ValidatedNumerics.jl is a suite of Julia packages for performing Validated Numerics in Julia, i.e. rigorous computations with finite-precision floating-point arithmetic, using interval arithmetic: quantities are treated as intervals that are propagated throughout a calculation. The final result is an interval that is guaranteed to contain the correct result, starting from the given initial data.
The aims of the package are both correctness and good performance.
To install the package, from within Julia do
julia> Pkg.add("ValidatedNumerics")Since version 0.9, ValidatedNumerics.jl is a meta-package that automatically installs and reexports the following packages from the JuliaIntervals GitHub organization:
-
IntervalArithmetic.jl: arithmetic and elementary functions on intervals -
IntervalRootFinding.jl: find roots of multidimensional functions in a guaranteed way -
IntervalOptimisation.jl: guaranteed global optimisation of multidimensional functions -
IntervalConstraintProgramming.jl: characterization of feasible sets of equations and inequalities via constraint propagation -
IntervalContractors.jl: contractors and reverse (or inverse) functions -
TaylorModels.jl: rigorous function approximation
The documentation for each of the above packages can be found here
The IEEE Std 1788-2015 - IEEE Standard for Interval Arithmetic was published in June 2015. We are working towards having ValidatedNumerics be conformant with this standard.
To do so, we have incorporated tests from the excellent ITF1788 test suite, originally written by Marco Nehmeier and Maximilian Kiesner, and converted to a common format and to output tests for Julia by Oliver Heimlich.
- Luis Benet, Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México (UNAM)
- David P. Sanders, Departamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de México (UNAM)
- Validated Numerics: A Short Introduction to Rigorous Computations, W. Tucker, Princeton University Press (2010)
- Introduction to Interval Analysis, R.E. Moore, R.B. Kearfott & M.J. Cloud, SIAM (2009)
- MPFI.jl, a Julia wrapper around the MPFI C library, a multiple-precision interval arithmetic library based on MPFR
- Intervals.jl, an alternative implementation of basic interval functions.
This project was begun during a masters' course in the postgraduate programs in Physics and in Mathematics at UNAM during the second semester of 2013 (in Python -- the ValidiPy package), and was reinitiated -- now in Julia -- in the first semester of 2015. We thank the participants of the courses for putting up with the half-baked material and contributing energy and ideas.
Financial support is acknowledged from DGAPA-UNAM PAPIME grants PE-105911 and PE-107114, and DGAPA-UNAM PAPIIT grants IN-117214 and 117117. LB acknowledges support through a Cátedra Marcos Moshinsky (2013). DPS acknowledges a sabbatical fellowship from CONACYT and thanks Alan Edelman and the Julia group at MIT for hosting his sabbatical visit during 2016.