I've looked at several issues involved in coding up my "ordered lists of disjoint atomic intervals" proposal.
The potential integration with other portions of Maxima is quite significant.
For example, one needs to be able to characterize many functions at the values MINF & INF.
This system should integrate with Maxima's functionality of _limits_.
For the elegant implementation of intervals, one needs the concept of interval "endpoints", which specify 3 things:
1. A _bound_, which may include the additional values MINF, INF;
2. A bit indicating whether the endpoint is _open_ or _closed_ ("(" v. "["; ")" v. "]"); and
3. A bit indicating the _direction_ of approach ("(" v. ")"; ")" v. "]").
An atomic interval consists of a lower endpoint, where the direction is from above, and
an upper endpoint, where the direction is from below.
Thus, the interval "[0 INF)" = "[0" intersect "INF)" is the non-negative reals, including 0, but not +infinity,
while "(0 INF)" = "(0" intersect "INF)" is the positive reals, not including 0, and not including +infinity.
(Intersection itself is commutative, but the interval "INF) [0" is more readably printed as "[0 INF)", etc.)
Mathematicians sometimes use the notation "0-" for "0)" and "0+" for "(0" in limits and integral bounds.
We need to extend functions to be able to compute not just f(x), but also f("x)"), f("x]"), f("(x"), and f("[x").
If f(x) is "increasing" (using Maxima terminology) and also continuous, then
f("x)") = "f(x) )"
f("x]") = "f(x) ]"
f("(x") = "( f(x)"
f("[x") = "[ f(x)"
If f(x) is "decreasing" (using Maxima terminology) and also continuous, then
f("x)") = "( f(x)"
f("x]") = "[ f(x)"
f("(x") = "f(x) )"
f("[x") = "f(x) ]"
In particular, we will need an AT/ATVALUE type of database for these extended values.
Consider the function invert(x):=1/x.
The domain of invert(x) is (MINF 0) union (0 INF).
We would like our database to record the following facts:
invert("(MINF")="0)",
invert("0)")="(MINF"),
invert("(0")="INF)", and
invert("INF)")="(0".
However, we cannot declare invert(x) a "decreasing" function, because although 1<2 and (1/1)>(1/2),
-1<1 and invert(-1)=-1 < invert(1)=+1.
But we can often gain precision in interval analysis by breaking invert(x) into 2 partial functions:
invert_negative(x):=1/x, defined only on the domain (MINF 0); and
invert_positive(x):=1/x, defined only on the domain (0 INF).
Now, both invert_negative(x) and invert_positive(x) can be declared "decreasing" functions.
We can now define invert(x) as:
invert(x):=if x<0 then invert_negative(x) elseif x>0 then invert_positive(x) else error("domain error for invert()");
By always "inlining"/"expanding" invert(x) as if it were a _macro_ instead of a function, the interval analysis can
sometimes gain additional precision by forcing the analysis to handle individual cases, depending upon the sign of x.
Also, now that we have characterized invert(x) on its entire domain, we no longer have to "special case" invert(x),
because "decreasing" tells us everything that we need to know about invert_negative(x) and invert_positive(x) to be
able to perform interval analysis.
Other complicated functions -- e.g., sin(x) -- can also be broken down for interval analysis:
sin(x):=sin_range_reduced(%pi-mod(%pi-x,2*%pi)),
where the domain of sin_range_reduced(x) is (-%pi %pi], and sin_range_reduced(x) can be declared "increasing".
*** See the Maxima manual description of mod(x,y). ***