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Interval package

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The GNU Octave interval package for real-valued [httphttps://octaveen.sourceforgewikipedia.netorg/intervalwiki/ Interval_arithmetic interval packagearithmetic] for .* Intervals are closed, connected subsets of the realnumbers. Intervals may be unbound (in either or both directions) or empty. In special cases <code>+inf</code> and <code>-valued inf</code> are used to denote boundaries of unbound intervals, but any member of the interval is a finite real number.* Classical functions are extended to interval functions as follows: The result of function f evaluated on interval x is an interval '''enclosure of all possible values''' of f over x where the function is defined. Most interval arithmetic functions in this package manage to produce a very accurate such enclosure.* The result of an interval arithmeticfunction is an interval in general. It might happen, that the mathematical range of a function consist of several intervals, but their union will be returned, e. g., 1 / [-1, 1] = [Entire].
The interval package provides data types and operations for__TOC__verified computing[[File:Interval-sombrero. It can be used to handle uncertainties, estimatearithmetic errors and produce reliable results. Also it can help findguaranteed solutions to numerical problems. Thanks to GNU MPFR manyarithmetic operations are provided with best possible accuracy. Theimplementation of intervals in inf-sup format is based on interval boundariesrepresented by binary64 numbers and is standard conforming to png|280px|thumb|left|Example: Plotting the (upcoming) standard for interval arithmetic enclosure of a function]][http<div style="clear:left"><// IEEE 1788].div>
== Features Distribution ==* [ Latest version at Octave Forge]** <code>pkg install -forge interval</code>** [ function reference]** [ package documentation] (user manual)'''Third-party'''* [ Debian GNU/Linux], [ Launchpad Ubuntu]* [ archlinux user repository]* Included in [ official Windows installer] and installed automatically with Octave (since version 4.0.1)* [ MacPorts] for Mac OS X* [ FreshPorts] for FreeBSD* [ Cygwin] for Windows* [ openSUSE build service]
== Development status ==* Free software licensed under the terms of the GNU General Public LicenseCompleteness* Many interval arithmetic * All required functions from [httphttps://octavestandards.sourceforgeieee.netorg/findstds/intervalstandard/overview1788-2015.htmlIEEE Std 1788-2015] with high, systemIEEE standard for interval arithmetic, are implemented. The standard was approved by IEEE-independent accuracySA on June 11, 2015. It will remain active for ten years. The standard was approved by ANSI in 2016.* Conforming to IEEE 1788 * Also, the minimalistic standard [http IEEE Std 1788.1-2017], IEEE standard for intervalarithmetic (simplified) is fully implemented. The standard was approved by IEEE-SA on December 6, 2017 (and published in January 2018).** In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.* Quality** Most arithmetic operations produce tight, correctly-rounded results. That is, the smallest possible interval with double-precision (binary64) endpoints, which encloses the exact result.** Includes [https:/ci/ large test suite]for arithmetic functions* Support for * For open bugs please refer to the [ vectors &type_of_search=bugs&only_group_id=1925&exact=1 bug tracker].* Performance** All elementary functions have been [ interval matrices with -Faster-Code-Execution.html vectorized function evaluation ] and broadcastingrun fast on large input data.* Easy usage* Arithmetic is performed with the [ GNU MPFR] library internally. Where possible, the optimized [http: Convenient interval constructors and // CRlibm] library is used.* Portability** Runs in GNU Octave function names≥ 3.8.2** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD
== Development Status Project ideas (TODOs) ==* To be considered in the future: Algorithms can be migrated from the C-XSC Toolbox (C++ code) from [] (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.* Interval version of <code>interp1</code>* Extend <code>subsasgn</code> to allow direct manipulation of inf and sup (and dec) properties. >> A = infsup ("[2, 4]"); >> A.inf = infsup ("[1, 3]") A = [1, 4] >> A.inf = 5 A = [Empty]:* While at it, also allow multiple subscripts in <code>subsasgn</code> >> A(:)(2:4)(2) = 42; # equivalent to A(3) = 42 >> A.inf(3) = 42; # also A(3).inf = 42 >> A.inf.inf = 42 # should produce error? >> A.inf.sup = 42 # should produce error?* Tight Enclosure of Matrix Multiplication with Level 3 BLAS [] []* Verified Convex Hull for Inexact Data [] []* Implement user-controllable output from the interval standard (e. g. via printf functions): a) It should be possible to specify the preferred overall field width (the length of s). b) It should be possible to specify how Empty, Entire and NaI are output, e.g., whether lower or upper case, and whether Entire becomes [Entire] or [-Inf, Inf]. c) For l and u, it should be possible to specify the field width, and the number of digits after the point or the number of significant digits. (partly this is already implemented by output_precision (...) / `format long` / `format short`) d) It should be possible to output the bounds of an interval without punctuation, e.g., 1.234 2.345 instead of [1.234, 2.345]. For instance, this might be a convenient way to write intervals to a file for use by another application.
