R

1 2

3

4

Figure 7.13: Partitioning of the face sinto regions R1, . . . , R4 in Example 7.5.

s1 s₂

s₃

s₄

Figure 7.14: Example optimal partitioning of the face s into terminal sub-faces s₁, . . . , s₄ in Example 7.5.

7 Computation of Topological Degree

performed, since anyn−2-dimensional cross-section that is chosen will intersect all solution sets. Such an example is illustrated in Figure 7.15.

f (x)=0₁ f (x)=0₂ f (x)=0₃

Figure 7.15: Optimal face partitioning in the maximal projection case (where the zero sets of each fi intersect each edge).

Here, finding a partitioning which is optimal is a significantly harder problem than for the typical case. We may briefly sketch a couple of possible approaches to this. The first is to observe that, although all solution sets project fully onto all edges of the face, it is possible to find sub-faces which do not have this maximal projection property. Any such sub-face, falling into the typical case, can be resolved by the former method. It therefore suffices to find a good way to partition the face into sub-faces with this property — this is in itself an optimal partitioning problem! So we have nested optimal partitioning problems; furthermore it is the case that an optimal partitioning into individually optimally-partitioned sub-faces does not necessarily constitute an overall partitioning that is optimal. This is clearly an extremely hard searching problem.

The other approach that we may mention, like the typical case, relies on enumerating the smallest possible subsets of the face which must contain a subdivision, except that instead of starting with a cross-section of the face, we start with a corner. From each corner, we can expand box subsets (which must initially be terminal) until they only just become non-terminal (where the last solution set enters minutely). As before, the remainder of the face can be handled repetetively. The difficulty here is that we can consider expanding these box subsets in more than one dimension (for faces of dimension 2 or more), so that there are more degrees of freedom in determining such minimally non-terminal regions.

(In the typical case, these regions were uniquely determined from any given cross–section,

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is dependent on the choice of initial region. The choice of each region can be described by n−2 variables, and we may need many regions, so we have a nonlinear minimisation problem. Since it is likely that the system of equations and inequalities which governs such an optimal partitioning may well be larger and more complicated than the original system under examination, the effort required to solve it would very likely not be worthwhile.

In conclusion, it is plausible to outline strategies that give very good face partitionings — optimal ones in many cases — but not strategies that give provably optimal ones in all cases.

Although the abstract problem of finding optimal partitionings is useful for motivating the design of practical algorithms, we have seen here that the problem of finding an optimal face partitioning can be more difficult to solve than the entire original degree computation. In terms of reducing computational effort, therefore, any attempt at a comprehensive optimal face–partitioning algorithm would be counterproductive — any further analysis here would thus only satisfy an esoteric, not a practical, interest.