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~~<div style="float: right; margin: 0 3em">[[File:Octave-interval.png|center]]</div>~~The GNU Octave interval package for real-valued [https://en.wikipedia.org/wiki/Interval_arithmetic interval arithmetic].
~~* Possible ~~== Project ideas (TODOs) ==~~*~~* To be considered in the future: Algorithms can be migrated from the C-XSC Toolbox (C++ code) from [http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_new.html] (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.~~*~~* Interval version of <code>interp1</code>~~*~~* Extend <code>subsasgn</code> to allow direct manipulation of inf and sup (and dec) properties.
~~:~~* Tight Enclosure of Matrix Multiplication with Level 3 BLAS [http://kam.mff.cuni.cz/conferences/swim2015/abstracts/ozaki.pdf] [http://kam.mff.cuni.cz/conferences/swim2015/slides/ozaki.pdf]~~:~~* Verified Convex Hull for Inexact Data [http://kam.mff.cuni.cz/conferences/swim2015/abstracts/ohta.pdf] [http://kam.mff.cuni.cz/conferences/swim2015/slides/ohta.pdf]~~:~~* Implement user-controllable output from the interval standard (e. g. via printf functions):
~~ ~~
~~== Similar software ==~~
~~ ~~
~~For C++ there is an interval library [https://github.com/nehmeier/libieeep1788/ libieeep1788] by Marco Nehmeier (member of IEEE P1788). It aims to be standard compliant with IEEE Std 1788-2015 and is designed in a modular way, supporting several interval data types and different flavors of interval arithmetic [http://www.youtube.com/watch?v=GOa9aWAZO_Q]. The GNU Octave interval package shares several unit tests with libieeep1788.~~
~~ ~~
~~For Julia there is an evolving interval library [https://github.com/dpsanders/ValidatedNumerics.jl ValidatedNumerics.jl] by Luis Benet and David P. Sanders. It is planned to become conforming to IEEE Std 1788-2015 (or to the basic standard 1788.1) in the future.~~
~~+~~`- src/~~ | | ~~ +- mpfr_function_d.cc ~~ ~~ – computes various arithmetic functions correctly rounded (using MPFR)~~ | | ~~ `- ... ~~ ~~ – other oct-file sources~~ `- test/ – interval arithmetic unit tests~~
~~ ~~ ... parameter checking ...~~ ~~ x.inf = ceil (x.inf);~~ ~~ x.sup = ceil (x.sup);
~~ ~~ ... parameter checking ...~~ ~~ l = mpfr_function_d ('plus', -inf, x.inf, y.inf);~~ ~~ u = mpfr_function_d ('plus', +inf, x.sup, y.sup);
~~ ~~ emptyresult = isempty (x) | isempty (y);~~ ~~ l(emptyresult) = inf;~~ ~~ u(emptyresult) = -inf;~~ ~~ …

Updated links and developer information, added reference to other IEEE 1788 software and new IEEE Std 1788.1

[[File:Interval-sombrero.png|280px|thumb|right|Plotting the interval enclosure of a function]]

== Distribution ==

* [~~http~~https://octave.sourceforge.~~net~~io/interval/~~index.html ~~Latest version at Octave Forge]

** <code>pkg install -forge interval</code>

** [~~http~~https://octave.sourceforge.~~net~~io/interval/overview.html function reference]** [~~http~~https://octave.sourceforge.~~net~~io/interval/package_doc/index.html package documentation] (user manual)

'''Third-party'''

* [https://tracker.debian.org/pkg/octave-interval Debian GNU/Linux], [https://launchpad.net/ubuntu/+source/octave-interval Launchpad Ubuntu]

* [https://aur.archlinux.org/packages/octave-interval/ archlinux user repository]

* Included in [https://ftp.gnu.org/gnu/octave/windows/ official Windows installer] and installed automatically with Octave (since version 4.0.1)

* [https://~~trac~~github.com/macports~~.org~~/~~browser~~macports-ports/~~trunk~~tree/~~dports~~master/math/octave-interval / MacPorts] for Mac OS X* [~~http~~https://www.freshports.org/math/octave-forge-interval/ FreshPorts] for FreeBSD

* [https://cygwin.com/cgi-bin2/package-grep.cgi?grep=octave-interval Cygwin] for Windows

* [https://build.opensuse.org/package/show/science/octave-forge-interval openSUSE build service]

== Development status ==

* Completeness

** All required functions from [https://standards.ieee.org/findstds/standard/1788-2015.html IEEE Std 1788-2015], IEEE standard for interval arithmetic, are implemented. The standard was approved by IEEE-SA on June 11, 2015. It will remain active for ten years. The standard was approved by ANSI in 2016.** Also, the minimalistic standard [https://standards.ieee.org/findstds/standard/1788.1-2017.html IEEE Std 1788.1-2017], IEEE standard for interval arithmetic (simplified) is fully implemented. The standard was approved by IEEE-SA on December 6, 2017 (and published in January 2018).** In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting , and solvers.

