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Interval package

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{{OctaveForge| name = interval| logo = [[File:Interval.png|100px]]| short description = Real-valued interval arithmetic.| version = 3.2.0| date = 2018-07-01| author 1 name = Oliver Heimlich| author 1 email = <oheim@posteo.de>| maintainer 1 name = Oliver Heimlich| maintainer 1 email = <oheim@posteo.de>| license = GPL-3.0+| group = Community package| doc 1 = [https://octave.sourceforge.io/interval/overview.html Function reference]| doc 2 = [https://octave.sourceforge.io/interval/package_doc/ User manual]| download 1 = [https://octave.sourceforge.io/download.php?package=interval-3.2.0.tar.gz interval-3.2.0.tar.gz]| repository 1 = https://octave.sourceforge.io/pkg-repository/interval/| dependency 1 = octave &ge; 3.8.0| dependency 2 = '''Runtime:''' mpfr (&ge; 3.1.0) [https://packages.debian.org/search?keywords=libmpfr4 libmpfr4 for Debian]| dependency 3 = '''Build:''' mpfr (&ge; 3.1.0) [https://packages.debian.org/search?keywords=libmpfr-dev libmpfr-dev for Debian]}} The GNU Octave interval package for real-valued [https://en.wikipedia.org/wiki/Interval_arithmetic interval arithmetic].* Intervals are closed, connected subsets of the real numbers. Intervals may be unbound (in either or both directions) or empty. In special cases <code>+inf</code> and <code>-inf</code> 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 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 manage to produce a very accurate such enclosure.* The result of an interval arithmetic function 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].  [[File:Interval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]<div style="clear:left"></div>
== Distribution ==
* [httphttps://octavetracker.sourceforgedebian.netorg/pkg/octave-intervalDebian GNU/Linux], [https://indexlaunchpad.html Latest version at Octave Forge]** <code>pkg install net/ubuntu/+source/octave-forge interval</code>Launchpad Ubuntu]** [httphttps://octaveaur.sourceforgearchlinux.netorg/packages/octave-interval/overview.html function referencearchlinux user repository]** Included in [httphttps://octaveftp.sourceforgegnu.netorg/gnu/intervaloctave/package_docwindows/index.html package documentationofficial Windows installer] and installed automatically with Octave (user manualsince version 4.0.1)* [https://tracgithub.com/macports.org/browsermacports-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 functions -2015], IEEE standard for interval arithmetic, are implemented. The standard was approved by IEEE-SA on June 11, the 2015. It will remain active for ten years. The standard is currently was approved by ANSI in recirculation ballot phase and quite stable2016.** Planned: more solvers*** <code>fsolve</code>Also,*** and possibly <code>roots</code>*** Algorithms can be migrated from the C-XSC Toolbox (C++ code) from minimalistic standard [httphttps://www2standards.mathieee.uni-wuppertal.deorg/wrswtfindstds/xscstandard/cxsc_new1788.1-2017.htmlIEEE Std 1788.1-2017] , IEEE standard for interval arithmetic (see nlinsyssimplified) is fully implemented.cppThe standard was approved by IEEE-SA on December 6, 2017 (and cpzero.cpp respectivelypublished in January 2018).** Planned: Improvement of the user documentation*** Explain In addition there are functions for interval matrix arithmetic concepts with examples from the package, 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 need more testingfr/www/crlibm/ CRlibm] library is used.
* Portability
** Runs in GNU Octave 3.8.2 and 4.0 release candidates** Known to run under GNU/Linux, Microsoft Windows, Mac OS X macOS, and FreeBSD == 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. >> A = infsup ("[2, 4]"); >> A.inf = infsup ("[1, 3]") A = [1, 4] >> 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 ===In 2008/2009 there was a Single Interval Mathematics Package (SIMP) for Octave, which has eventually become unmaintained at Octave Forge. The simp package contains a few basic interval arithmetic operations on scalar or vector intervals. It does not consider inaccurate built-in arithmetic functions, round-off, conversion and representational errors. As a result its syntax is very easy, but the arithmetic fails to produce guaranteed enclosures. It is recommended to use the interval package as a replacement for simp. However, function names and interval constructors are not compatible between the packages.
=== INTLAB ===
This interval package is ''not'' meant to be a replacement for INTLAB and any compatibility with it is pure coincidence. Since both are compatible with GNU Octave, they happen to agree on many function names and programs written for INTLAB may possibly run with this interval package as well. Some fundamental differences that I am currently aware of:
* INTLAB is non-free software, it grants none of the [http://www.gnu.org/philosophy/free-sw.html four essential freedoms] of free software
* INTLAB is not conforming to IEEE Std 1788 -2015 and the parsing of intervals from strings uses a different format—especially for the uncertain form
* INTLAB supports intervals with complex numbers and sparse interval matrices, but no empty intervals
* INTLAB uses inferior accuracy for most arithmetic operations, because it focuses on speed
* 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 ====
Simple programs written for INTLAB should run without modification with this interval package. The following table lists common functions that use a different name in INTLAB.
{|
! interval package
! INTLAB
|-
| infsup (x)
| intval (x)
|-
| wid (x)
| diam (x)
|-
| subset (a, b)
| in (a, b)
|-
| interior (a, b)
| in0 (a, b)
|-
| isempty (x)
| isnan (x)
|-
| disjoint (a, b)
| emptyintersect (a, b)
|-
| hdist (a, b)
| qdist (a, b)
|-
| disp (x)
| disp2str (x)
|-
| infsup (s)
| str2intval (s)
|-
| isa (x, "infsup")
| isintval (x)
|}
 
