# Changes

,  04:15, 10 June 2019
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Remove redundant Category:Packages.
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The [https://sourceforge.net/p/octave/interval/ interval package] provides data types and fundamental operations for real valued interval arithmetic based on the common floating-point format “binary64” a. k. a. double-precision.  It aims to be standard compliant with the (upcoming) [http://standards.ieee.org/develop/project/1788.html IEEE 1788] and therefore implements the ''set-based'' interval arithmetic flavor. '''Interval arithmetic''' produces mathematically proven numerical results.
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The GNU Octave interval package for real-valued [https://en.wikipedia.org/wiki/Interval_arithmetic interval arithmetic].
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* 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.
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* 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.
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* 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].
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== Motivation ==
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__TOC__
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[[File:Interval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]
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<div style="clear:left"></div>
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{{quote|Give a digital computer a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. … An interval computation yields a pair of numbers, an upper and a lower bound, which are guaranteed to enclose the exact answer. Maybe you still don’t know the truth, but at least you know how much you don’t know.|Brian Hayes|[http://dx.doi.org/10.1511/2003.6.484 DOI: 10.1511/2003.6.484]}}
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== Distribution ==
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** <code>pkg install -forge interval</code>
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** [https://octave.sourceforge.io/interval/overview.html function reference]
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** [https://octave.sourceforge.io/interval/package_doc/index.html package documentation] (user manual)
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'''Third-party'''
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* [https://aur.archlinux.org/packages/octave-interval/ archlinux user repository]
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* Included in [https://ftp.gnu.org/gnu/octave/windows/ official Windows installer] and installed automatically with Octave (since version 4.0.1)
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* [https://github.com/macports/macports-ports/tree/master/math/octave-interval/ MacPorts] for Mac OS X
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* [https://cygwin.com/cgi-bin2/package-grep.cgi?grep=octave-interval Cygwin] for Windows
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* [https://build.opensuse.org/package/show/science/octave-forge-interval openSUSE build service]
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{| class="wikitable" style="margin: auto"
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== Development status ==
!Standard floating point arithmetic
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* Completeness
!Interval arithmetic
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** 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.
|-
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** 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).
| style = "vertical-align: top" |
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** In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.
octave:1> 19 * 0.1 - 2 + 0.1
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* Quality
ans =  1.3878e-16
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** Most arithmetic operations produce tight, correctly-rounded results. That is, the smallest possible interval with double-precision (binary64) endpoints, which encloses the exact result.
| style = "vertical-align: top" |
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** Includes [https://github.com/oheim/ITF1788 large test suite] for arithmetic functions
octave:1> x = infsup ("0.1");
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** 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].
octave:2> 19 * x - 2 + x
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* Performance
ans ⊂ [-3.1918911957973251e-16, +1.3877787807814457e-16]
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** All elementary functions have been [https://octave.org/doc/interpreter/Vectorization-and-Faster-Code-Execution.html vectorized] and run fast on large input data.
|}
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** Arithmetic is performed with the [http://www.mpfr.org/ GNU MPFR] library internally. Where possible, the optimized [http://web.archive.org/web/20170128033523/http://lipforge.ens-lyon.fr/www/crlibm/ CRlibm] library is used.
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* Portability
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** Runs in GNU Octave ≥ 3.8.2
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** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD
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== Project ideas (TODOs) ==
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* 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.
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* Interval version of <code>interp1</code>
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* Extend <code>subsasgn</code> to allow direct manipulation of inf and sup (and dec) properties.
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>> A = infsup ("[2, 4]");
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>> A.inf = infsup ("[1, 3]")
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A = [1, 4]
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>> A.inf = 5
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A = [Empty]
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:* While at it, also allow multiple subscripts in <code>subsasgn</code>
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>> A(:)(2:4)(2) = 42; # equivalent to A(3) = 42
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>> A.inf(3) = 42; # also  A(3).inf = 42
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>> A.inf.inf = 42 # should produce error?
