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<div style="float: right; margin: 0 3em">[The GNU Octave interval package for real-valued [Filehttps:Octave-//en.wikipedia.org/wiki/Interval_arithmetic intervalarithmetic].* Intervals are closed, connected subsets of the real numbers. Intervals may be unbound (in either or both directions) or empty.png|center]]In special cases <code>+inf</code> and <code>-inf</divcode>are used to denote boundaries of unbound intervals, but any member of the interval is a finite real number.* Classical functions are extended to interval functions as follows: The GNU Octave result of function f evaluated on interval x is an interval '''enclosure of all possible values''' of f over x where the function is defined. Most interval arithmetic functions in this package for real-valued manage to produce a very accurate such enclosure.* The result of an interval arithmeticfunction is an interval in general. It might happen, that the mathematical range of a function consist of several intervals, but their union will be returned, e. g., 1 / [-1, 1] = [Entire].
__TOC__[[File:Interval-sombrero.png|280px|thumb|rightleft|Example: Plotting the interval enclosure of a function]] <div style="clear:left"></div>
== Distribution ==
* [httphttps://octave.sourceforge.netio/interval/index.html Latest version at Octave Forge]
** <code>pkg install -forge interval</code>
** [httphttps://octave.sourceforge.netio/interval/overview.html function reference]** [httphttps://octave.sourceforge.netio/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://tracftp.macportsgnu.org/browsergnu/octave/windows/ official Windows installer] and installed automatically with Octave (since version 4.0.1)* [https://github.com/macports/macports-ports/trunktree/dportsmaster/math/octave-interval / MacPorts] for Mac OS X* [httphttps://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 baseperformed with the [http://www.mpfr.org/ GNU MPFR] library internally. Where possible, so there might be hidden bugsthe optimized [http://web.archive.org/web/20170128033523/http://lipforge.ens-lyon. Also some advanced functions will always need more testingfr/www/crlibm/ CRlibm] library is used.
* Portability
** Runs in GNU Octave 3.8.2 and 4.0.0** Known to run under GNU/Linux, Microsoft Windows, Mac OS X macOS, and FreeBSD* 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>** The [http://capd.ii.uj.edu.pl/ CAPD package] contains an algorithm for QR factorization. Analyze. Migrate.** Extend <code>subsasgn</code> to allow direct manipulation of inf and sup (and dec) properties.
>> A = infsup ("[2, 4]");
>> A.inf = infsup ("[1, 3]")
>> A.inf = 5
A = [Empty]
*:* While at it, also allow multiple subscripts in <code>subsasgn</code>
>> A(:)(2:4)(2) = 42; # equivalent to A(3) = 42
>> A.inf(3) = 42; # also A(3).inf = 42
>> A.inf.inf = 42 # should produce error?
>> A.inf.sup = 42 # should produce error?
* 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):
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,
e.g., whether lower or upper case, and whether Entire becomes [Entire] or [-Inf, Inf].
c) For l and u, it should be possible to specify the field width,
and the number of digits after the point or the number of significant digits.
(partly this is already implemented by output_precision (...) / `format long` / `format short`)
d) It should be possible to output the bounds of an interval without punctuation,
e.g., 1.234 2.345 instead of [1.234, 2.345]. For instance, this might be a
convenient way to write intervals to a file for use by another application.
== 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 intervalspkg load interval A1 = infsup (2, 3); B1 = hull (-4, A1A2); C1 = midrad (0, 2); # Interval package intervals pkg load intervalA1 + 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)
|}
 
== 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.
== Developer Information ==
=== Source Code Repository ===
https://sourceforge.net/p/octave/interval/ci/default/tree/
 
=== Dependencies ===
apt-get install liboctave-dev mercurial make automake libmpfr-dev
=== Build ===
'''Build dependencies'''
* Octave** Version 3.8.0 or greater** No need to compile from source, but you need development files e.g. package <code>liboctaveapt-dev</code> in Debian.* Mercurial* Texinfo* MPFR** Version 3.1.0 or greater** No need to compile from source, but you need development files e.g. package <code>get install libmpfr-dev</code> in Debian.* 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* ITF1788** Purpose: Compilation of unit-test** Installation:**# Clone the git repository from https://github.com/oheim/ITF1788**# Install python 3 and the dependencies described by ITF1788's <code>setup.py</code> file**# Set up an environment variable ITF1788_HOME to point to your local git workspace, for example put the line <code>export IFT1788_HOME=/home/user/ITF1788</code> into your <code>.bashrc</code>.* LilyPond, Inkscape, Poppler** Purpose: Generate / convert images for the manual** Installation: Use your distribution's package manager (look for packages called <code>lilypond</code> <code>autoconf automake inkscapezopfli</code> <code>poppler-utils</code>)
=== Architecture ===
| | `- ... – further functions on decorated intervals
| `- ... – a few global functions that don't operate on intervals
+`- src/ | | +- mpfr_function_d.cc – computes various arithmetic functions correctly rounded (using MPFR) | | `- ... – other oct-file sources `- test/ – interval arithmetic unit tests
=== Best practices ===
* Methods with more than 1 parameter must convert non-interval parameters to intervals using the class constructor. This allows the user to mix non-interval parameters with interval parameters and the function treats any inputs as intervals. Invalid values will be handled by the class constructors.
