Interval comparisons
According to Motion 2 on process structure we should settle issues
at level 1 of that motion before descending into the details of
the lower levels. An outstanding unresolved issue at level 1 is
the definition of relation operators.
What we have so far are the relational operators in the Vienna Proposal
section 5.4 (henceforth "Vienna 5.4"). Here == is the name given to
the interval extension of the equality relation among reals. Is this
satisfactory?
A consequence of Vienna 5.4:
if E computes f and if x in xx, then E(xx) == hull(f(x)) is false.
Remember, == is the interval extension of equality.
This problem can be avoided. The remedy that I propose has the additional
advantage that it explains the backward definitions in Vienna 3.11. This
is necessary because the backward definition of interval division is
not universally agreed to (see, e.g. Motion 5).
The cause of the difficulty is that IA focuses exclusively on EVALUATION,
whereas numerical analysis (which is the intended application of
interval arithmetic) is concerned with SOLVING.
In numerical analysis one can only solve by iterated evaluation.
In interval arithmetic one can make solving the basic step.
Evaluation becomes available as a special case of solving:
Proposal: given expression E for computing function f, xx, and yy,
define IA as solving
E(xx) = yy
in the sense of making xx or yy or both as narrow as the equation allows.
Properties:
(1) Interval arithmetic is the special case where yy = [-oo,+oo].
In that case xx stays unchanged and yy narrows to the result of E(xx)
according to interval arithmetic as traditionally understood.
(2) If yy = hull(y) for some real y, then xx narrows to the hull of
the set of solutions to f(x) = y. The SIVIA algorithm, for example,
computes this hull. It solves, for example, algebraic equations this way.
(3) The above are extremes for yy. Solving E(xx) = yy makes sense for any
combination of intervals xx and yy of whatever widths, thus making the
problem a common generalization of evaluation and solving. Such generality
can be standard interval arithmetic. It seems beyond conventional
numerical analysis. In this general case SIVIA works unchanged.
(4) It makes just as much sense to solve E(xx) <= yy and E(xx) >= yy:
narrow xx or yy or both as much as the inequalities allow. SIVIA works
unchanged.
As final property: it defines the meaning of
xx <= yy and of xx >= yy
as a special case of (4). To solve these is to narrow xx or yy or both
as much as the inequalities allow. As a relational operation: true or
false according to whether the inequality is solvable.
But then we have, at variance with Vienna 5.4, that the relational
operation evaluates to true except when xx and yy are disjoint with xx
to the right of yy.
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