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>> A.inf.sup = 42 # should produce error?
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* 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]
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* 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]
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* Implement user-controllable output from the interval standard (e. g. via printf functions):
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a) It should be possible to specify the preferred overall field width (the length of s).
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b) It should be possible to specify how Empty, Entire and NaI are output,
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e.g., whether lower or upper case, and whether Entire becomes [Entire] or [-Inf, Inf].
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c) For l and u, it should be possible to specify the field width,
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and the number of digits after the point or the number of significant digits.
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(partly this is already implemented by output_precision (...) / `format long` / `format short`)
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d) It should be possible to output the bounds of an interval without punctuation,
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e.g., 1.234 2.345 instead of [1.234, 2.345]. For instance, this might be a
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convenient way to write intervals to a file for use by another application.
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Floating-point arithmetic, as specified by [http://en.wikipedia.org/wiki/IEEE_floating_point IEEE 754], is available in almost every computer system today. It is wide-spread, implemented in common hardware and integral part in programming languages. For example, the double-precision format is the default numeric data type in GNU Octave. Benefits are obvious: The results of arithmetic operations are well-defined and comparable between different systems and computation is highly efficient.
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== Compatibility ==
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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:
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However, there are some downsides of floating-point arithmetic in practice, which will eventually produce errors in computations.
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* [https://github.com/JuliaIntervals/IntervalArithmetic.jl IntervalArithmetic.jl package] (Julia)
* Floating-point arithmetic is often used mindlessly by developers. [http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html] [http://www.cs.berkeley.edu/~wkahan/Mindless.pdf]
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* [https://github.com/jinterval/jinterval JInterval library] (Java)
* The binary data types categorically are not suitable for doing financial computations. Very often representational errors are introduced when using “real world” decimal numbers. [http://en.wikipedia.org/wiki/Decimal_computer]
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* [https://github.com/nadezhin/libieeep1788 ieeep1788 library] (C++) created by Marco Nehmeier, later forked by Dmitry Nadezhin
* Even if the developer would be proficient, most developing environments / technologies limit floating-point arithmetic capabilities to a very limited subset of IEEE 754: Only one or two data types, no rounding modes, missing functions … [http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf]
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* Results are hardly predictable. [https://hal.archives-ouvertes.fr/hal-00128124/en/] All operations produce the best possible accuracy ''at runtime'', this is how a floating point works. Contrariwise, financial computer systems typically use a [http://en.wikipedia.org/wiki/Fixed-point_arithmetic fixed-point arithmetic] (COBOL, PL/I, …), where overflow and rounding can be precisely predicted ''at compile-time''.
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* Results are system dependent. All but the most basic floating-point operations are not guaranteed to be accurate and produce different results depending on low level libraries and hardware. [http://www.gnu.org/software/libc/manual/html_node/Errors-in-Math-Functions.html#Errors-in-Math-Functions] [http://developer.amd.com/tools-and-sdks/cpu-development/libm/]
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* If you do not know the technical details (cf. first bullet) you ignore the fact that the computer lies to you in many situations. For example, when looking at numerical output and the computer says “<code>ans = 0.1</code>,” this is not absolutely correct. In fact, the value is only ''close enough'' to the value 0.1. Additionally, many functions produce limit values (∞ × −∞ = −∞, ∞ ÷ 0 = ∞, ∞ ÷ −0 = −∞, log (0) = −∞), which is sometimes (but not always!) useful when overflow and underflow occur.
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Interval arithmetic addresses above problems in its very special way and introduces new possibilities for algorithms. For example, the [http://en.wikipedia.org/wiki/Interval_arithmetic#Interval_Newton_method interval newton method] is able to find ''all'' zeros of a particular function.
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=== Octave Forge simp package ===
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In 2008/2009 there was a Single Interval Mathematics Package (SIMP) for Octave, which has eventually become unmaintained at Octave Forge.
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== Theory ==
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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.
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=== Moore's fundamental theroem of interval arithmetic ===
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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.
Let '''''y''''' = ''f''('''''x''''') be the result of
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interval-evaluation of ''f'' over a box '''''x''''' = (''x''<sub>1</sub>, … , ''x''<sub>''n''</sub>)
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using any interval versions of its component library functions. Then
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# In all cases, '''''y''''' contains the range of ''f'' over '''''x''''', that is, the set of ''f''('''''x''''') at points of '''''x''''' where it is defined: '''''y''''' ⊇ Rge(''f'' | '''''x''''') = {''f''(''x'') | ''x'' ∈ '''''x''''' ∩ Dom(''f'') }