if (not (isa (x, "infsup")))
x = infsup (x);
endif
if (not (isa (y, "infsup")))
y = infsup (y);
endif
if (not (isa (x, "infsupdec")))
x = infsupdec (x);
endif
if (not (isa (y, "infsupdec")))
y = infsupdec (y); endif* Methods of class <code>infsupdec</code> as well as methods of class <code>infsup</code> that are not overridden by <code>infsupdec</code> must check parameters using the <code>isnai</code> function and return the NAI value if it is present. This will propagate NAI values through any function evaluations. if (isnai (x)) result = x; return
endif
==== Use of Octave functions ====
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 result x = ceil (x) ... parameter checking ... result x.inf = infsup (ceil (x.inf), ; x.sup = ceil (x.sup));
endfunction
 
If Octave functions would introduce arithmetic/rounding errors, there are interfaces to MPFR (<code>mpfr_function_d</code>) and crlibm (<code>crlibm_function</code>), which can produce guaranteed boundaries.
==== Vectorization & Indexing ====
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 result x = plus (x, y) ... 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; result = infsup (l, u);
endfunction
== VERSOFT ==The [http://uivtx.cs.cas.cz/~rohn/matlab/ VERSOFT] software package (by Jiří Rohn) has been released under a free software license (Expat license) and algorithms may be migrated into the interval package. {|! Function! Status! Information|-|colspan="3"|Real (or complex) data only: Matrices|-|verbasis|style="color:red"| trapped| depends on <code style="color:red">verfullcolrank</code>|-|vercondnum|style="color:red"| trapped| depends on <code style="color:red">versingval</code>|-|verdet|style="color:red"| trapped| depends on <code>vereig</code>|-|verdistsing|style="color:red"| trapped| depends on <code style="color:red">versingval</code>|-|verfullcolrank|style="color:red"| trapped| depends on <code>verpinv</code>|-|vernorm2|style="color:red"| trapped| depends on <code style="color:red">versingval</code>|-|vernull (experimental)| unknown| depends on <code style="color:red">verlsq</code>; todo: compare with local function inside <code style="color:green">verintlinineqs</code>|-|verorth|style="color:red"| trapped| depends on <code style="color:red">verbasis</code> and <code style="color:red">verthinsvd</code>|-|verorthproj|style="color:red"| trapped| depends on <code style="color:red">verpinv</code> and <code style="color:red">verfullcolrank</code>|-|verpd|style="color:red"| trapped| depends on <code>isspd</code> (by Rump, to be checked) and <code style="color:red">vereig</code>|-|verpinv|style="color:red"| trapped| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code style="color:red">verthinsvd</code>|-|verpmat|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>|-|verrank|style="color:red"| trapped| depends on <code style="color:red">versingval</code> and <code style="color:red">verbasis</code>|-|verrref|style="color:red"| trapped| depends on <code style="color:red">verfullcolrank</code> and <code style="color:red">verpinv</code>|-|colspan="3"|Real (or complex) data only: Matrices: Eigenvalues and singular values|-|vereig|style="color:red"| trapped| depends on proprietary <code>verifyeig</code> function from INTLAB, depends on complex interval arithmetic|-|<s>vereigback</s>|style="color:green"| free, migrated (for real eigenvalues)| dependency <code>norm</code> is already implemented|-|verspectrad|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3"|Real (or complex) data only: Matrices: Decompositions|-|verpoldec|style="color:red"| trapped| depends on <code style="color:red">verthinsvd</code>|-|verrankdec|style="color:red"| trapped| depends on <code style="color:red">verfullcolrank</code> and <code style="color:red">verpinv</code>|-|verspectdec|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|verthinsvd|style="color:red"| trapped| implemented in <code>vereig</code>|-|colspan="3"|Real (or complex) data only: Matrix functions|-|vermatfun|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3"|Real data only: Linear systems (rectangular) |-|<s>verlinineqnn</s>|style="color:green"| free, migrated| use <code>glpk</code> as a