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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.~~net~~io/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” a~~3.~~ k~~2.~~ a~~0. ~~double~~tar.gz interval-~~precision~~3.2.0.tar. ~~ It aims to be standard compliant with the (upcoming) [http~~gz]| repository 1 = https://~~standards~~octave.~~ieee~~sourceforge.~~org~~io/~~develop~~pkg-repository/~~project~~interval/~~1788~~| dependency 1 = octave ≥ 3.~~html IEEE 1788] and therefore implements the ~~8.0| dependency 2 = '''Runtime:'~~set-based~~'' ~~interval arithmetic flavor~~mpfr (≥ 3.1.0) [https://packages.debian. ~~ ~~org/search?keywords=libmpfr4 libmpfr4 for Debian]| dependency 3 = '''~~Interval arithmetic~~Build:''' ~~produces mathematically proven numerical results~~mpfr (≥ 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 [~~http~~https://~~octave~~en.~~sourceforge~~wikipedia.~~net~~org/~~fenv~~wiki/ ~~fenv package~~Interval_arithmetic interval arithmetic].* Intervals are closed, connected subsets of the real numbers. Intervals may be unbound (~~2~~in either or both directions) ~~download a [https:~~or empty. In special cases <code>+inf</code> and <code>-inf</~~sourceforge~~code> 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>inst~~mathematical 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].
~~{{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 answer~~Launchpad Ubuntu]* [https://aur. ~~Maybe you still don’t know the truth, but at least you know how much you don’t know~~archlinux.~~|Brian Hayes|~~org/packages/octave-interval/ archlinux user repository]* Included in [~~http~~https://~~dx~~ftp.~~doi~~gnu.org/~~10~~gnu/octave/windows/ official Windows installer] and installed automatically with Octave (since version 4.0.~~1511~~1)* [https://github.com/macports/macports-ports/tree/master/math/octave-interval/~~2003~~MacPorts] for Mac OS X* [https://www.~~6~~freshports.~~484 DOI~~org/math/octave-forge-interval/ FreshPorts] for FreeBSD* [https: ~~10~~//cygwin.~~1511~~com/~~2003~~cgi-bin2/package-grep.~~6~~cgi?grep=octave-interval Cygwin] for Windows* [https://build.~~484~~opensuse.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 + 0~~2017], IEEE standard for interval arithmetic (simplified) is fully implemented. The standard was approved by IEEE-SA on December 6, 2017 (and published in January 2018).** In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.~~1~~~~ ans = 1~~* Quality** Most arithmetic operations produce tight, correctly-rounded results.~~3878e~~That is, the smallest possible interval with double-~~16~~precision (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 ("0~~bug tracker].~~1");~~~~ ~~* Performance** All elementary functions have been [https://octave~~:2> 19 * x ~~.org/doc/interpreter/Vectorization-and-Faster-Code- ~~2 + x~~Execution.html vectorized] and run fast on large input data.~~ ans ⊂ ~~** Arithmetic is performed with the [~~-3~~http://www.mpfr.org/ GNU MPFR] library internally.~~1918911957973251e-16~~Where possible, ~~+1~~the optimized [http://web.archive.org/web/20170128033523/http://lipforge.~~3877787807814457e~~ens-~~16~~lyon.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 point ~~== Project ideas (TODOs) ==* To be considered in the future: Algorithms can be migrated from the C-XSC Toolbox (C++ code) from [http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_new.html] (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.* Interval version of <code>interp1</code>* Extend <code>subsasgn</code> to allow direct manipulation of inf and sup (and dec) properties. >> A = infsup ("[2, ~~as specified by ~~4]"); >> A.inf = infsup ("[1, 3]") A = [1, 4] >> A.inf = 5 A = [Empty]:* While at it, also allow multiple subscripts in <code>subsasgn</code> >> A(:)(2:4)(2) = 42; # equivalent to A(3) = 42 >> A.inf(3) = 42; # also A(3).inf = 42 >> A.inf.inf = 42 # should produce error? >> A.inf.sup = 42 # should produce error?* Tight Enclosure of Matrix Multiplication with Level 3 BLAS [http://kam.mff.cuni.cz/conferences/swim2015/abstracts/ozaki.pdf] [http://kam.mff.cuni.cz/conferences/swim2015/slides/ozaki.pdf]* Verified Convex Hull for Inexact Data [http://kam.mff.cuni.cz/conferences/swim2015/abstracts/ohta.pdf] [http://~~en~~kam.~~wikipedia~~mff.