== Motivation Compatibility ==The interval package's main goal is to be compliant with IEEE Std 1788-2015, so it is compatible with other standard-conforming implementations (on the set of operations described by the standard document). Other implementations, which are known to aim for standard conformance are:
{{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* [https://github. … An interval computation yields a pair of numbers, an upper and a lower bound, which are guaranteed to enclose the exact answercom/JuliaIntervals/IntervalArithmetic. Maybe you still don’t know the truth, but at least you know how much you don’t knowjl IntervalArithmetic.|Brian Hayes|jl package] (Julia)* [httphttps://dx.doigithub.orgcom/10.1511jinterval/2003.6.484 DOIjinterval JInterval library] (Java)* [https: 10//github.1511com/nadezhin/2003.6.484libieeep1788 ieeep1788 library]}}(C++) created by Marco Nehmeier, later forked by Dmitry Nadezhin
{| class="wikitable" style="margin: auto"!Standard floating point arithmetic!Interval arithmetic|-| style = "vertical-align: top" | <span styleOctave Forge simp package ="opacity:.5">octave:1> </span>19 * 0.1 - 2 + 0.1 ans = 1.3878e-16| style = "vertical-align: top" | <span style="opacity:.5">octave:1> <In 2008/span>x = infsup 2009 there was a Single Interval Mathematics Package ("0.1"SIMP); <span style="opacity:.5">octave:2> </span>19 * x - 2 + x ans ⊂ [-3.191891195797326e-16for Octave, +1which has eventually become unmaintained at Octave Forge.3877787807814457e-16]|}
FloatingThe simp package contains a few basic interval arithmetic operations on scalar or vector intervals. It does not consider inaccurate built-point in arithmeticfunctions, as specified by [ IEEE 754], is available in almost every computer system today. It is wideround-spreadoff, implemented in common hardware conversion and integral part in programming languagesrepresentational errors. For exampleAs a result its syntax is very easy, but 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 efficientfails to produce guaranteed enclosures.
However, there are some downsides of floating-point arithmetic in practice, which will eventually produce errors in computations.* Floating-point arithmetic It is often used mindlessly by developers. [] [] []* The binary data types categorically are not suitable for doing financial computations. Very often representational errors are introduced when using “real world” decimal numbers. []* Even if the developer would be proficient, most developing environments / technologies limit floating-point arithmetic capabilities recommended to a very limited subset of IEEE 754: Only one or two data types, no rounding modes, missing functions … []* Results are hardly predictable. [] All operations produce use the best possible accuracy ''at runtime'', this is how interval package as a floating point worksreplacement for simp. Contrariwise, financial computer systems typically use a [ fixed-point arithmetic] (COBOL, PL/I, …)However, where overflow function names and rounding can be precisely predicted ''at compile-time''.* Results are system dependent. All but the most basic floating-point operations interval constructors are not guaranteed to be accurate and produce different results depending on low level libraries and hardware. [] []* 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, compatible between 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 occurpackages.
