RobustArithmetic: Verified Interval Arithmetic and Rigorous Enclosures in Pure R
Verified interval arithmetic for R, in the inf-sup (endpoint)
representation of the set-based flavor of the interval standard. Every
operation returns an enclosure that provably contains the exact result:
outward rounding is obtained from the predecessor and successor formulas of
Rump, Zimmermann, Boldo and Melquiond (2009)
<doi:10.1007/s10543-009-0218-z>, which are valid under round-to-nearest and
therefore need no change to the floating-point rounding mode. That mode is
not reachable from R, and changing it would not be a local act: it is
per-thread state of the processor, so it would govern every floating-point
operation executed afterwards on that thread, in this package or anywhere
else. Elementary functions are provided at two
levels: a fast level over the system math library, widened by a declared
slack derived from published accuracy measurements, and a rigorous level
over 'Rmpfr' with a directed-rounding bridge, reached by an escalation
ladder of precisions when a verdict would otherwise fall inside the slack.
On top of the kernel the package builds natural and centered interval
extensions of expressions, a monotonicity test, the Hansen-Sengupta
interval Newton operator with extended division and epsilon-inflated
candidate verification, and a subdivision (paving) engine whose only
failure mode is a named abstention with its budget printed. Conformance
with IEEE Std 1788.1-2017 <doi:10.1109/IEEESTD.2018.8277144> is not
claimed, and the reason is the standard's own: its subclause 1.5 makes
conformance a list of requirements that an implementation shall satisfy,
with no partial grade to claim. What this
package follows, measured one requirement at a time and stated in the
package documentation, is the interval type and the decoration system of
clause 5, 22 of the 39 arithmetic operations of Table 4.1, and the seven
numeric functions of Table 4.3. What it does not provide is the
cancellative operations, the interval comparison relations, the text input
and output of subclause 6.8, the interchange representation of subclause
7.3, and the tightest accuracy that subclause 6.5.2 requires of the basic
operations, which here are one unit in the last place wider at each end.
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