The approach taken for binary functions with constant interval arithmetic may be used with linear interval arithmetic. As before, we focus on binary grid functions. When presented with a binary function, we will cut it into sections where each section may be extended to a grid function which fits into a class.
As with unary functions, we will partition the domain based on monotonicity, so we may handle the upper and lower bounds independently. Concavity is also used when partitioning:
An upper bound will be determined for 
in two stages: first, a bilinear upper bound 
of
will be determined; then, a linear upper bound of h will be constructed;
this upper bound will be an upper bound of g.
A function is bilinear if
both 
and 
are linear.
An approximate upper bound 
is constructed from g:
is the bilinear Lagrange intepolating polynomial
of the set G, a subgrid of g:
.
The location of the subgrid g is constrained by which
concavity class g belongs to:
;
if this is not the case we may extend g.
Where g is concave up,
we assume that G is finer than g:
,
the mixed partial at (p,q) is matched.
When 
, is an upper bound.
This is shown by the following proof by contradiction;
the diagram given accompanies the proof.
fails, at (x',y'), to be an upper bound of g:
, which has an upper bound 
,
given by with 
,
since 
.
Since 
is an upper bound of
for 
,
is an upper bound of
.
It follows that 
is an upper bound
of , since both functions are linear;
so 
,
which contradicts our initial assumption.
When 
,
is again an upper bound,
and may be proven with a similar argument, which follows.
fails, at (x',y'), to be an upper bound of g:
, which has an upper bound ,
given by with ,
since .
Since is an upper bound of
for ,
is an upper bound of
.
It follows that is an upper bound
of , since both functions are linear;
so ,
which contradicts our initial assumption.
When 
,
is again an upper bound,
and may be proven with the preceding argument.
Alternatively, one may consider g'(x,y) = g(y,x),
after ensuring that 
.
The proof does not hold for the last case,
when .
In any of the other three cases,
we may take 
;
in this last case, we must further
test g to determine an upper bound.
We will not dwell on this since another
method will be presented shortly.
After h is determined, we construct a linear upper bound
of 
from
.
Given that
.
Previous subsections detail how an upper bound of a unary
function may be found.
| Jeff Tupper | March 1996 |