Interval arithmetic addresses above problems in its very special way === INTLAB ===This interval package is ''not'' meant to be a replacement for INTLAB and any compatibility with it is pure coincidence. Since both are compatible with GNU Octave, they happen to agree on many function names and introduces new possibilities programs written for algorithmsINTLAB may possibly run with this interval package as well. For exampleSome fundamental differences that I am currently aware of:* INTLAB is non-free software, it grants none of the [ interval newton methodfree-sw.html four essential freedoms] of free software* INTLAB is able not conforming to find ''all'' zeros IEEE Std 1788-2015 and the parsing of intervals from strings uses a particular function.different format—especially for the uncertain form* INTLAB supports intervals with complex numbers and sparse interval matrices, but no empty intervals* INTLAB uses inferior accuracy for most arithmetic operations, because it focuses on speed* Basic operations can be found in both packages, but the availability of special functions depends
== Theory ==
<div style="display:flex; align-items: flex-start"><div style="margin-right: 2em">{{Code|Computation with this interval package|<syntaxhighlight lang= Online Introductions "octave">pkg load intervalA1 =infsup (2, 3);B1 =hull (-4, A2);C1 =midrad (0, 2);
[http:A1 + B1 * C1</syntaxhighlight>}}</ Interval analysis in MATLAB] Note: The div><div>{{Code|Computation with INTLAB toolbox for Matlab is not entirely compatible with this interval package for GNU Octave|<syntaxhighlight lang="octave">startintlabA2 = infsup (2, 3);B2 = hull (-4, cf. [[#Compatibility]]. HoweverA2);C2 = midrad (0, basic operations can be compared and should be compatible for common intervals.2);
=== Moore's fundamental theroem of interval arithmetic ===A2 + B2 * C2Let '''''y''''' = ''f''('''''x''''') be the result ofinterval-evaluation of ''f'' over a box '''''x''''' = (''x''<sub/syntaxhighlight>1}}</subdiv>, … , ''x''<sub>''n''</subdiv>)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 on '''''x''''', that is Dom(''f'') ⊇ '''''x'''''.# If in addition, each library operation in ''f'' is everywhere continuous on its inputs, while evaluating '''''y''''', then ''f'' is everywhere continuous on '''''x'''''.# 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.
== What to expect == Known differences ====The interval arithmetic provided by Simple programs written for INTLAB should run without modification with this interval package is '''slow''', but '''accurate'''. The following table lists common functions that use a different name in INTLAB.{|''Why is the ! interval package slow?''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 []. 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., [ interval power function], atan2, or [[#Reverse_arithmetic_operations! INTLAB|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.| infsup (x)For example, multiplication of matrices with many elements becomes unfeasible as it takes a lot of time.| intval (x)|-{| class="wikitable" style="margin: auto; align:right"wid (x)|+ Approximate runtime diam (wall clock time in secondsx) <span style="font|-weight: normal">— Results have been produced with GNU Octave 3.8.2 and Interval package 0.1.4 on an Intel® Core™ i5-4340M CPU | subset (2.9–3.6 GHza, b)</span>!rowspan="2"|Interval matrix sizein (a, b)!valign="top" style="min|-width:6em"|<code>plus</code>!valign="top" style="min-width:6em"|<code>times</code>interior (a, b)!valign="top" style="min-width:6em"|<code>log</code>in0 (a, b)!valign="top" style="min-width:6em"|<code>pow</code>!valign="top" style="min-width:6em"|<code>mtimes</code>!valign="top" style="min-width:6em"|<code>mtimes</code>isempty (x)!valign="top" style="min-width:6em"|<code>inv</code>isnan (x)
!style="font-weight:normal"|tightest<br/>accuracydisjoint (a, b)!style="font-weight:normal"|tightest<br/>accuracy!style="font-weight:normal"|tightest<br/>accuracy!style="font-weight:normal"|tightest<br/>accuracy!style="font-weight:normal"|valid<br/>accuracy!style="font-weight:normal"|tightest<br/>accuracy!style="font-weight:normal"|valid<br/>accuracyemptyintersect (a, b)
| 10 × 10hdist (a, b)|align="right"| < 0.001|align="right"| < 0.001|align="right"| 0.001|align="right"| 0.008|align="right"| 0.001|align="right"| 0.002|align="right"| 0.025qdist (a, b)
| 100 × 100disp (x)|align="right"| 0.003|align="right"| 0.010|align="right"| 0.055|align="right"| 0.61|align="right"| 0.012|align="right"| 0.53|align="right"| 0.30disp2str (x)
| 500 × 500infsup (s)|align="right"| 0.060|align="right"| 0.24|align="right"| 1.3|align="right"| 15|align="right"| 0.30|align="right"| 63|align="right"| 4.2str2intval (s)
| isa (x, "infsup")
| isintval (x)
''Why is the interval package accurate?''== Developer Information ===== Source Code Repository ===The GNU Octave built-in floating-point routines are not useful for interval arithmetichttps: Their results depend on hardware, system libraries and compilation options//sourceforge. The net/p/octave/interval package handles all arithmetic functions with the help of the [http:/ci/default/www.mpfr.orgtree/ GNU MPFR library]. With MPFR it is possible to compute system-independent, valid and tight enclosures of the correct results for most functions. However, it should be noted that some reverse operations and matrix operations do not exists in GNU MPFR and therefore cannot be computed with the same accuracy.
