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Interval package

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The {{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.netio/interval/overview.html Function reference]| doc 2 = [https:/p/octave.sourceforge.io/interval/ interval package_doc/ User manual]| download 1 = [https://octave.sourceforge.io/download.php?package] provides data types and fundamental operations for real valued =interval arithmetic based on the common floating-point format “binary64” a3. k2. a0. doubletar.gz interval-precision3.2.0.tar. It aims to be standard compliant with the (upcoming) [httpgz]| repository 1 = https://standardsoctave.ieeesourceforge.orgio/developpkg-repository/projectinterval/1788| dependency 1 = octave &ge; 3.html IEEE 1788] and therefore implements the 8.0| dependency 2 = '''Runtime:'set-based'' interval arithmetic flavormpfr (&ge; 3.1.0) [https://packages.debian. org/search?keywords=libmpfr4 libmpfr4 for Debian]| dependency 3 = '''Interval arithmeticBuild:''' produces mathematically proven numerical resultsmpfr (&ge; 3.1.0) [https://packages.debian.org/search?keywords=libmpfr-dev libmpfr-dev for Debian]}}
Warning: The GNU Octave interval package has not yet been released. If you want to experience the development version, you may (1) install the (currently deprecated) for real-valued [httphttps://octaveen.sourceforgewikipedia.netorg/fenvwiki/ fenv packageInterval_arithmetic interval arithmetic].* Intervals are closed, connected subsets of the real numbers. Intervals may be unbound (2in either or both directions) download a [https:or empty. In special cases <code>+inf</code> and <code>-inf</sourceforgecode> are used to denote boundaries of unbound intervals, but any member of the interval is a finite real number.net/p/octave/* Classical functions are extended to interval functions as follows: The result of function f evaluated on interval x is an interval/ci/default/tarball snapshot version '''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, (3) navigate to that the <code>instmathematical range of a function consist of several intervals, but their union will be returned, e. g., 1 /</code> subfolder and run octave[-1, 1] = [Entire].
[[File:Interval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]<div style== Motivation =="clear:left"></div>
{{quote|Give a digital computer a problem in arithmetic== Distribution ==* [https://tracker.debian.org/pkg/octave-interval Debian GNU/Linux], and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer[https://launchpad. … An net/ubuntu/+source/octave-interval computation yields a pair of numbers, an upper and a lower bound, which are guaranteed to enclose the exact answerLaunchpad Ubuntu]* [https://aur. Maybe you still don’t know the truth, but at least you know how much you don’t knowarchlinux.|Brian Hayes|org/packages/octave-interval/ archlinux user repository]* Included in [httphttps://dxftp.doignu.org/10gnu/octave/windows/ official Windows installer] and installed automatically with Octave (since version 4.0.15111)* [https://github.com/macports/macports-ports/tree/master/math/octave-interval/2003MacPorts] for Mac OS X* [https://www.6freshports.484 DOIorg/math/octave-forge-interval/ FreshPorts] for FreeBSD* [https: 10//cygwin.1511com/2003cgi-bin2/package-grep.6cgi?grep=octave-interval Cygwin] for Windows* [https://build.484opensuse.org/package/show/science/octave-forge-interval openSUSE build service]}}
{| class="wikitable" style="margin: auto"Development status ==!Standard floating point arithmetic* Completeness!Interval ** 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.| style = "vertical-align: top" | octave** Also, the minimalistic standard [https://standards.ieee.org/findstds/standard/1788.1> 19 * 0-2017.html IEEE Std 1788.1 - 2 + 02017], 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.1 ans = 1* Quality** Most arithmetic operations produce tight, correctly-rounded results.3878eThat is, the smallest possible interval with double-16precision (binary64) endpoints, which encloses the exact result.| style = "vertical-align** Includes [https: top" |//github.com/oheim/ITF1788 large test suite] for arithmetic functions octave** 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> x = infsup ("0bug tracker].1"); * Performance** All elementary functions have been [https://octave:2> 19 * x .org/doc/interpreter/Vectorization-and-Faster-Code- 2 + xExecution.html vectorized] and run fast on large input data. ans ⊂ ** Arithmetic is performed with the [-3http://www.mpfr.org/ GNU MPFR] library internally.1918911957973251e-16Where possible, +1the optimized [http://web.archive.org/web/20170128033523/http://lipforge.3877787807814457eens-16lyon.fr/www/crlibm/ CRlibm]library is used.* Portability** Runs in GNU Octave ≥ 3.8.2|}** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD
Floating== Project ideas (TODOs) ==* To be considered in the future: Algorithms can be migrated from the C-point arithmetic, as specified by XSC Toolbox (C++ code) from [http://enwww2.math.wikipediauni-wuppertal.orgde/wikiwrswt/IEEE_floating_point IEEE 754xsc/cxsc_new.html](nlinsys.cpp and cpzero.cpp), is available in almost every computer system todayhowever 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. It is wide-spreadinf = 5 A = [Empty]:* While at it, implemented also allow multiple subscripts in common hardware and integral part in programming languages<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. For example, the doublepdf]* Implement user-precision format is controllable output from the default numeric data type in GNU Octaveinterval standard (e. g. Benefits are obviousvia printf functions): The results a) It should be possible to specify the preferred overall field width (the length of arithmetic operations s). b) It should be possible to specify how Empty, Entire and NaI are welloutput, e.g., whether lower or upper case, and whether Entire becomes [Entire] or [-defined Inf, Inf]. c) For l and comparable between different systems u, it should be possible to specify the field width, and computation the number of digits after the point or the number of significant digits. (partly this is highly efficientalready 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.
However, there are some downsides of floating-point arithmetic in practice, which will eventually produce errors in computations.== Compatibility ==* Floating-point arithmetic The interval package's main goal 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]* 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]* Even if the developer would to be proficient, most developing environments / technologies limit floating-point arithmetic capabilities to a very limited subset of compliant with IEEE 754: Only one or two data types, no rounding modes, missing functions … [http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf]* Results are hardly predictable. [https://hal.archives-ouvertes.fr/halStd 1788-00128124/en/] All operations produce the best possible accuracy ''at runtime''2015, this so it is how a floating point works. Contrariwise, financial computer systems typically use a [http://en.wikipedia.org/wiki/Fixedcompatible with other standard-point_arithmetic fixed-point arithmetic] conforming implementations (COBOL, PL/I, …), where overflow and rounding can be precisely predicted ''at compile-time''.* If you do not know on the set of operations described by the technical details (cf. first bulletstandard document) you ignore the fact that the computer lies to you in many situations. For exampleOther implementations, 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'' which are known 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.aim for standard conformance are:
Interval arithmetic addresses above problems in its very special way and introduces new possibilities for algorithms* [https://github.com/JuliaIntervals/IntervalArithmetic. For example, the jl IntervalArithmetic.jl package] (Julia)* [httphttps://engithub.wikipediacom/jinterval/jinterval JInterval library] (Java)* [https://github.orgcom/wikinadezhin/Interval_arithmetic#Interval_Newton_method interval newton methodlibieeep1788 ieeep1788 library] is able to find ''all'' zeros of a particular function.(C++) created by Marco Nehmeier, later forked by Dmitry Nadezhin
== Theory =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.
=== Moore's fundamental theroem of The simp package contains a few basic interval arithmetic ===Let '''''y''''' = ''f''('''''x''''') be the result ofinterval-evaluation of ''f'' over a box '''''x''''' = (''x''<sub>1</sub>, … , ''x''<sub>''n''</sub>)using any interval versions of its component library functions. Then# 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'') }# If also each library operation in ''f'' is everywhere defined on its inputs, while evaluating '''''y''''', then ''f'' is everywhere defined operations on '''''x''''', that is Dom(''f'') ⊇ '''''x'''''scalar or vector intervals.# If It does not consider inaccurate built-in additionarithmetic functions, each library operation in ''f'' is everywhere continuous on its inputs, while evaluating '''''y'''''round-off, then ''f'' is everywhere continuous on '''''x'''''conversion and representational errors.# If some library operation in ''f'' is nowhere defined on As a result its inputs, while evaluating '''''y''''', then ''f'' syntax is nowhere defined on '''''x'''''very easy, that is Dom(''f'') ∩ '''''x''''' = Øbut the arithmetic fails to produce guaranteed enclosures.