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# If also each library operation in ''f'' is everywhere defined on its inputs, while evaluating '''''y''''', then ''f'' is everywhere defined on '''''x''''', that is Dom(''f'') ⊇ '''''x'''''.
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# If in addition, each library operation in ''f'' is everywhere continuous on its inputs, while evaluating '''''y''''', then ''f'' is everywhere continuous on '''''x'''''.
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# If some library operation in ''f'' is nowhere defined on its inputs, while evaluating '''''y''''', then ''f'' is nowhere defined on '''''x''''', that is Dom(''f'') ∩ '''''x''''' = Ø.
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=== Online Introductions ===
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=== INTLAB ===
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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:
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* INTLAB is non-free software, it grants none of the [http://www.gnu.org/philosophy/free-sw.html four essential freedoms] of free software
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* 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
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* INTLAB supports intervals with complex numbers and sparse interval matrices, but no empty intervals
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* INTLAB uses inferior accuracy for most arithmetic operations, because it focuses on speed
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* Basic operations can be found in both packages, but the availability of special functions depends
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[http://www.maths.manchester.ac.uk/~higham/narep/narep416.pdf  Interval analysis in MATLAB] Note: The INTLAB toolbox for Matlab is not compatible with the interval package for GNU Octave. For example, INTLAB additionally supports intervals in mid-rad form with complex values, INTLAB has no representation of an empty interval and INTLAB provides several other features that this interval package is missing. However, basic operations can be compared and should be compatible for common intervals.
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== What to expect ==
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<div style="display:flex; align-items: flex-start">
The interval arithmetic provided by this interval package is '''slow''', but '''accurate'''.
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<div style="margin-right: 2em">
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{{Code|Computation with this interval package|<syntaxhighlight lang="octave">
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A1 = infsup (2, 3);
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B1 = hull (-4, A2);
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''Why is the interval package slow?''
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A1 + B1 * C1
All arithmetic interval operations are simulated in high-level octave language using C99 or multi-precision floating-point routines, which is a lot slower than a hardware implementation [https://books.google.de/books?id=JTc4XdXFnQIC&pg=PA61]. Building interval arithmetic operations from floating-point routines is easy for simple monotonic functions, e. g., addition and subtraction, but is complex for others, e. g., [http://exp.ln0.de/heimlich-power-2011.htm interval power function], atan2, or [[#Reverse_arithmetic_operations|reverse functions]]. For some interval operations it is not even possible to rely on floating-point routines, since not all required routines are available in C99 or BLAS.
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</syntaxhighlight>
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}}
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</div><div>
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{{Code|Computation with INTLAB|<syntaxhighlight lang="octave">
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startintlab
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A2 = infsup (2, 3);
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B2 = hull (-4, A2);
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Great algorithms and optimizations exist for floating-point matrix arithmetic in GNU Octave. Good interval versions of these still have to be found and implemented. For example, multiplication of matrices with many elements becomes unfeasible as it takes a lot of time. (For that purpose one could use the [[#Related work|INTLAB toolbox]], which uses a less accurate arithmetic)
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A2 + B2 * C2
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</syntaxhighlight>
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}}
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</div>
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</div>
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{| class="wikitable" style="margin: auto; align:right"
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==== Known differences ====
|+ Approximate runtime (wall clock time in seconds) on an<br/>Intel® Core™ i5-4340M CPU (2.9 GHz)
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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.
!Matrix dimension
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{|
!Interval <code>plus</code>
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! interval package
!Interval <code>log</code>
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! INTLAB
!Interval <code>mtimes</code>
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|-
!Interval <code>inv</code>
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| infsup (x)
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| intval (x)
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|-
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| wid (x)
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| diam (x)
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|-
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| subset (a, b)
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| in (a, b)
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|-
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| interior (a, b)
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| in0 (a, b)
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|-
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| isempty (x)
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| isnan (x)