== Installation = Dependencies ===The interval package is available at Octave Forge [] and can be installed from within GNU Octave (version ≥ 3.8.2). During installation parts of the package are compiled for the target system, which requires the GNU MPFR development libraries (version ≥ 3.1.0) to be installed. <span style="opacity:.5">octave:1> </span>pkg apt-get install liboctave-forge interval <span style="opacity:.5">octave:2> </span>pkg load intervaldev mercurial make automake libmpfr-dev
=== Build ===The ''development version'' may be obtained from its mercurial repository. For convenience contains a Makefile target allows running which controls the package from sourcebuild process.Some common targets are: hg clone http:* <code>make release<//hgcode> Create a release tarball and the HTML documentation for [[Octave Forge]] (takes a while).* <>make check</p/octave/interval octavecode> Run the full test-intervalsuite to verify that code changes didn't break anything (takes a while). cd octave-interval; * <code>make run</code> Quickly start Octave with minimal recompilation and functions loaded from the workspace (for interactive testing of code changes).
== Quick start introduction =='''Build dependencies'''<code>apt-get install libmpfr-dev autoconf automake inkscape zopfli</code>
=== Input and output Architecture ===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 interval constructors <code>infsupdec</code> (create an interval from boundaries), <code>midrad</code> (create an interval from midpoint and radius) and <code>hull</code> (create an interval enclosure for a list of mixed arguments: numbers, intervals or interval literals). The 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.
{{Code|Create In a nutshell the package provides two new data types to users: bare intervals for performing interval arithmetic|and decorated intervals. The data types are implemented as:* class <code>infsup<syntaxhighlight lang="octave"/code>infsupdec (1bare interval) # [1]_cominfsupdec with attributes <code>inf</code> (1, 2lower interval boundary) # [1, 2]_cominfsupdec and <code>sup</code> ("3", "4"upper interval boundary) # [3, 4]_com* class <code>infsupdec </code> ("1.1"decorated interval) # [1.0999999999999998, 1.100000000000001]_cominfsupdec which extends the former and adds attribute <code>dec</code> ("5.8e-17") # [5.799999999999999e-17, 5.800000000000001e-17]_commidrad (12, 3interval decoration) # [9, 15]_commidrad ("4.2", "1e-7") # [4.199999899999999, 4.200000100000001]_comhull (3, 42, "19.3", "-2.3") # [-2.300000000000001, +42]_comhull ("pi", "e") # [2.718281828459045, 3.1415926535897936]_com</syntaxhighlight>}}
The default text representation Almost all functions in the package are implemented as methods of intervals is not guaranteed to be exactthese classes, because this would massively spam console outpute. g. For example, the exact text representation of <code>realmin@infsup/sin</code> would be over 700 decimal places long! Howeverimplements 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 default text representation package. The source code is correct organized as it guarantees to contain the actual boundaries.follows:
{{Warning +- doc/ – package manual +- inst/ |Decimal fractions as well as numbers of high magnitude +- @infsup/ | | +- infsup.m – class constructor for bare intervals | | +- sin.m – sine function for bare intervals (> 2<sup>53<uses mpfr_function_d internally) | | `- ... – further functions on bare intervals | +- @infsupdec/sup> | | +- infsupdec.m – class constructor for decorated intervals | | +- sin.m – sine function for decorated intervals (uses @infsup/sin internally) should always be passed as | | `- ... – further functions on decorated intervals | `- ... – a string to the constructorfew global functions that don't operate on intervals `- src/ +- mpfr_function_d. Otherwise it is possible, that GNU Octave introduces conversion errors when the numeric literal is converted into floatingcc – computes various arithmetic functions correctly rounded (using MPFR) `-point format '''before''' it is passed to the constructor.}}.. – other oct-file sources