== Quick start introduction ==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.
=== Input and output INTLAB ===Before exercising This interval arithmeticpackage 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 objects must be created from package as well. Some fundamental differences that I am currently aware of:* INTLAB is non-interval datafree software, it grants none of the [http://www.gnu. There are interval constants <code>empty<org/code> and <code>entire<philosophy/code> free-sw.html four essential freedoms] of free software* INTLAB is not conforming to IEEE Std 1788-2015 and the class constructors <code>infsup</code> parsing of intervals from strings uses a different format—especially for bare the uncertain form* INTLAB supports intervals with complex numbers and <code>infsupdec</code> sparse interval matrices, but no empty intervals* INTLAB uses inferior accuracy for decorated intervals. The class constructors are very sophisticated and most arithmetic operations, because it focuses on speed* Basic operations can be used with several kinds found in both packages, but the availability 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.special functions depends
octave:1> infsup (1)
ans = [1]
octave:2> infsup (1, 2)
ans = [1, 2]
octave:3> infsup ("3", "4")
ans = [3, 4]
octave:4> infsup ("1.1")
ans ⊂ [1.0999999999999998, 1.1000000000000001]
octave:5> infsup ("[5, 6.5]")
ans = [5, 6.5]
octave:6> infsup ("[5.8e-17]")
ans ⊂ [5.799999999999999e-17, 5.800000000000001e-17]
It is possible to access the exact numeric interval boundaries with the functions <code>inf</codediv style="display:flex; align-items: flex-start"> and <codediv style="margin-right: 2em">sup</code>. The shown text representation of intervals can be created {{Code|Computation with this interval package|<codesyntaxhighlight lang="octave">intervaltotext</code>. The default text representation is not guaranteed to be exact pkg load intervalA1 = infsup (see function <code>intervaltoexact</code> for that purpose2, 3);B1 = hull (-4, because this would massively spam console output. For exampleA2);C1 = midrad (0, 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.2);
A1 + B1 * C1</syntaxhighlight>}}</div><div>{{Code|Computation with INTLAB|<syntaxhighlight lang="octave:7"> startintlabA2 = infsup (12, 1 + eps3); ans ⊂ [1B2 = hull (-4, 1.0000000000000003]A2); octave:8> infsup C2 = midrad (10, 1 + 2 * eps) ans ⊂ [1, 1.0000000000000005];
Warning: Decimal fractions as well as numbers of high magnitude (A2 + B2 * C2</syntaxhighlight> 2}}<sup/div>53</supdiv>) 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.