== 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 ===
The repository contains a Makefile which controls the build process. Some common targets are:
* <code>make release</code> Create a release tarball and the HTML documentation for [[Octave Forge]] (takes a while).
* <code>make check</code> Run the full test-suite to verify that code changes didn't break anything (takes a while).
* <code>make run</code> Quickly start Octave with minimal recompilation and functions loaded from the workspace (for interactive testing of code changes).
 
'''Build dependencies'''
<code>apt-get install libmpfr-dev autoconf automake inkscape zopfli</code>
 
=== Architecture ===
 
In a nutshell the package provides two new data types to users: bare intervals and decorated intervals. The data types are implemented as:
* class <code>infsup</code> (bare interval) with attributes <code>inf</code> (lower interval boundary) and <code>sup</code> (upper interval boundary)
* class <code>infsupdec</code> (decorated interval) which extends the former and adds attribute <code>dec</code> (interval decoration).
 
Almost all functions in the package are implemented as methods of these classes, e. g. <code>@infsup/sin</code> implements the sine function for bare intervals. Most code is kept in m-files. Arithmetic operations that require correctly-rounded results are implemented in oct-files (C++ code), these are used internally by the m-files of the package. The source code is organized as follows:
 
+- doc/ – package manual
+- inst/
| +- @infsup/
| | +- infsup.m – class constructor for bare intervals
| | +- sin.m – sine function for bare intervals (uses mpfr_function_d internally)
| | `- ... – further functions on bare intervals
| +- @infsupdec/
| | +- infsupdec.m – class constructor for decorated intervals
| | +- sin.m – sine function for decorated intervals (uses @infsup/sin internally)
| | `- ... – 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
 
=== Best practices ===
==== Parameter checking ====
* All methods must check <code>nargin</code> and call <code>print_usage</code> if the number of parameters is wrong. This prevents simple errors by the user.
* 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 = Similar software infsupdec (y); 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 x = ceil (x) ... parameter checking ... x.inf = 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 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; endfunction
For C++ there is an open source interval library == VERSOFT ==The [httpshttp://githubuivtx.cs.cas.comcz/nehmeier~rohn/libieeep1788matlab/ libieeep1788VERSOFT] software package (by Marco Nehmeier Jiří Rohn) has been released under a free software license (member of IEEE P1788Expat license). It aims to and algorithms may be standard compliant with IEEE 1788 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 migrated into the interval package shares several unit tests with libieeep1788.
For C++{|! 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, Pascal 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 Fortran there 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 interval library | 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://www2www.mathti3.unituhh.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|-wuppertal|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.decas.cz/~xscrohn/publist/xsc-sprachensingreg.html XSCpdf]|-|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. It is not standard compliant with IEEE 1788No migration needed. Some parts of |-|<s>verchol (experimental)</s>|style="color:green"| free, migrated| migrated version has been named after the GNU standard Octave interval package have been derived from Cfunction <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|-XSC| 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>|-|}
For MATLAB there is a proprietary interval arithmetic toolbox [http://www.ti3.tu-harburg.de/rump/intlab/ INTLAB] by Siegfried Rump. It had been free of charge for academic use in the past, but no longer is. Its origin dates back to 1999, so it is well tested and comprises a lot of functionality, especially for vector / matrix operations. INTLAB is compatible with GNU Octave since Version 9 [http://www.ti3.tuhh.de/rump/intlab/Octave/octave.html].
[[Category:Octave-Forge]]

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