{{Code|Beware of the conversion pitfall|=== Best practices ======= Parameter checking ====* All methods must check <code>nargin</code> and call <code>print_usage<syntaxhighlight lang="octave"/code>## The numeric constant “0.3” is an approximation of if the## decimal number 0of parameters is wrong.3This prevents simple errors by the user. An * Methods with more than 1 parameter must convert non-interval around this approximation## will not contain parameters to intervals using the decimal number 0.3.infsupdec (0.3) # [0.29999999999999998, 0class constructor.29999999999999999]_com## However, passing This allows the decimal number 0.3 as a string## user to mix non-interval parameters with interval parameters and the interval constructor function treats any inputs as intervals. Invalid values will create an interval which## actually encloses be handled by the decimal numberclass constructors.infsupdec if (not (isa (x, "0.3infsup") # [0.29999999999999998)) x = infsup (x); endif if (not (isa (y, 0.3000000000000001]_com"infsup")))</syntaxhighlight> y = infsup (y);}} endif
if (not (isa (x, "infsupdec"))) x =infsupdec (x); endif if (not (isa (y, "infsupdec"))) y === Interval vectors and matrices ====infsupdec (y);Vectors and matrices of intervals can be created by passing numerical matrices, char vectors or cell arrays to the interval constructors. With cell arrays it is also possible to mix several types of boundaries. endif
Interval matrices behave like normal matrices in GNU ==== Use of Octave and functions ====Octave functions may be used as long as they don't introduce arithmetic errors. For example, the ceil function can be used for broadcasting and vectorized safely since it is exact on binary64 numbers. function evaluationx = ceil (x) ... parameter checking ... x.inf = ceil (x.inf); x.sup = ceil (x.sup); endfunction
{{Code|Create interval matrices|If Octave functions would introduce arithmetic/rounding errors, there are interfaces to MPFR (<code>mpfr_function_d<syntaxhighlight lang="octave"/code>M = infsup (magic (3)) # [8] [1] [6] # [3] [5] [7] # [4] [9] [2]infsup and crlibm (magic (3), magic (3) + 1) # [8, 9] [1, 2] [6, 7] # [3, 4] [5, 6] [7, 8] # [4, 5] [9, 10] [2, 3]infsupdec (["0.1"; "0.2"; "0.3"; "0.4"; "0.5"]) # [0.09999999999999999, 0.1000000000000001]_com # [0.19999999999999998, 0.2000000000000001]_com # [0.29999999999999998, 0.3000000000000001]_com # [0.39999999999999996, 0.4000000000000001]_com # [0.5]_cominfsup ({1, eps; "4<code>crlibm_function</7", "pi"}, {2, 1; "e", "0xff"}code>) # [1, 2] [2.220446049250313e-16, 1] # [0which can produce guaranteed boundaries.5714285714285713, 2.7182818284590456] [3.141592653589793, 255]</syntaxhighlight>}}
=== Arithmetic operations = Vectorization & Indexing ====The interval packages comprises many interval arithmetic operationsAll 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. A complete list can inf, y.inf, x.sup, y.sup may be found in its [http://octavevectors or matrices) and then uses an indexing expression to adjust values where empty intervals would have produces problematic values. function x = plus (x, y) function reference]parameter checking ... l = mpfr_function_d ('plus', -inf, x.inf, y.inf); u = mpfr_function_d ('plus', +inf, x. Function names match GNU Octave standard functions where applicable and follow recommendations by IEEE 1788 otherwisesup, cfy. [[#IEEE_1788_indexsup); emptyresult = isempty (x) |IEEE 1788 index]].isempty (y); l(emptyresult) = inf; u(emptyresult) = -inf; endfunction
Arithmetic functions in a set-based interval arithmetic follow these rules== VERSOFT ==The [http: Intervals are sets//uivtx.cs. They are subsets of the set of real numberscas. The interval version of an elementary function such as sincz/~rohn/matlab/ VERSOFT] software package (by Jiří Rohn) has been released under a free software license (''x''Expat license) 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 and algorithms may be migrated into the interval where the function is defined and the result must be an enclosure of all possible values that may occurpackage.