|-

|-
| 10
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| disjoint (a, b)
|align="right"| < 0.1
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| emptyintersect (a, b)
|align="right"| < 0.1
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|align="right"| 0.4
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|align="right"| 3.3

|-

|-
| 20
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| hdist (a, b)
|align="right"| < 0.1
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| qdist (a, b)
|align="right"| < 0.1
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|align="right"| 2.3
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|style="text-align:right; font-weight: bold; color:orange"| 32.3

|-

|-
| 30
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| disp (x)
|align="right"| < 0.1
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| disp2str (x)
|align="right"| < 0.1
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|align="right"| 7.0
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|style="text-align:right; font-weight: bold; color:red"| 142.1

|-

|-
| 40
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| infsup (s)
|align="right"| < 0.1
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| str2intval (s)
|align="right"| 0.1
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|align="right"| 15.6
−
|

|-

|-
| 50
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| isa (x, "infsup")
|align="right"| 0.1
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| isintval (x)
|align="right"| 0.2
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|style="text-align:right; font-weight: bold; color:orange"| 29.1
−
|

|}

|}
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''Why is the interval package accurate?''
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== Developer Information ==
The GNU Octave built-in floating-point routines are not useful for interval arithmetic: Their results depend on hardware, system libraries and compilation options.
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=== Source Code Repository ===
The interval package handles all arithmetic functions with the help of the [http://www.mpfr.org/ GNU MPFR library]. With MPFR it is possible to compute system-independent, valid and tight enclosures of the correct results.
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https://sourceforge.net/p/octave/interval/ci/default/tree/
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== Quick start introduction ==
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=== Dependencies ===
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apt-get install liboctave-dev mercurial make automake libmpfr-dev
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=== Input and output ===
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=== Build ===
Before exercising interval arithmetic, interval objects must be created from non-interval data. There are interval constants <code>empty</code> and <code>entire</code> and the class constructors <code>infsup</code> for bare intervals and <code>infsupdec</code> for decorated intervals. The class constructors are very sophisticated and can be used with several kinds of parameters: Interval boundaries can be given by numeric values or string values with decimal numbers. Also it is possible to use so called interval literals with square brackets.
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The repository contains a Makefile which controls the build process. Some common targets are:
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* <code>make release</code> Create a release tarball and the HTML documentation for [[Octave Forge]] (takes a while).
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* <code>make check</code> Run the full test-suite to verify that code changes didn't break anything (takes a while).
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* <code>make run</code> Quickly start Octave with minimal recompilation and functions loaded from the workspace (for interactive testing of code changes).
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octave:1> infsup (1)
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'''Build dependencies'''
ans = 
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<code>apt-get install libmpfr-dev autoconf automake inkscape zopfli</code>
octave:2> infsup (1, 2)
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ans = [1, 2]
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octave:3> infsup ("3", "4")
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ans = [3, 4]
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octave:4> infsup ("1.1")
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ans ⊂ [1.0999999999999998, 1.1000000000000001]
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octave:5> infsup ("[5, 6.5]")
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ans = [5, 6.5]
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octave:6> infsup ("[5.8e-17]")
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ans ⊂ [5.799999999999999e-17, 5.800000000000001e-17]
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It is possible to access the exact numeric interval boundaries with the functions <code>inf</code> and <code>sup</code>. The shown text representation of intervals can be created with <code>intervaltotext</code>. The default text representation is not guaranteed to be exact (see function <code>intervaltoexact</code> for that purpose), because this would massively spam console output. For example, the exact text representation of <code>realmin</code> would be over 700 decimal places long! However, the default text representation is correct as it guarantees to contain the actual boundaries and is accurate enough to separate different boundaries.
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=== Architecture ===
−
octave:7> infsup (1, 1 + eps)
+
In a nutshell the package provides two new data types to users: bare intervals and decorated intervals. The data types are implemented as:
ans ⊂ [1, 1.0000000000000003]
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* class <code>infsup</code> (bare interval) with attributes <code>inf</code> (lower interval boundary) and <code>sup</code> (upper interval boundary)