~~org~~cuni.cz/~~wiki~~conferences/~~IEEE_floating_point IEEE 754~~swim2015/slides/ohta.pdf]* Implement user-controllable output from the interval standard (e. g. via printf functions): a) It should be possible to specify the preferred overall field width (the length of s). b) It should be possible to specify how Empty, ~~is available in almost every computer system today~~Entire and NaI are output, e.g. ~~It is wide~~, whether lower or upper case, and whether Entire becomes [Entire] or [-~~spread~~Inf, ~~implemented in common hardware and integral part in programming languages~~Inf]. c) For ~~example~~l and u, it should be possible to specify the field width, and the number of digits after the point or the ~~extended precision ~~number of significant digits. (partly this is already implemented by output_precision (...) / `format ~~is ~~long` / `format short`) d) It should be possible to output the ~~default numeric data type in GNU Octave~~bounds of an interval without punctuation, e.g., 1.234 2. ~~Benefits are obvious: The results ~~345 instead of ~~arithmetic operations are well-defined and comparable between different systems and computation is highly efficient~~[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, … [http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf]~~~~* Results are hardly predictable. [https://hal.archives-ouvertes.fr/hal~~Std 1788-~~00128124/en/] All operations produce the best possible accuracy ''at runtime''~~2015, ~~this ~~so it is ~~how floating~~compatible with other standard-~~point works. Contrariwise, financial computer systems typically use a [http://en.wikipedia.org/wiki/Fixed-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 ~~technical details, cf. first bullet, you ignore ~~set of operations described by the ~~fact that the computer lies to you in many situations~~standard document). ~~For example~~Other 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)* [~~http~~https://~~en~~github.~~wikipedia~~com/jinterval/jinterval JInterval library] (Java)* [https://github.~~org~~com/~~wiki~~nadezhin/~~Interval_arithmetic#Interval_Newton_method interval newton method~~libieeep1788 ieeep1788 library] ~~is able to find ''all'' zeros of a particular function.~~(C++) created by Marco Nehmeier, later forked by Dmitry Nadezhin
~~=== Moore's fundamental theroem of ~~The simp package contains a few basic interval arithmetic ~~===~~~~Let '''''y''''' = ''f''('''''x''''') be the result of~~~~interval-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 ~~addition~~arithmetic 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''''' = Ø~~but 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.
~~ 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~~<~~/code~~div style="display:flex; align-items: flex-start"> ~~and ~~<~~code~~div style="margin-right: 2em">~~sup</code>. The default text representation of intervals can be created ~~{{Code|Computation with this interval package|<~~code~~syntaxhighlight lang="octave">~~intervaltotext</code>. The default text representation is not guaranteed to be exact ~~pkg load intervalA1 = infsup (~~see function <code>intervaltoexact</code>~~2, 3);B1 = hull (-4, ~~because this would massively spam console output. For example~~A2);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 (~~1~~2, ~~1 + eps~~3);~~ ans ⊂ [1~~B2 = hull (-4, ~~1.0000000000000003]~~A2);~~ octave:8> infsup ~~C2 = midrad (~~1~~0, ~~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~~</~~sup~~div>~~) 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 straightforward~~package. ~~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 numbers~~INTLAB.~~* 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>, where~~package~~** <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 directions~~x) ~~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 10~~x)~~.~~ ~~ octave:13> infsup ("0x1.999999999999Ap~~|-~~4")~~~~ ans ⊂ [.1, .10000000000000001]~~~~ octave:14> infsup ~~| wid (~~"1/3", "7/9"~~x)~~ ans ⊂ [.33333333333333331, .7777777777777778]~~~~ octave:15> infsup ~~| diam (~~"121.2?"~~x)~~ ans ⊂ [121.14999999999999, 121.25]~~~~ octave:16> infsup ("5?32e2")~~~~ ans = [~~|-~~2700, +3700]~~~~ octave:17> infsup ~~| subset (~~"-42??u")~~~~ ans = [-42~~a, ~~+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)~~b)~~ M = 3×3 interval matrix~~~~ ~~~~ [8] [1] [6]~~~~ [3] [5] [7]~~~~ [4] [9] [2]~~~~ ~~~~ octave:19> infsup ~~| in (~~magic (3)~~a, ~~magic (3) + 1~~b)~~ 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 ~~| interior (~~["0.1"; "0.2"; "0.3"; "0.4"; "0.5"])~~~~ ans ⊂ 5×1 interval vector~~~~ ~~~~ [.09999999999999999, .10000000000000001]~~~~ [.19999999999999998~~a, ~~.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 ~~| in0 (~~4~~a, ~~4~~b) ~~= 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 (<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 ~~(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~~