* Quality

** Most arithmetic operations produce tight, correctly-rounded results. That is, the smallest possible interval with double-precision (binary64) endpoints, which encloses the exact result.** Includes ~~tests ~~[https://github.com/oheim/ITF1788 large test suite] for ~~all functions, many tests for basic ~~arithmetic functions** ~~No known ~~For open bugs please refer to the [https://savannah.gnu.org/search/?words=forge+interval&type_of_search=bugs&only_group_id=1925&exact=1 bug tracker].* Performance** All elementary functions have been [https://octave.org/doc/interpreter/Vectorization-and-Faster-Code-Execution.html vectorized] and run fast on large input data. ~~The package ~~** Arithmetic is ~~quite new and still has a small user base~~performed with the [http://www.mpfr.org/ GNU MPFR] library internally. Where possible, ~~so there might be hidden bugs~~the optimized [http://web.archive.org/web/20170128033523/http://lipforge.ens-lyon. ~~Also some advanced functions will always need more testing~~fr/www/crlibm/ CRlibm] library is used.

* Portability

** Runs in GNU Octave ≥ 3.8.2

** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD

>> A = infsup ("[2, 4]");

>> A.inf = infsup ("[1, 3]")

>> A.inf.inf = 42 # should produce error?

>> A.inf.sup = 42 # should produce error?

a) It should be possible to specify the preferred overall field width (the length of s).

b) It should be possible to specify how Empty, Entire and NaI are output,

== Compatibility ==

The interval package's main goal is to be compliant with IEEE Std 1788-2015, so it is compatible with other standard-conforming implementations (on the set of operations described by the standard document).Other implementations, which are known to aim for standard conformance are: * [https://github.com/JuliaIntervals/IntervalArithmetic.jl IntervalArithmetic.jl package] (Julia)* [https://github.com/jinterval/jinterval JInterval library] (Java)* [https://github.com/nadezhin/libieeep1788 ieeep1788 library] (C++) created by Marco Nehmeier, later forked by Dmitry Nadezhin

=== Octave Forge simp package ===

* Basic operations can be found in both packages, but the availability of special functions depends

<div style="display:flex; align-items: flex-start"><div style="margin-right: 2em">{{Code|~~In GNU Octave the ~~Computation with this interval package ~~can also be run alongside INTLAB.~~|<syntaxhighlight lang="octave">~~ # INTLAB intervals~~pkg load interval~~ ~~A1 = infsup (2, 3);~~ ~~B1 = hull (-4, ~~A1~~A2);~~ ~~C1 = midrad (0, 2);~~ # Interval package intervals~~~~ pkg load interval~~A1 + B1 * C1</syntaxhighlight>}}</div><div>{{Code|Computation with INTLAB|<syntaxhighlight lang="octave">startintlab~~ ~~A2 = infsup (2, 3);~~ ~~B2 = hull (-4, A2);~~ ~~C2 = midrad (0, 2);~~ pkg unload interval~~~~ ~~~~ # Computation with INTLAB~~~~ A1 + B1 * C1~~~~ # Computation without INTLAB~~~~ ~~A2 + B2 * C2

</syntaxhighlight>

}}

</div>

</div>

==== Known differences ====

| isintval (x)

|}

== Developer Information ==

'''Build dependencies'''

<code>apt-get install libmpfr-dev autoconf automake ~~git python3-ply python3-yaml ~~inkscape zopfli</code> ~~* Octave~~~~* Mercurial~~~~* Texinfo~~~~* MPFR~~~~* Octave package: doctest~~~~** Purpose: Verification of the examples in the manual and in the function documentation~~~~** Installation: Use <code>pkg install -forge doctest</code> inside Octave~~~~* Octave package: generate_html~~~~** Purpose: Generate HTML documenation for publication on Octave Forge (only needed for release)~~~~** Installation: Use <code>pkg install -forge generate_html</code> inside Octave~~~~* Autoconf & Automake~~~~** Purpose: Preparation of bundled crlibm library for a release~~~~* ITF1788~~~~** Purpose: Compilation of unit-test~~~~** Installation: The git repository is automatically cloned by the Makefile~~~~* Inkscape~~~~** Purpose: Generate / convert images for the manual~~~~* Zopfli~~~~** Purpose: Optimize the size of the compressed release tarball~~

=== Architecture ===

| | `- ... – further functions on decorated intervals

| `- ... – a few global functions that don't operate on intervals

=== Best practices ===

Octave functions may be used as long as they don't introduce arithmetic errors. For example, the ceil function can be used safely since it is exact on binary64 numbers.

function x = ceil (x)

endfunction

All functions should be implemented using vectorization and indexing. This is very important for performance on large data. For example, consider the plus function. It computes lower and upper boundaries of the result (x.inf, y.inf, x.sup, y.sup may be vectors or matrices) and then uses an indexing expression to adjust values where empty intervals would have produces problematic values.

function x = plus (x, y)

endfunction

240

edits

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