octave:8> infsup (1, 1 + 2 * eps)
+
* class <code>infsupdec</code> (decorated interval) which extends the former and adds attribute <code>dec</code> (interval decoration).
ans ⊂ [1, 1.0000000000000005]
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Warning: Decimal fractions as well as numbers of high magnitude (> 2<sup>53</sup>) should always be passed as a string to the constructor. Otherwise it is possible, that GNU Octave introduces conversion errors when the numeric literal is converted into floating-point format '''before''' it is passed to the constructor.
+
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:
−
octave:9> infsup (<span style = "color:red">0.2</span>)
+
+- doc/                        – package manual
ans ⊂ [.20000000000000001, .20000000000000002]
+
+- inst/
octave:10> infsup (<span style = "color:green">"0.2"</span>)
+
|  +- @infsup/
ans ⊂ [.19999999999999998, .20000000000000002]
+
|  |  +- infsup.m            – class constructor for bare intervals
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|  |  +- sin.m              – sine function for bare intervals (uses mpfr_function_d internally)
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|  |  `- ...                – further functions on bare intervals
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|  +- @infsupdec/
+
|  |  +- infsupdec.m        – class constructor for decorated intervals
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|  |  +- sin.m              – sine function for decorated intervals (uses @infsup/sin internally)
+
|  |  `- ...                – further functions on decorated intervals
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|  `- ...                     – a few global functions that don't operate on intervals
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`- src/
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+- mpfr_function_d.cc      – computes various arithmetic functions correctly rounded (using MPFR)
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`- ...                     – other oct-file sources
−
For convenience it is possible to implicitly call the interval constructor during all interval operations if at least one input already is an interval object.
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=== Best practices ===
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==== Parameter checking ====
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* 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.
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* 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.
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if (not (isa (x, "infsup")))
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x = infsup (x);
+
endif
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if (not (isa (y, "infsup")))
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y = infsup (y);
+
endif
−
octave:11> infsup ("17.7") + 1
+
if (not (isa (x, "infsupdec")))
ans ⊂ [18.699999999999999, 18.700000000000003]
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x = infsupdec (x);
octave:12> ans + "[0, 2]"
+
endif
ans ⊂ [18.699999999999999, 20.700000000000003]
+
if (not (isa (y, "infsupdec")))
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y = infsupdec (y);
+
endif
−
==== Specialized interval constructors ====
+
==== Use of Octave functions ====
Above mentioned interval construction with decimal numbers or numeric data is straightforward. Beyond that, there are more ways to define intervals or interval boundaries.
+
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.
* Hexadecimal-floating-constant form: Each interval boundary may be defined by a hexadecimal number (optionally containing a point) and an exponent field with an integral power of two as defined by the C99 standard ([http://www.open-std.org/jtc1/sc22/WG14/www/docs/n1256.pdf ISO/IEC9899, N1256, §6.4.4.2]). This can be used as a convenient way to define interval boundaries in double-precision, because the hexadecimal form is much shorter than the decimal representation of many numbers.
+
function x = ceil (x)
* Rational literals: Each interval boundary may be defined as a fraction of two decimal numbers. This is especially useful if interval boundaries shall be tightest enclosures of fractions, that would be hard to write down as a decimal number.
+
... parameter checking ...
* Uncertain form: The interval as a whole can be defined by a midpoint or upper/lower boundary and an integral number of [http://en.wikipedia.org/wiki/Unit_in_the_last_place “units in last place” (ULPs)] as an uncertainty. The format is <code>''m''?''ruE''</code>, where
+
x.inf = ceil (x.inf);
** <code>''m ''</code> is a mantissa in decimal,
+
x.sup = ceil (x.sup);
** <code>''r ''</code> is either empty (which means ½ ULP) or is a non-negative decimal integral ULP count or is the <code>?</code> character (for unbounded intervals),
+
endfunction
** <code>''u ''</code> is either empty (symmetrical uncertainty of ''r'' ULPs in both directions) or is either <code>u</code> (up) or <code>d</code> (down),
−
** <code>''E ''</code> is either empty or an exponent field comprising the character <code>e</code> followed by a decimal integer exponent (base 10).
−
octave:13> infsup ("0x1.999999999999Ap-4")
+
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.
ans ⊂ [.1, .10000000000000001]
−
octave:14> infsup ("1/3", "7/9")
−
ans ⊂ [.33333333333333331, .7777777777777778]
−
octave:15> infsup ("121.2?")
−
ans ⊂ [121.14999999999999, 121.25]
−
octave:16> infsup ("5?32e2")
−
ans = [-2700, +3700]
−
octave:17> infsup ("-42??u")
−
ans = [-42, +Inf]
−
==== Interval vectors and matrices ====
+
==== Vectorization & Indexing ====
Vectors and matrices of intervals can be created by passing numerical matrices, char vectors or cell arrays to the <code>infsup</code> constructor. With cell arrays it is also possible to mix several types of boundaries.
+
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.
octave:18> M = infsup (magic (3))
+
function x = plus (x, y)
M = 3×3 interval matrix
+
... parameter checking ...
+
l = mpfr_function_d ('plus', -inf, x.inf, y.inf);
    