~~In the following example, all decoration information is lost when the ~~== Developer Information ===== Source Code Repository ===https://sourceforge.net/p/octave/interval ~~is possibly divided by zero, i. e., the overall function is not guaranteed to be defined for all possible inputs.~~/ci/default/tree/
~~ octave:1> infsupdec (3, 4)~~~~ ans ~~= ~~[3, 4]_com~~~~ octave:2> ans + 12~~~~ ans ~~= ~~[15, 16]_com~~= Dependencies === ~~octave:3> ans / "[0, 2]"~~~~ ans = [7.5, Inf]_trv~~apt-get install liboctave-dev mercurial make automake libmpfr-dev
~~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~~'Build dependencies'''~~) is essentially the natural extension to sets of the corresponding point~~<code>apt-get install libmpfr-~~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.~~dev autoconf automake inkscape zopfli</code>
~~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 much slower than multiplication followed by addition. However, it is more accurate and therefore preferred in some situations.~~=== Architecture ===
~~ octave~~In a nutshell the package provides two new data types to users:~~1~~bare intervals and decorated intervals. The data types are implemented as:* class <code> ~~sin (~~infsup </code> (~~0.5~~bare interval)~~)~~~~ ans ⊂ [.47942553860420294, .47942553860420307]~~~~ octave:2~~with attributes <code>inf</code> ~~pow (infsup ~~(~~2~~lower interval boundary)~~, infsup ~~and <code>sup</code> (~~3, 4)~~upper interval boundary)~~ ans = [8, 16]~~~~ octave:3~~* class <code>infsupdec</code> ~~atan2 ~~(~~infsup (1~~decorated interval)~~, infsup ~~which extends the former and adds attribute <code>dec</code> (~~1~~interval decoration)~~)~~~~ ans ⊂ [.785398163397448, ~~.~~7853981633974487]~~
~~=== Reverse arithmetic operations ===~~~~[[File:Reverse-power-~~Almost all functionsin the package are implemented as methods of these classes, e.~~png|400px|thumb|right|Reverse power operations~~g. ~~The relevant subset of the function's domain, where ''x''~~<~~sup~~code>~~''y''~~@infsup/sin</~~sup~~code>~~ ∈ [2, 3]~~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 ~~outlined and hatched.]]~~organized as follows:
~~Some arithmetic functions also provide reverse mode operations~~ +- doc/ – package manual +- inst/ | +- @infsup/ | | +- infsup. ~~That is inverse functions with interval constraints~~m – class constructor for bare intervals | | +- sin. ~~For example the <code>sqrrev</code> can compute the inverse of the <code>sqr</code> ~~m – sine function for bare intervals (uses mpfr_function_d internally) | | `- ... – further functions on bare intervals | +- @infsupdec/ | | +- infsupdec.m – class constructor for decorated intervals | | +- sin. ~~The syntax is <code>X = sqrrev ~~m – sine function for decorated intervals (~~C, X~~uses @infsup/sin internally)~~</code> and will compute the enclosure of all numbers ''x'' ∈ X ~~ | | `- ... – further functions on decorated intervals | `- ... – a few global functions that ~~fulfill the constraint ~~don'~~'x''² ∈ C~~t operate on intervals `- src/ +- mpfr_function_d.cc – computes various arithmetic functions correctly rounded (using MPFR) `- ... – other oct-file sources
~~In the following example, we compute the constraints for base ~~=== Best practices ======= Parameter checking ====* All methods must check <code>nargin</code> and ~~exponent of the power function ~~call <code>~~pow~~print_usage</code> if the number of parameters is wrong. This prevents simple errors by the user.* Methods with more than 1 parameter must convert non-interval parameters to intervals using the class constructor. This allows the user to mix non-interval parameters with interval parameters and the function treats any inputs as ~~shown in ~~intervals. Invalid values will be handled by the ~~figure~~class constructors. ~~octave:1> x = powrev1 ~~if (not (~~infsup ~~isa (x, "~~[1.1, 1.45]~~infsup")~~, ~~)) x = infsup (~~2, 3)~~x); ~~x ⊂ [1.6128979635153644, 2.7148547265657923]~~endif ~~octave:2> y = powrev2 ~~if (not (~~infsup ~~isa (y, "~~[2.14, 2.5]~~infsup")~~, ~~)) y = infsup (~~2, 3)~~y); ~~y ⊂ [.7564707973660297, 1.4440113978403293]~~endif
~~More such operations are <code>mid</code> (approximation ~~==== Use of ~~the interval~~Octave functions ====Octave functions may be used as long as they don'~~s midpoint)~~t introduce arithmetic errors. For example, ~~<code>wid</code> ~~the ceil function can be used safely since it is exact on binary64 numbers. function x = ceil (~~approximation of the interval's width~~x)~~, <code>rad</code> ~~ ... parameter checking ... x.inf = ceil (~~approximation of the interval's radius~~x.inf)~~, <code>mag</code> and <code>mig</code>~~; x.sup = ceil (x.sup); endfunction
