During the past week Adam and Jan worked on improving the quality of the enclosure based hybrid solver by developing a suite of property based tests for the underlying enclosure arithmetic. One observation was that enclosures may have (surprisingly) weak properties,. They may, for example, not satisfy seemingly obvious monotonicity properties. A concrete example may be observed when using affine Chebyshev approximation (in the sense of
Darulova and Kuncak) for quadratic terms to approximate the quadratic term x^2. For x in [0,1] one obtains [x,x-0.25] and for x in [0,0.5] the result is [0.5*x,0.5*x-0.0625]. An illustration of this is provided below:
The latter is everywhere point-wise thinner than the former, however the former does not contain the latter as an enclosure. In the coming days the property based testing of the solver will continue as well as work to extend the arithmetic operations supported in the arithmetic.