+
u = mpfr_function_d ('plus', +inf, x.sup, y.sup);
    
−
    
−
−
octave:19> infsup (magic (3), magic (3) + 1)
−
ans = 3×3 interval matrix
−
−
[8, 9]    [1, 2]  [6, 7]
−
[3, 4]   [5, 6]  [7, 8]
−
[4, 5]  [9, 10]  [2, 3]
−
−
octave:20> infsup (["0.1"; "0.2"; "0.3"; "0.4"; "0.5"])
−
ans ⊂ 5×1 interval vector
−
−
[.09999999999999999, .10000000000000001]
−
[.19999999999999998, .20000000000000002]
−
[.29999999999999998, .30000000000000005]
−
[.39999999999999996, .40000000000000003]
−
[.5]
−
−
octave:21> infsup ({1, eps; "4/7", "pi"}, {2, 1; "e", "0xff"})
−
ans ⊂ 2×2 interval matrix

[1, 2]  [2.220446049250313e-16, 1]
+
emptyresult = isempty (x) | isempty (y);
[.5714285714285713, 2.7182818284590456]   [3.1415926535897931, 255]
+
l(emptyresult) = inf;
+
u(emptyresult) = -inf;
+

+
endfunction
−
When matrices are resized using subscripted assignment, any implicit new matrix elements will carry an empty interval.
+
== VERSOFT ==
octave:22> M (4, 4) = 42
+
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.
M = 4×4 interval matrix
−
−
              [Empty]
−
              [Empty]
−
              [Empty]
−
[Empty]  [Empty]  [Empty]      
−
Note: Whilst most functions (<code>size</code>, <code>isvector</code>, <code>ismatrix</code>, ) work as expected on interval data types, the function <code>'''isempty'''</code> is evaluated element-wise and checks if an interval equals the empty set.
+
{|
octave:23> builtin ("isempty", empty ()), isempty (empty ())
+
! Function
ans = 0
+
! Status
ans = 1
+
! Information

+
|-
=== Decorations ===
+
|colspan="3"|Real (or complex) data only: Matrices
With the subclass <code>infsupdec</code> it is possible to extend interval arithmetic with a decoration system. Every interval and intermediate result will additionally carry a decoration, which may provide additional information about the final result. The following decorations are available:
+
|-

+
|verbasis
{| class="wikitable" style="margin: auto"
+
|style="color:red"| trapped
!Decoration
+
| depends on <code style="color:red">verfullcolrank</code>
!Bounded
+
|-
!Continuous
+
|vercondnum
!Defined
+
|style="color:red"| trapped
!Definition
+
| 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
+
|-
+
+
|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>
+
|-
+
+
| 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

|-

|-
| com<br/>(common)
+
|colspan="3"|Real data only: Uncommon problems
| style="text-align: center" | ✓
−
| style="text-align: center" |
−
| style="text-align: center" | ✓
−
| '''''x''''' is a bounded, nonempty subset of Dom(''f''); ''f'' is continuous at each point of '''''x'''''; and the computed interval ''f''('''''x''''') is bounded

|-

|-
| dac<br/>(defined &amp; continuous)
+
| plusminusoneset
|
+
|style="color:green"| free
| style="text-align: center" | ✓
−
| style="text-align: center" |
−
| '''''x''''' is a nonempty subset of Dom(''f''); and the restriction of ''f'' to '''''x''''' is continuous

|-

|-
| def<br/>(defined)
+
| verabsvaleqn
|
+
|style="color:green"| free
|
+
| to be migrated
| style="text-align: center" |
−
| '''''x''''' is a nonempty subset of Dom(''f'')

|-

|-
| trv<br/>(trivial)
+
| verabsvaleqnall
|
+
|style="color:green"| free
|
+
|
−
| always true (so gives no information)

|-

|-
| ill<br/>(ill-formed)
+
| verbasintnpprob
|
+
|style="color:red"| trapped
|
+
| depends on <code style="color:red">verregsing</code>
|
−
| Not an interval, at least one interval constructor failed during the course of computation
−
|}
−

−
In the following example, all decoration information is lost when the interval is possibly divided by zero, i. e., the overall function is not guaranteed to be defined for all possible inputs.
−

−
octave:1> infsupdec (3, 4)
−
ans = [3, 4]_com
−
octave:2> ans + 12
−
ans = [15, 16]_com
−
octave:3> ans / "[0, 2]"
−
ans = [7.5, Inf]_trv
−

−
=== Arithmetic operations ===
−
The interval packages comprises many interval arithmetic operations. Function names match GNU Octave standard functions where applicable, and follow recommendations by IEEE 1788 otherwise. It is possible to look up all functions by their corresponding IEEE 1788 name in the [http://sourceforge.net/p/octave/interval/ci/default/tree/INDEX index].
−