replacement for <code>linprog</code>|-|verlinsys|style="color:red"| trapped| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code style="color:red">verpinv</code>, <code style="color:red">verfullcolrank</code>, and <code style="color:red">verbasis</code>|-|verlsq|style="color:red"| trapped| depends on <code style="color:red">verpinv</code> and <code style="color:red">verfullcolrank</code>|-|colspan="3"|Real data only: Optimization|-|verlcpall|style="color:green"| free| depends on <code>verabsvaleqnall</code>|-|<s>verlinprog</s>|style="color:green"| free, migrated| use <code>glpk</code> as a replacement for <code>linprog</code>; dependency <code>verifylss</code> is implemented as <code>mldivide</code>|-|verlinprogg|style="color:red"| trapped| depends on <code>verfullcolrank</code>|-|verquadprog| unknown| use <code>quadprog</code> from the optim package; use <code>glpk</code> as a replacement for <code>linprog</code>; dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code>isspd</code> (by Rump, to be checked, algorithm in [http://www.ti3.tuhh.de/paper/rump/Ru06c.pdf])|-|colspan="3"|Real (or complex) data only: Polynomials|-|verroots|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3"|Interval (or real) data: Matrices|-|verhurwstab|style="color:red"| trapped| depends on <code style="color:red">verposdef</code>|-|verinverse|style="color:green"| free| depends on <code style="color:green">verintervalhull</code>, to be migrated|-|<s>verinvnonneg</s>|style="color:green"| free, migrated|-|verposdef|style="color:red"| trapped| depends on <code>isspd</code> (by Rump, to be checked) and <code style="color:red">verregsing</code>|-|verregsing|style="color:red"| trapped| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code>isspd</code> (by Rump, to be checked) and <code>verintervalhull</code>; see also [http://uivtx.cs.cas.cz/~rohn/publist/singreg.pdf]|-|colspan="3"|Interval (or real) data: Matrices: Eigenvalues and singular values|-|vereigsym|style="color:red"| trapped| main part implemented in <code>vereig</code>, depends on <code style="color:red">verspectrad</code>|-|vereigval|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>|-|<s>vereigvec</s>|style="color:green"| free, migrated|-|verperrvec|style="color:green"| free| the function is just a wrapper around <code style="color:green">vereigvec</code>?!?|-|versingval|style="color:red"| trapped| depends on <code style="color:red">vereigsym</code>|-|colspan="3"|Interval (or real) data: Matrices: Decompositions|-|verqr (experimental)|style="color:green"| free| <code>qr</code> has already been implemented using the Gram-Schmidt process, which seems to be more accurate and faster than the Cholsky decomposition or Householder reflections used in verqr. No migration needed.|-|<s>verchol (experimental)</s>|style="color:green"| free, migrated| migrated version has been named after the standard Octave function <code>chol</code>|-|colspan="3"|Interval (or real) data: Linear systems (square)|-|verenclinthull|style="color:green"| free| to be migrated|-|verhullparam|style="color:green"| free| depends on <code>verintervalhull</code>, to be migrated|-|verhullpatt|style="color:green"| free| depends on <code>verhullparam</code>, to be migrated|-|verintervalhull|style="color:green"| free| to be migrated|-|colspan="3"|Interval (or real) data: Linear systems (rectangular)|-|verintlinineqs|style="color:green"| free| depends on <code style="color:green">verlinineqnn</code>|-|veroettprag|style="color:green"| free|-|vertolsol|style="color:green"| free| depends on <code style="color:green">verlinineqnn</code>|-|colspan="3"|Interval (or real) data: Matrix equations (rectangular)|-|vermatreqn|style="color:green"| free|-|colspan="3"|Real data only: Uncommon problems|-| plusminusoneset|style="color:green"| free|-| verabsvaleqn|style="color:green"| free| to be migrated|-| verabsvaleqnall|style="color:green"| free| depends on <code>verabsvaleqn</code>, see also [http://uivtx.cs.cas.cz/~rohn/publist/absvaleqnall.pdf], to be migrated|-| verbasintnpprob|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>|-|}  [[Category:Octave-Forge]]

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