By default arithmetic functions are computed with best possible accuracy {|! Function! Status! Information|-|colspan="3"|Real (which is more than what is guaranteed by GNU Octave core functionsor complex). The result will therefore be a tight and very accurate enclosure of the true mathematical value in most cases. Details on each function's accuracy can be found in its documentation, which is accessible with GNU Octave's help only: Matrices|-|verbasis{{Code|Examples of using interval arithmetic functionsstyle="color:red"| trapped|depends on <syntaxhighlight langcode style="octavecolor:red">verfullcolrank</code>sin (infsupdec (0.5)) # [.4794255386042029, .4794255386042031]_com|-pow (infsupdec (2), infsupdec (3, 4)) # [8, 16]_com|vercondnumatan2 (infsupdec (1), infsupdec (1))|style="color:red"| trapped # [.7853981633974482, .7853981633974484]_com| depends on <code style="color:red">versingval</code>midrad (magic (3), 0.5) * pascal (3)|- # [13.5, 16.5]_com [25, 31]_com [42, 52]_com|verdet # [13.5, 16.5]_com [31, 37]_com [55, 65]_com # [13.5, 16.5]_com [25, 31]_com [38, 48]_com|style="color:red"| trapped| depends on <code>vereig</syntaxhighlightcode>}}|-|verdistsing|style=== Numerical operations ==="color:red"| trappedSome operations | depends on intervals do not return an interval enclosure, but a single number (in double-precision). Most important are <codestyle="color:red">infversingval</code> and |-|verfullcolrank|style="color:red"| trapped| depends on <code>supverpinv</code>, which return the lower and upper interval boundaries.|-|vernorm2|style="color:red"| trappedMore such operations are | depends on <codestyle="color:red">midversingval</code> |-|vernull (approximation of the interval's midpointexperimental), | unknown| depends on <codestyle="color:red">widverlsq</code> (approximation of the interval's width), ; todo: compare with local function inside <codestyle="color:green">radverintlinineqs</code> (approximation of the interval's radius), |-|verorth|style="color:red"| trapped| depends on <codestyle="color:red">magverbasis</code> (interval's magnitude) and <codestyle="color:red">migverthinsvd</code> (interval's mignitude).|-|verorthproj|style="color:red"| trapped| depends on <code style="color:red">verpinv</code> and <code style= Boolean operations =="color:red">verfullcolrank</code>|-|verpd|style="color:red"| trappedInterval comparison operations produce boolean results. While some comparisons are especially for intervals | depends on <code>isspd</code> (subsetby Rump, interior, ismember, isempty, disjoint, …to be checked) others are extensions of simple numerical comparison. For example, the less-or-equal comparison is mathematically defined as ∀and <subcode style="color:red">''a''vereig</subcode> ∃|-|verpinv|style="color:red"| trapped| dependency <subcode>''b''verifylss</subcode> ''a'' ≤ ''b'' ∧ ∀is implemented as <subcode>''b''mldivide</subcode> ∃; depends on <subcode style="color:red">''a''verthinsvd</subcode> ''a'' ≤ ''b''.|-|verpmat |style="color:red"| trapped| depends on <span code style="opacitycolor:.5red">octave:1> verregsing</spancode>infsup (1, 3) <= infsup (2, 4) ans = 1|-|verrank|style="color:red"| trapped| depends on <code style="color:red">versingval</code> and <code style= Matrix operations ===Above mentioned operations can also be applied element-wise to interval vectors and matrices. Many operations use [http"color:red">verbasis<//>|-and-Faster-Code-Execution.html#Vectorization-and-Faster-Code-Execution vectorization techniques].|verrref|style="color:red"| trappedIn addition, there are matrix operations | depends on interval matrices. These operations comprise<code style="color: dot product, matrix multiplication, vector sums (all with tightest accuracy), matrix inversion, matrix powers, and solving linear systems (the latter are less accurate). As a result of missing hardware red">verfullcolrank</ low-level library support code> and missing optimizations, these operations are relatively slow compared to familiar operations in floating-point arithmetic. {{Code|Examples of using interval matrix functions|<syntaxhighlight langcode style="octavecolor:red">verpinv</code>|-A |colspan= infsup ([1, 2, "3; 4, 0, 0; 0, 0, 1]); A "|Real (2, 3or complex) data only: Matrices: Eigenvalues and singular values|-|vereig|style= "[0, 6]color:red"| trapped # [1] [2] [3]| depends on proprietary <code>verifyeig</code> function