−
Arithmetic functions in a set-based interval arithmetic follow these rules: Intervals are sets. They are subsets of the set of real numbers. The interval version of an elementary function such as sin(''x'') is essentially the natural extension to sets of the corresponding point-wise function on real numbers. That is, the function is evaluated for each number in the interval where the function is defined and the result must be an enclosure of all possible values that may occur.
−

−
One operation that should be noted is the <code>fma</code> function (fused multiply and add). It computes '''''x''''' × '''''y''''' + '''''z''''' in a single step and is more accurate (because of no intermediate result) and therefore preferred in some situations.
−

−
octave:1> sin (infsup (0.5))
−
ans ⊂ [.47942553860420294, .47942553860420301]
−
octave:2> pow (infsup (2), infsup (3, 4))
−
ans = [8, 16]
−
octave:3> atan2 (infsup (1), infsup (1))
−
ans ⊂ [.7853981633974482, .7853981633974484]
−

−
=== Reverse arithmetic operations ===
−
[[File:Reverse-power-functions.png|400px|thumb|right|Reverse power operations. A relevant subset of the function's domain is outlined and hatched. In this example we use ''x''<sup>''y''</sup> ∈ [2, 3].]]
−

−
Some arithmetic functions also provide reverse mode operations. That is inverse functions with interval constraints. For example the <code>sqrrev</code> can compute the inverse of the <code>sqr</code> function on intervals. The syntax is <code>sqrrev (C, X)</code> and will compute the enclosure of all numbers ''x'' ∈ X that fulfill the constraint ''x''² ∈ C.
−

−
In the following example, we compute the constraints for base and exponent of the power function <code>pow</code> as shown in the figure.
−
octave:1> x = powrev1 (infsup ("[1.1, 1.45]"), infsup (2, 3))
−
x ⊂ [1.6128979635153646, 2.7148547265657915]
−
octave:2> y = powrev2 (infsup ("[2.14, 2.5]"), infsup (2, 3))
−
y ⊂ [.7564707973660299, 1.4440113978403284]
−

−
=== Numerical operations ===
−
Some operations on intervals do not return an interval enclosure, but a single number (in double-precision). Most important are <code>inf</code> and <code>sup</code>, which return the lower and upper interval boundaries.
−

−
More such operations are <code>mid</code> (approximation of the interval's midpoint), <code>wid</code> (approximation of the interval's width), <code>rad</code> (approximation of the interval's radius), <code>mag</code> and <code>mig</code>.
−

−
=== Boolean operations ===
−
Interval comparison operations produce boolean results. While some comparisons are especially for intervals (subset, interior, ismember, isempty, disjoint, …) others are extensions of simple numerical comparison. For example, the less-or-equal comparison is mathematically defined as ∀<sub>''a''</sub> ∃<sub>''b''</sub> ''a'' ≤ ''b'' ∧ ∀<sub>''b''</sub> ∃<sub>''a''</sub> ''a'' ≤ ''b''.
−

−
octave:1> infsup (1, 3) <= infsup (2, 4)
−
ans =  1
−

−
=== Matrix operations ===
−
Above mentioned operations can also be applied element-wise to interval vectors and matrices. Many operations use [http://www.gnu.org/software/octave/doc/interpreter/Vectorization-and-Faster-Code-Execution.html#Vectorization-and-Faster-Code-Execution vectorization techniques].
−

−
In addition, there are matrix operations on interval matrices. These operations comprise: tight dot product, tight matrix multiplication, tight vector sums, matrix inversion, matrix powers, and solving linear systems. As a result of missing hardware / low-level library support and missing optimizations, these operations are quite slow compared to familiar operations in floating-point arithmetic.
−

−
octave:1> A = infsup ([1, 2, 3; 4, 0, 0; 0, 0, 1]); A (2, 3) = "[0, 6]"
−
A = 3×3 interval matrix
−
−
        
−
    [0, 6]
−
        
−
−
octave:2> B = inv (A)
−
B = 3×3 interval matrix
−
−
    [.25]      [-1.5, 0]
−
[.5]  [-.125]  [-1.5, -.75]
−
                  
−
−
octave:3> A * B
−
ans = 3×3 interval matrix
−
−
    [-1.5, +1.5]
−
        [-6, +6]
−
              
−
−

−
octave:4> A = infsup (magic (3))
−
A = 3×3 interval matrix
−
−
    
−
    
−
    
−
−
octave:5> c = A \ [3; 4; 5]
−
c ⊂ 3×1 interval vector
−
−
[.18333333333333323, .18333333333333344]
−
[.43333333333333318, .43333333333333346]
−
[.18333333333333323, .18333333333333344]
−
−
octave:6> A * c
−
ans ⊂ 3×1 interval vector
−
−
[2.9999999999999982, 3.0000000000000018]
−
[3.9999999999999982, 4.0000000000000018]
−
[4.9999999999999973, 5.0000000000000018]
−

