Interval arithmetic is a method of representing uncertain or inexact real quantities by intervals a, b and extending arithmetic and elementary functions so that evaluating an expression over intervals produces an interval enclosure that is guaranteed to contain all pointwise results for inputs within those intervals, a property foundational to validated numerics and Numerical analysis. According to the IEEE 1788 standard, the model is defined over “common intervals” with support for empty and unbounded intervals via a set‑based flavor and a system of decorations to record semantic information such as domain intersections. IEEE SA;
ANSI Webstore. (
standards.ieee.org)
Origins and formalization
- –Early systematic development is credited to Ramon E. Moore, whose 1966 monograph introduced the core arithmetic and its use for rigorous error control in computation; later syntheses include his co‑authored 2009 text. [Interval Analysis](book://Ramon E. Moore|Interval Analysis|Prentice-Hall|1966);
Introduction to Interval Analysis. (
catalogue.nla.gov.au)
- –Foundational extensions and monographs that shaped practice include Neumaier’s treatment of systems of equations and interval matrices, emphasizing inclusion properties and algorithmic use.
Interval Methods for Systems of Equations. (
Basic definitions and operations
- –A real interval X = x̲, x̄ represents all x with x̲ ≤ x ≤ x̄; real numbers are identified with degenerate intervals r, r. Addition and subtraction are endpoint‑wise: x̲, x̄ + y̲, ȳ = x̲ + y̲, x̄ + ȳ; multiplication and division use extrema over endpoint combinations, with division undefined when the denominator contains 0 in the set‑based model unless a domain intersection is taken.
Interval arithmetic;
IEEE SA. (
- –Correctness requires outward rounding of endpoints so the computed enclosure provably contains the exact result in floating‑point environments such as IEEE 754.
MPFR library. (
Fundamental properties and theorems
- –The inclusion property (also called inclusion isotonicity) ensures that if X ⊆ X′ and Y ⊆ Y′ then f(X, Y) ⊆ f(X′, Y′) for inclusion‑isotone interval extensions; as a consequence, the natural interval extension (obtained by replacing real operations and functions with their interval counterparts) yields an enclosure of the true range of a real function over a box.
BIT Numerical Mathematics. (
- –Interval versions of the mean‑value and Taylor theorems provide computable enclosures essential to many algorithms, including root isolation and verified solvers.
Standards and models
- –IEEE Std 1788‑2015 specifies operations, decorations, flavors (set‑based by default), and exception handling for interval arithmetic, interoperating with Floating-point arithmetic as a layer between hardware and language. A simplified subset for double precision is codified in IEEE 1788.1‑2017.
Core algorithms and methods
- –Interval Newton methods combine interval enclosures with derivative bounds to contract boxes and prove existence/uniqueness of zeros, often accelerating convergence relative to simple bisection.
- –Branch‑and‑bound global optimization uses interval range bounds and monotonicity tests to discard subregions that cannot contain optima, yielding rigorous bounds and, upon termination, certified optima.
Limitations and typical effects
- –The dependency problem arises when a variable appears multiple times in an expression; naive interval evaluation may overestimate ranges because correlations are ignored. Remedies include reformulation, subdivision, and advanced models.
- –The wrapping effect, prominent in validated ODE integration and iterative linear transformations, inflates enclosures as aligned boxes are repeatedly mapped under rotations or shears. Analyses formalize conditions under which wrapping occurs and techniques to mitigate it.
Extensions and related frameworks
- –Affine arithmetic augments intervals with first‑order correlation terms, typically reducing overestimation for composed computations at increased cost; it has seen applications in graphics and uncertainty quantification.
An Introduction to Affine Arithmetic;
Computer Graphics Forum (2003). (
tcam.sbmac.org.br)
- –Generalized, Kaucher, or directed intervals extend the algebra to include “improper” intervals, enabling inner/outer operations and alternative semantics; modal interval analysis adds logical quantifiers to intervals for reasoning about feasibility.
JUCS (1995);
Directed intervals resource. (
zenodo.org)
Software and implementations
- –MATLAB/Octave’s INTLAB provides a comprehensive environment for interval arithmetic and verified numerics, including linear algebra, nonlinear systems, ODEs, and global optimization.
INTLAB site;
- –C++ users commonly employ Boost.Interval, a header‑only library providing interval types and policies for rounding and checking.
Boost.Interval docs;
- –The Julia ecosystem offers IntervalArithmetic.jl, which supports IEEE 1788 semantics (including decorations) and integrates with other validated‑numerics packages; recent releases are archived with DOIs.
IntervalArithmetic.jl docs;
- –Multiple‑precision interval libraries include MPFI (based on MPFR) and GNU MPRIA (rational intervals), enabling arbitrary‑precision enclosures.
GNU MPRIA;
Applications
- –Verified numerical solvers use interval arithmetic to produce mathematically certified results in non‑linear equations, ODEs, and PDE‑related tasks, often as part of Computer-assisted proof workflows.
Validated Numerics. (
- –In Global optimization, interval branch‑and‑bound has been applied to industrial problems and to provide rigorous bounds for nonconvex programs.
- –Computer‑assisted proofs in dynamics and geometry have leveraged interval arithmetic, notably Warwick Tucker’s rigorous computations establishing the existence of the Lorenz attractor, later influencing further verified ODE work.
C. R. Acad. Sci. Paris 1999 notice. (
Practical considerations
- –Correct rounding is essential; implementations rely on directed rounding modes or exact rational endpoints to maintain outward rounding, ensuring that computed bounds are conservative with respect to floating‑point error.
- –To reduce overestimation from dependency and wrapping, practitioners use algebraic reformulation, variable splitting, Taylor models, affine arithmetic, and adaptive subdivision, often combined with Automatic differentiation for tight derivative bounds.