from INTLAB, depends on complex interval arithmetic # [4] [0] [0, 6]|- # [0] [0] [1]|<s>vereigback</s>B |style= inv "color:green"| free, migrated (Afor real eigenvalues) # [0] [0.25] [-1.5, 0]| dependency <code>norm</code> is already implemented # [0.5] [|-0.125] [-1.5, -0.75] # [0] [0] [1]|verspectradA * B|style="color:red"| trapped # [1] [0] [-1.5, +1.5]| main part implemented in <code>vereig</code> # [0] [1] [|-6, +6] # [0] [0] [1] A |colspan= infsup "3"|Real (magic (3)or complex)data only: Matrices: Decompositions # [8] [1] [6]|- # [3] [5] [7]|verpoldec # [4] [9] [2]|style="color:red"| trappedc | depends on <code style= A \ [3; 4; 5]"color:red">verthinsvd</code> # [0.18333333333333315, 0.18333333333333358]|- # [0.43333333333333318, 0.43333333333333346]|verrankdec # [0.18333333333333307, 0.18333333333333355]|style="color:red"| trappedA * c| depends on <code style="color:red">verfullcolrank</code> and <code style="color:red">verpinv</code> # [2.9999999999999964, 3.0000000000000036]|- # [3.9999999999999964, 4.000000000000004]|verspectdec # [4.999999999999997, 5.000000000000003]|style="color:red"| trapped| main part implemented in <code>vereig</syntaxhighlightcode>}}|-|verthinsvd|style==== Notes on linear systems ===="color:red"| trappedA system of linear equations | implemented in the form A''x'' <code>vereig</code>|-|colspan= 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 "3"|Real (possibly severalor complex) linear systems. Some characteristics should definitely be known when linear interval systems are solveddata only: Matrix functions|-|vermatfun|style="color:red"| trapped* If the linear system is underdetermined and has infinitely many solutions, the interval solution will be unbound | main part implemented 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.<code>vereig</code>* 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 |colspan="3"|Real data only: Linear systems have solutions or not.(rectangular) * 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.|-|<s>verlinineqnn</s>However|style="color:green"| free, solving linear systems with interval arithmetic can produce useful results in many cases and automatically carries migrated| use <code>glpk</code> as a guaranty replacement for error boundaries. Additionally, it can give better information than the floating<code>linprog</code>|-point variants for some cases.|verlinsys|style="color:red"| trapped{{Code|Standard floating point arithmetic versus interval arithmetic dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on ill-conditioned linear systems|<syntaxhighlight langcode style="octavecolor:red">A verpinv</code>, <code style= [1, 0; 2, 0];## This linear system has no solutionsA \ [3; 0] # warning"color: matrix singular to machine precisionred">verfullcolrank</code>, rcond and <code style= 0"color:red">verbasis</code> # 0.60000|- # 0.00000|verlsq## This linear system has many solutions|style="color:red"| trappedA \ [4; 8]| depends on <code style="color:red">verpinv</code> and <code style="color:red">verfullcolrank</code> # 4|- # 0 ## The empty interval vector proves that there is no solutioninfsup (A) \ [|colspan="3; 0]"|Real data only: Optimization # [Empty]|- # [Empty]|verlcpall## The unbound interval vector indicates that there may be many solutions|style="color:green"| freeinfsup (A) \ [4; 8]| depends on <code>verabsvaleqnall</code> # [4] # [Entire]|-|<s>verlinprog</syntaxhighlights>}}|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>== Advanced topics ==|-|verlinprogg|style=== Error handling ==="color:red"| trapped| depends on <code>verfullcolrank</code>Due to the nature of set|-based interval arithmetic, one should not observe errors (in the sense of raised GNU Octave error messages) during computation unless operations are evaluated for incompatible data types. Arithmetic operations which are not defined for (parts of) their input, simply ignore anything that is outside of their domain.