octave:9> infsup (<span style = "color:red">0.2</span>) ans ⊂ [.20000000000000001, .20000000000000002] octave:10> infsup (<span style = "color:green">"0.2"</span>) ans ⊂ [.19999999999999998, .20000000000000002] 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.  octave:11> infsup ("17.7") + 1 ans ⊂ [18.699999999999999, 18.700000000000003] octave:12> ans + "[0, 2]" ans ⊂ [18.699999999999999, 20.700000000000003] ==== Specialized interval constructors Known differences ====Above mentioned Simple programs written for INTLAB should run without modification with this interval construction with decimal numbers or numeric data is straightforwardpackage. Beyond The following table lists common functions that, there are more ways to define intervals or interval boundaries.* Hexadecimal-floating-constant form: Each interval boundary may be defined by a hexadecimal number (optionally containing use 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 different name in double-precision, because the hexadecimal form is much shorter than the decimal representation of many numbersINTLAB.* 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.{|* 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>, wherepackage** <code>''m ''</code> is a mantissa in decimal,! INTLAB** <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),** <code>''u ''</code> is either empty | infsup (symmetrical uncertainty of ''r'' ULPs in both directionsx) 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 | intval (base 10x).  octave:13> infsup ("0x1.999999999999Ap|-4") ans ⊂ [.1, .10000000000000001] octave:14> infsup | wid ("1/3", "7/9"x) ans ⊂ [.33333333333333331, .7777777777777778] octave:15> infsup ("121.2?") ans ⊂ [121.14999999999999, 121.25] octave:16> infsup | diam ("5?32e2"x) ans = [|-2700, +3700] octave:17> infsup | subset ("-42??u") ans = [-42, +Inf] ==== Interval vectors and matrices ====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. octave:18> M = infsup (magic (3)) M = 3×3 interval matrix [8] [1] [6] [3] [5] [7] [4] [9] [2] octave:19> infsup (magic (3)a, magic (3) + 1b) 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 | in (["0.1"; "0.2"; "0.3"; "0.4"; "0.5"]) ans ⊂ 5×1 interval vector [.09999999999999999, .10000000000000001] [.19999999999999998a, .20000000000000002] [.29999999999999998, .30000000000000005] [.39999999999999996, .40000000000000003] [.5] octave:21> infsup ({1, eps; "4/7", "pi"}, {2, 1; "e", "0xff"}b) ans ⊂ 2×2 interval matrix [1, 2] [2.220446049250313e|-16, 1] [.5714285714285713, 2.7182818284590456] [3.1415926535897931, 255] When matrices are resized using subscripted assignment, any implicit new matrix elements will carry an empty interval. octave:22> M | interior (4a, 4b) = 42 M = 4×4 interval matrix [8] [1] [6] [Empty] [3] [5] [7] [Empty] [4] [9] [2] [Empty] [Empty] [Empty] [Empty] [42] Note: Whilst most functions | in0 (<code>size</code>a, <code>isvector</code>, <code>ismatrix</code>, …b) 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 (x)), isempty (empty | isnan (x)) ans = 0 ans = 1 === Decorations ===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: {| class="wikitable" style="margin: auto"!Decoration!Bounded!Continuous!Defined!Definition
|-
| com<br/>disjoint (commona, b)| style="text-align: center" | ✓| style="text-align: center" | ✓| style="text-align: center" | ✓| '''''x''''' is emptyintersect (a bounded, nonempty subset of Dom(''f''); ''f'' is continuous at each point of '''''x'''''; and the computed interval ''f''('''''x'''''b) is bounded
|-
| dac<br/>hdist (defined &amp; continuousa, b)|| style="text-align: center" | ✓| style="text-align: center" | ✓| '''''x''''' is qdist (a nonempty subset of Dom(''f'', b); and the restriction of ''f'' to '''''x''''' is continuous
|-
| def<br/>disp (definedx)||| style="text-align: center" | ✓| '''''disp2str (x''''' is a nonempty subset of Dom(''f'')
|-
| trv<br/>infsup (trivials)|||| always true str2intval (so gives no informations)
|-
| ill<br/>isa (ill-formedx, "infsup")|||| Not an interval, at least one interval constructor failed during the course of computationisintval (x)
|}
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.== Developer Information ==  octave:1> infsupdec (3, 4) ans = [3, 4]_com octave:2> ans + 12 ans = [15, 16]_com= Source Code Repository === octavehttps:3> ans / "[0, 2]" ans = [7/sourceforge.5, Inf]_trvnet/p/octave/interval/ci/default/tree/
=== Arithmetic operations Dependencies ===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 index {{Citation needed}}. apt-get install liboctave-dev mercurial make automake libmpfr-dev
Arithmetic functions in === Build ===The repository contains a set-based interval arithmetic follow these rules: Intervals are setsMakefile which controls the build process. They Some common targets are subsets of :* <code>make release</code> Create a release tarball and the set of real numbers. The interval version of an elementary function such as sinHTML documentation for [[Octave Forge]] (''x''takes a while) is essentially .* <code>make check</code> Run the natural extension full test-suite to sets of the corresponding point-wise function on real numbersverify that code changes didn't break anything (takes a while). That is, * <code>make run</code> Quickly start Octave with minimal recompilation and functions loaded from the function is evaluated workspace (for each number in the interval where the function is defined and the result must be an enclosure interactive testing of all possible values that may occurcode changes).