~~=== Boolean operations ===~~~~Interval comparison operations produce boolean results. While some comparisons are especially for intervals (subset~~If Octave functions would introduce arithmetic/rounding errors, ~~interior, ismember, isempty, disjoint, …) others ~~there are ~~extensions of simple numerical comparison. For example, the less ~~interfaces to MPFR (~~or equal) comparison is mathematically defined as ∀~~<~~sub~~code>~~''a''~~mpfr_function_d</~~sub~~code>~~ ∃~~) and crlibm (<~~sub~~code>~~''b''~~crlibm_function</~~sub~~code>~~ ''a'' ≤ ''b'' ∧ ∀<sub>''b''</sub> ∃<sub>''a''</sub> ''a'' ≤ ''b''~~), which can produce guaranteed boundaries.
~~ octave~~{|! 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:~~1~~red"| trapped| depends on <code style="color:red"> ~~infsup ~~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 (~~2, 3~~experimental) | unknown| depends on <code style="color:red">verlsq</ ~~0~~code>; todo: compare with local function inside <code style="color:green">verintlinineqs</code>~~ ans ~~|-|verorth|style="color:red"| trapped| depends on <code style= ~~[Empty]~~"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~~ octave~~|style="color:~~2~~red"| trapped| depends on <code>isspd</code> ~~infsup ~~(~~0~~by Rump, to be checked) ~~^ infsup ~~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 (~~0~~or complex)data only: Matrices: Eigenvalues and singular values~~ ans ~~|-|vereig|style= ~~[Empty]~~"color:red"| trapped ~~However, the interval constructors can produce errors depending ~~| depends on ~~the input. The ~~proprietary <code>~~infsup~~verifyeig</code> ~~constructor will fail if the ~~function from INTLAB, depends on complex interval ~~boundaries are invalid. Contrariwise~~arithmetic|-|<s>vereigback</s>|style="color:green"| free, ~~the ~~migrated (for real eigenvalues)| dependency <code>~~infsupdec~~norm</code> ~~constructor will only issue a warning and return a [NaI], which will propagate and survive through computations.~~is already implemented|-|verspectrad~~ octave~~|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3~~> infsup ~~"|Real (~~3, 2~~or complex) ~~+ 1~~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>~~ error~~|-|verspectdec|style="color: ~~illegal interval boundaries~~red"| trapped| main part implemented in <code>vereig</code>|-|verthinsvd|style="color: ~~infimum greater than supremum~~red"| trapped| implemented in <code>vereig</code>|-~~ ''… ~~|colspan="3"|Real (~~call stack~~or complex) ~~…''~~data only: Matrix functions~~ octave~~|-|vermatfun|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3~~> infsupdec ~~"|Real data only: Linear systems (~~3, 2~~rectangular) ~~+ 1~~~~ warning~~|-|<s>verlinineqnn</s>|style="color: ~~illegal interval boundaries~~green"| free, migrated| use <code>glpk</code> as a replacement for <code>linprog</code>|-|verlinsys|style="color: ~~infimum greater than supremum~~red"| trapped~~ ans ~~| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code style= ~~[NaI]~~"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= ~~Related work ~~"color:red">verpinv</code> and <code style="color:red">verfullcolrank</code>|-|colspan="3"|Real data only: Optimization|-|verlcpall|style="color:green"| free~~For MATLAB there ~~| 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 ~~popular interval arithmetic toolbox ~~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.~~tu-harburg~~tuhh.de/paper/rump/~~intlab~~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</ ~~INTLAB] ~~code> (by ~~Siegfried ~~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> (~~member of IEEE P1788~~by Rump, to be checked)and <code>verintervalhull</code>; see also [http://uivtx.cs.cas.cz/~rohn/publist/singreg. ~~It had been free ~~pdf]|-|colspan="3"|Interval (~~as ~~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 ~~beer~~, 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) ~~for academic use in ~~data: Matrices: Decompositions|-|verqr (experimental)|style="color:green"| free| <code>qr</code> has already been implemented using the ~~past~~Gram-Schmidt process, ~~but no longer is~~which 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 ~~1999~~be migrated|-|verhullparam|style="color:green"| free| depends on <code>verintervalhull</code>, ~~so it is well tested and comprises a lot of functionality~~to 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"| free~~For C++ there is an interval library ~~| depends on <code>verabsvaleqn</code>, see also [~~https~~http://~~github~~uivtx.~~com~~cs.cas.cz/~~nehmeier~~~rohn/~~libieeep1788~~publist/ ~~libIEEE1788~~absvaleqnall.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.~~

