4.2 Nomenclature and Types

A rational interval is a closed connected set of rational numbers, it is represented in MPRIA by its endpoints which are GMP rational numbers. The C data type for these objects is mpri_t.

MPRIA functions operate on valid rational intervals, while their behaviour remains undefined with non-valid rational intervals; a valid rational interval is defined as follows¹:

• A valid rational interval can have finite or infinite endpoints, but its left endpoint is not larger than its right endpoint and cannot be +infinity (+1/0) while the right endpoint cannot be -infinity (-1/0). Whenever the left and right endpoints are equal to a same rational q, the valid rational interval reduces then to the singleton interval [q,q] which represents exactly the rational q; conversely, any rational q is perfectly represented by the singleton interval [q,q].

MPRIA functions may return intervals that are not valid as input value; their semantic is defined as follows²:

• Whenever the left endpoint or the right endpoint is NaN (0/0), it indicates that an invalid operation has been performed and that the resulting rational interval has no mathematical meaning.
• Whenever the left endpoint is strictly greater than the right endpoint, it means that the resulting rational interval is the empty interval.

Some functions on rational intervals return a rational number. Among such functions, there are mpri_get_left and mpri_get_right that respectively return the left and right endpoints of a rational interval, and there is mpri_diam_abs that computes the width of a rational interval.

Rational numbers (or rationals for short) and rational arithmetic functions are brought as is from the GMP library. The C data type for rationals is mpq_t, while their related functions start with the prefix mpq_ (see Rational Number Functions in The GNU MP Manual).

For rational intervals, because their endpoints are numbers exactly representable that are meant to enclose a result not exactly representable, the notion of precision is essentially related to their width which is meant to be arbitrarily small. The precision of a rational interval designs the integer binary logarithm of the reciprocal of its width; as such, it expresses in bits. The corresponding C data type is mpri_prec_t.

When a MPRIA function implements some sort of convergent algorithm to return rational intervals, besides passing a precision parameter in bits to terminate the computation, a surrounding mode parameter specifies whether to place the best convert either at the left endpoint, at the right endpoint or arbitrarily. The C data type for these modes is mpri_srnd_t. Typically it concerns implementations based on the Euclidean algorithm (which are omnipresent).

Some MPRIA functions that involve heavy computations admit as last parameter an assignment mode which specifies whether to assign either only the left endpoint, only the right endpoint, or the two endpoints. The C data type for these modes is mpri_asgmt_t. Those functions are considered as low-level and are both appended with the capitalised suffix _ASGMT and wrapped by a macro that assigns the two endpoints.