One operation that should be noted is the '''Build dependencies'''<code>fmaapt-get install libmpfr-dev autoconf automake inkscape zopfli</code> function (fused multiply and add). It computes '''''x''''' × '''''y''''' + '''''z''''' in a single step and is much slower than multiplication followed by addition. However, it is more accurate and therefore preferred in some situations.
octave:1> sin (infsup (0.5)) ans ⊂ [.47942553860420294, .47942553860420307] octave:2> pow (infsup (2), infsup (3, 4)) ans = [8, 16] octave:3> atan2 (infsup (1), infsup (1)) ans ⊂ [.785398163397448, .7853981633974487]== Architecture ===
=== Reverse arithmetic operations ===[[FileIn a nutshell the package provides two new data types to users:Reverse-power-functionsbare intervals and decorated intervals.png|400px|thumb|right|Reverse power operations. A relevant subset of 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 function's domain is outlined former and hatched. In this example we use ''x''adds attribute <supcode>''y''dec</supcode> ∈ [2, 3](interval decoration).]]
Some arithmetic Almost all functions also provide reverse mode operationsin the package are implemented as methods of these classes, e. That is inverse functions with interval constraintsg. For example the <code>sqrrev@infsup/sin</code> can compute implements the inverse of the <code>sqr</code> sine function on for bare intervals. The syntax Most code is <code>sqrrev kept in m-files. Arithmetic operations that require correctly-rounded results are implemented in oct-files (C++ code), X)</code> and will compute these are used internally by the enclosure m-files of all numbers ''x'' ∈ X that fulfill the constraint ''x''² ∈ Cpackage.The source code is organized as follows:
In the following example, we compute the constraints for base and exponent of the power function <code>pow< +- doc/ – package manual +- inst/ | +- @infsup/code> as shown in the figure. octave:1> x = powrev1 (| | +- infsup ("[1.1, 1m – class constructor for bare intervals | | +- sin.45]"), infsup m – sine function for bare intervals (2, 3)uses mpfr_function_d internally) x ⊂ [1| | `- .6128979635153644, 2.7148547265657923]. – further functions on bare intervals | +- @infsupdec/ octave:2> y = powrev2 | | +- infsupdec.m – class constructor for decorated intervals | | +- sin.m – sine function for decorated intervals (uses @infsup ("[2/sin internally) | | `- ... – further functions on decorated intervals | `- ..14, 2.5]"), infsup – a few global functions that don't operate on intervals `- src/ +- mpfr_function_d.cc – computes various arithmetic functions correctly rounded (2, 3)using MPFR) y ⊂ [ `- ..7564707973660297, 1.4440113978403293] – other oct-file sources
=== Numerical operations Best practices ===Some operations on intervals do not return an interval enclosure, but a single number (in double-precision). Most important are ==== Parameter checking ====* All methods must check <code>infnargin</code> and call <code>supprint_usage</code>, which return 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 lower user to mix non-interval parameters with interval parameters and upper interval boundariesthe 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
More such operations are <code>mid</code> if (approximation of the interval's midpointnot (isa (x, "infsupdec"), <code>wid</code> )) x = infsupdec (approximation of the interval's widthx); endif if (not (isa (y, <code>rad</code> "infsupdec"))) y = infsupdec (approximation of the interval's radiusy), <code>mag</code> and <code>mig</code>.; endif
=== Boolean operations = Use of Octave functions ====Interval comparison operations produce boolean results. While some comparisons are especially for intervals (subset, interior, ismember, isempty, disjoint, …) others are extensions of simple numerical comparisonOctave functions may be used as long as they don't introduce arithmetic errors. For example, the less-or-equal comparison ceil function can be used safely since it is mathematically defined as ∀<sub>''a''</sub> ∃<sub>''b''</sub> ''a'' ≤ ''b'' ∧ ∀<sub>''b''</sub> ∃<sub>''a''</sub> ''a'' ≤ ''b''exact on binary64 numbers. function x = ceil (x) ... parameter checking ... x.inf = ceil (x.inf); x.sup = ceil (x.sup); endfunction
octave:1If Octave functions would introduce arithmetic/rounding errors, there are interfaces to MPFR (<code> infsup mpfr_function_d</code>) and crlibm (1, 3<code>crlibm_function</code>) <= infsup (2, 4) ans = 1which can produce guaranteed boundaries.