−
==== Notes on linear systems ====
−
A system of linear equations in the form A''x'' = b with intervals can be seen as a range of ''classical'' linear systems, which can be solved simultaneously. Whereas classical algorithms compute an approximation for a single solution of a single linear system, interval algorithms compute an enclosure for all possible solutions of (possibly several) linear systems. Some characteristics should definitely be known when linear interval systems are solved:
−
* If the linear system is underdetermined and has infinitely many solutions, the interval solution will be unbound in at least one of its coordinates. Contrariwise, from an unbound result it can not be concluded whether the linear system is underdetermined or has solutions.
−
* If the interval result is empty in at least one of its coordinates, the linear system is guaranteed to be underdetermined and has no solutions. Contrariwise, from a non-empty result it can not be concluded whether all or some of the systems have solutions or not.
−
* Wide intervals within the matrix A can easily lead to a superposition of cases, where the rank of A is no longer unique. If the linear interval system contains cases of linear independent equations as well as linear dependent equations, the resulting enclosure of solutions will inevitably be very broad.
−
* Due to the [http://en.wikipedia.org/wiki/Interval_arithmetic#Dependency_problem dependency problem in interval arithmetic], it may happen that the current solving algorithm produces poor results for some inputs.
−

−
However, solving linear systems with interval arithmetic can produce useful results in many cases and automatically carries a guaranty for error boundaries. Additionally, it can give better information than the floating-point variants for some cases.
−

−
{| class="wikitable" style="margin: auto"
−
!Standard floating point arithmetic
−
!Interval arithmetic

|-

|-
| style = "vertical-align: top" |
−
octave:1> A = [1, 0; 2, 0];
−
octave:2> A \ [3; 0]    # no solution
−
warning: matrix singular to machine precision, rcond = 0
−
ans =
−
−
0.60000
−
0.00000
−
−
octave:3> A \ [4; 8]    # many solutions
−
ans =
−
−
4
−
0
−
| style = "vertical-align: top" |
−
octave:4> A = infsup (A);
−
octave:5> A \ [3; 0]    # no solution
−
ans = 2×1 interval vector
−
−
[Empty]
−
[Empty]
−
−
octave:6> A \ [4; 8]    # many solutions
−
ans = 2×1 interval vector
−
−

−
[Entire]

|}

|}
−
=== Error handling ===
−
Due to the nature of set-based interval arithmetic, you should never observe errors (in the sense of raised GNU Octave error messages) during computation. If you do, there either is a bug in the code or there are unsupported data types.
−
−
octave:1> infsup (2, 3) / 0
−
ans = [Empty]
−
octave:2> infsup (0) ^ infsup (0)
−
ans = [Empty]
−
−
However, the interval constructors can produce errors depending on the input. The <code>infsup</code> constructor will fail if the interval boundaries are invalid. Contrariwise, the <code>infsupdec</code> constructor will only issue a warning and return a [NaI], which will propagate and survive through computations.
−
−
octave:3> infsup (3, 2) + 1
−
error: illegal interval boundaries: infimum greater than supremum
−
''… (call stack) …''
−
octave:3> infsupdec (3, 2) + 1
−
warning: illegal interval boundaries: infimum greater than supremum
−
ans = [NaI]
−
−
== Related work ==
−
−
For MATLAB there is a popular interval arithmetic toolbox [http://www.ti3.tu-harburg.de/rump/intlab/ INTLAB] by Siegfried Rump (member of IEEE P1788). It had been free (as in free beer) 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 not compatible with GNU Octave. I don't know if INTLAB is or will be compliant with IEEE 1788.
−
−
For C++ there is an interval library [https://github.com/nehmeier/libieeep1788/ libIEEE1788] by Marco Nehmeier (member of IEEE P1788). It aims to be standard compliant with IEEE 1788.
−
−
For Java there is a library [https://java.net/projects/jinterval/ jinterval] by Dmitry Nadezhin (member of IEEE P1788). It aims to be standard compliant with IEEE 1788, but is not complete yet.
−
[[Category:Octave-Forge]]
+
[[Category:Octave Forge]]
1,561

edits