Use OctaveForge template.

[[File:Interval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]<div style=~~= Motivation ==~~"clear:left"></div>

== ~~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.

=== ~~Input and output ~~INTLAB ===~~Before exercising ~~This interval ~~arithmetic~~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 ~~objects must be created, typically from ~~package as well. Some fundamental differences that I am currently aware of:* INTLAB is non-~~interval data~~free 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

|-

| ~~com<br/>~~disjoint (~~common~~a, 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 & continuous~~a, 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 (~~defined~~x)|~~|~~~~| style="text-align: center" | ✓~~~~| '''''~~disp2str (x~~''''' is a nonempty subset of Dom(''f''~~)

|-

| ~~trv<br/>~~infsup (~~trivial~~s)|~~|~~~~|~~~~| always true ~~str2intval (~~so gives no information~~s)

|-

| ~~ill<br/>~~isa (~~ill-formed~~x, "infsup")|~~|~~~~|~~~~| Not an interval, at least one interval constructor failed during the course of computation~~isintval (x)

|}

=== ~~Arithmetic operations ~~Build ===The ~~interval packages comprises many interval arithmetic operations~~repository contains a Makefile which controls the build process. ~~Function names match GNU ~~Some common targets are:* <code>make release</code> Create a release tarball and the HTML documentation for [[Octave ~~standard functions where applicable, and follow recommendations by IEEE 1788 otherwise~~Forge]] (takes a while). ~~It is possible ~~* <code>make check</code> Run the full test-suite to ~~look up all ~~verify that code changes didn't break anything (takes a while).* <code>make run</code> Quickly start Octave with minimal recompilation and functions ~~by their corresponding IEEE 1788 name in ~~loaded from the ~~index {{Citation needed}}~~workspace (for interactive testing of code changes).

if (not (isa (x, "infsupdec"))) x =~~== Numerical operations ===~~infsupdec (x); endif~~Some operations on intervals do ~~ if (not ~~return an interval enclosure~~(isa (y, ~~but a single number. Most important are <code>inf</code> and <code>sup</code>, which return the interval boundaries in double precision.~~"infsupdec"))) y = infsupdec (y); endif

==== Vectorization & Indexing ====All functions should be implemented using vectorization and indexing. This is very important for performance on large data. For example, consider the plus function. It computes lower and upper boundaries of the result (x.inf, y.inf, x.sup, y.sup may be vectors or matrices) and then uses an indexing expression to adjust values where empty intervals would have produces problematic values. ~~octave:1> infsup ~~function x = plus (~~1~~x, ~~3~~y) ~~<~~ ... parameter checking ... l = ~~infsup ~~mpfr_function_d (~~2~~'plus', ~~4~~-inf, x.inf, y.inf); u = mpfr_function_d ('plus', +inf, x.sup, y.sup); ~~ans ~~ emptyresult = isempty (x) | isempty (y); l(emptyresult) = inf; u(emptyresult) = -inf; … ~~1~~endfunction

==~~= Error handling =~~VERSOFT ==~~Due to the nature of set-based interval arithmetic, you should never observe errors during computation~~The [http://uivtx.cs.cas. ~~If you do, there either is ~~cz/~rohn/matlab/ VERSOFT] software package (by Jiří Rohn) has been released under a ~~bug in ~~free software license (Expat license) and algorithms may be migrated into the ~~code or there are unsupported data types~~interval package.

[[Category:Octave~~-~~Forge]]

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