=== Matrix operations = Vectorization & Indexing ====Above mentioned operations can also All functions should be applied element-wise to interval vectors implemented using vectorization and matricesindexing. Many operations use [http://wwwThis is very important for performance on large data.gnuFor example, consider the plus function.org/software/octave/doc/interpreter/Vectorization-It computes lower and-Faster-Code-Executionupper boundaries of the result (x.html#Vectorization-and-Faster-Code-Execution vectorization techniques]inf, yIn additioninf, there are matrix operations on interval matricesx. These operations comprise: exact dot productsup, exact matrix multiplication, exact vector sums, (not-exacty.sup may be vectors or matrices) 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 then uses an indexing expression to familiar operations in floating-point arithmeticadjust values where empty intervals would have produces problematic values. function x = plus (x, y)''Technical background information: The interval package simulates a [http://books ... parameter checking .google.de/books?hl=de&id=I7X9EVfeV5EC&q=accumulator Kulisch accumulator] in software to produce tightly rounded results for vector and matrix operations.'' octave:1> A l = infsup mpfr_function_d ([1'plus', 2-inf, 3; 4x.inf, 0, 0; 0, 0, 0], [1, 2, 3; 4, 0, 6y.inf); 0, 0, 1]) A u = 3×3 interval matrix [1] [2] [3] [4] [0] [0, 6] [0] [0] [0mpfr_function_d ('plus', 1] octave:2> B = inv (A) B = 3×3 interval matrix [0] [.25] [-Inf+inf, 0] [.5] [-x.125] [-Infsup, -y.75] [0] [0] [1, Inf] octave:3> A * B ans = 3×3 interval matrix [1] [0] [Entire] [0] [1] [Entire] [0] [0] [0, Inf]sup);
emptyresult = isempty (x) | isempty (y);
l(emptyresult) = inf;
u(emptyresult) = -inf;
endfunction
octave:4> A = infsup (magic (3)) A = 3×3 interval matrix [8] [1] [6] [3] [5] [7]VERSOFT == [4] [9] The [2] octavehttp:5> c = A \ [3; 4; 5] c ⊂ 3×1 interval vector [//uivtx.18333333333333323, cs.18333333333333344] [.43333333333333318, cas.43333333333333352cz/~rohn/matlab/ VERSOFT] [.18333333333333323, .18333333333333344] octave:6> A * c ans ⊂ 3×1 software package (by Jiří Rohn) has been released under a free software license (Expat license) and algorithms may be migrated into the interval vector [2package.9999999999999982, 3.0000000000000018] [3.9999999999999982, 4.0000000000000027] [4.9999999999999973, 5.0000000000000027]
{|! 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= Error handling "color:red"| trapped| depends on <code style="color:red">verpinv</code> and <code style="color:red">verfullcolrank</code>|-|verpd|style="color:red"| trappedDue | depends on <code>isspd</code> (by Rump, to the nature of setbe 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|-based |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, you should never observe errors migrated (in the sense of raised GNU Octave error messagesfor real eigenvalues) during computation. If you do, there either | dependency <code>norm</code> is a bug already implemented|-|verspectrad|style="color:red"| trapped| main part implemented in the <code>vereig</code >|-|colspan="3"|Real (or there are unsupported complex) data types.only: Matrices: Decompositions|-|verpoldec|style="color:red"| trapped| depends on <code style="color:red">verthinsvd</code>|-|verrankdec octave|style="color:1red"| trapped| depends on <code style="color:red"> infsup (2, 3) verfullcolrank</code> and <code style="color:red">verpinv</ 0code> ans |-|verspectdec|style= [Empty]"color:red"| trapped| main part implemented in <code>vereig</code>|-|verthinsvd octave|style="color:2red"| trapped| implemented in <code> infsup vereig</code>|-|colspan="3"|Real (0or complex) ^ infsup data only: Matrix functions|-|vermatfun|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3"|Real data only: Linear systems (0rectangular) ans |-|<s>verlinineqnn</s>|style= [Empty]"color:green"| free, migrated| use <code>glpk</code> as a replacement for <code>linprog</code>|-|verlinsys|style="color:red"| trappedHowever, the interval constructors can produce errors depending | dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on the input. The <codestyle="color:red">infsupverpinv</code> constructor will fail if the interval boundaries are invalid. Contrariwise, the <codestyle="color:red">infsupdecverfullcolrank</code> constructor will only issue a warning and return a [NaI], which will propagate and survive through computations.<code style="color:red">verbasis</code>|-|verlsq octave|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> infsup (3by Rump, to be checked, 2algorithm in [http://www.ti3.tuhh.de/paper/rump/Ru06c.pdf]) + 1 error: illegal interval boundaries: infimum greater than supremum|- ''… |colspan="3"|Real (call stackor complex) …''data only: Polynomials octave|-|verroots|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3> infsupdec "|Interval (3, 2or real) + 1data: Matrices warning|-|verhurwstab|style="color:red"| trapped| depends on <code style="color: illegal interval boundariesred">verposdef</code>|-|verinverse|style="color: infimum greater than supremumgreen"| free ans | depends on <code style= [NaI]"color:green">verintervalhull</code>, to be migrated|-|<s>verinvnonneg</s>|style="color:green"| free, migrated|-|verposdef|style= Related work "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 For MATLAB there | dependency <code>verifylss</code> is a popular interval arithmetic toolbox implemented as <code>mldivide</code>; depends on <code>isspd</code> (by Rump, to be checked) and <code>verintervalhull</code>; see also [http://wwwuivtx.ti3cs.tu-harburgcas.decz/rump~rohn/intlabpublist/ INTLABsingreg.pdf] by Siegfried Rump |-|colspan="3"|Interval (member of IEEE P1788or real). It had been 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 (as in or real) data: Matrices: Decompositions|-|verqr (experimental)|style="color:green"| free beer) for academic use in | <code>qr</code> has already been implemented using the pastGram-Schmidt process, but no longer iswhich seems to be more accurate and faster than the Cholsky decomposition or Householder reflections used in verqr. No migration needed. Its origin dates back |-|<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 1999be migrated|-|verhullparam|style="color:green"| free| depends on <code>verintervalhull</code>, so it is well tested and comprises a lot of functionalityto be migrated|-|verhullpatt|style="color:green"| free| depends on <code>verhullparam</code>, especially for vector 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</ matrix operations. INTLAB is not compatible with GNU Octave. I don't know if INTLAB is code>|-|colspan="3"|Interval (or will 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 compliant with IEEE 1788.migrated|-| verabsvaleqnall|style="color:green"| freeFor C++ there is an interval library | depends on <code>verabsvaleqn</code>, see also [httpshttp://githubuivtx.comcs.cas.cz/nehmeier~rohn/libieeep1788publist/ libIEEE1788absvaleqnall.pdf] by Marco Nehmeier (member of IEEE P1788). It aims , to be standard compliant with IEEE 1788, but is not complete yet.migrated|-| verbasintnpprob|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>|-|}
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]]

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