|verquadprog| unknownHowever, | use <code>quadprog</code> from the interval constructors can produce errors depending on the input. The optim package; use <code>glpk</code> as a replacement for <code>infsuplinprog</code> constructor will fail if the interval boundaries are invalid. Contrariwise, the (preferred) ; dependency <code>infsupdecverifylss</code>, is implemented as <code>midradmldivide</code> and ; depends on <code>hullisspd</code> constructors will only issue a warning and return a (by Rump, to be checked, algorithm in [NaI] object, which will propagate and survive through computationshttp://www.ti3.tuhh. NaI stands for “not an interval”de/paper/rump/Ru06c.pdf]) {{Code|Effects of set-based interval arithmetic on partial functions and the NaI object|<syntaxhighlight langcolspan="octave3">|Real (or complex) data only: Polynomials## Evaluation of a function outside of its domain returns an empty interval|-infsupdec (2) / 0 # [Empty]_trv|verrootsinfsupdec (0) ^ infsupdec (0) # [Empty]_trv|style="color:red"| trapped| main part implemented in <code>vereig</code>## Illegal interval construction creates a NaI|-infsupdec |colspan="3"|Interval (3, 2or real) # [NaI]data: Matrices|-## NaI even survives through computations|verhurwstabans + 1 # [NaI]|style="color:red"| trapped| depends on <code style="color:red">verposdef</syntaxhighlightcode>}}|-|verinverse|style="color:green"| freeThere are some situations where the interval package cannot decide whether an error occurred or not and issues a warning. The user may choose to ignore these warnings or handle them as errors, see | depends on <codestyle="color:green">help warningverintervalhull</code> for instructions., to be migrated|-|<s>verinvnonneg</s>{| classstyle="wikitablecolor:green" | free, migrated|-|verposdef|style="margincolor: autored"| trapped!Warning ID!Reason!Possible consequences| depends on <code>isspd</code> (by Rump, to be checked) and <code style="color:red">verregsing</code>
| interval:PossiblyUndefinedverregsing|style="vertical-aligncolor:topred" | Interval construction with boundaries in decimal formattrapped| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code>isspd</code> (by Rump, to be checked) and the constructor can't decide whether the lower boundary is smaller than the upper boundary<code>verintervalhull</code>; see also [http://uivtx. Both boundaries are very close and lie between subsequent binary64 numberscs.|style="vertical-align:top" | The constructed interval is a valid and tight enclosure of both numberscas. If the lower boundary was actually greater than the upper boundary, this illegal interval is not considered an errorcz/~rohn/publist/singreg.pdf]
| interval:ImplicitPromote|stylecolspan="vertical-align:top3" | An interval operation has been evaluated on both, a bare and a decorated interval. The bare interval has been converted into a decorated interval in order to produce a decorated result. Note: This warning does not occur if a bare interval literal string gets promoted into a decorated interval, e. g., <code>infsupdec (1, 2) + "[3, 4]"</code> does not produce this warning whereas <code>infsupdec Interval (1, 2or real) + infsup (3, 4)</code> does. A bare interval can be explicitly promoted with the newdec [httpdata://] function.|style="vertical-alignMatrices:top" | The implicit conversion applies the best possible decoration for the bare interval. If the bare interval has been produced from an interval arithmetic computation, this branch of computation is not covered by the decoration information Eigenvalues and the final decoration could be considered wrong. For example, <code>infsupdec (1, 2) + infsup (0, 1) ^ 0</code> would ignore that 0<sup>0</sup> is undefined.singular values
| interval:NaIvereigsym|style="vertical-aligncolor:topred" | An error has occured during interval construction and the NaI object has been produced. The warning text contains further details. A NaI can be explicitly created with the nai [] function.trapped|style="vertical-align:top" | Nothing bad is going to happen, because the semantics of NaI are well defined by IEEE 1788. However, the user might choose to cancel the algorithm immediately when the NaI is encountered for the first time.|} === Decorations ===The interval package provides a powerful decoration system for intervals, as specified by the IEEE standard for interval arithmetic. By default any interval carries a decoration, which collects additional information about the course of function evaluation on the interval data. Only the (unfavored) main part implemented in <code>infsupvereig</code> constructor creates bare, undecorated intervals and the depends on <codestyle="color:red">intervalpartverspectrad</code> operation may be used to demote decorated intervals into bare, undecorated ones. It is highly recommended to always use the decorated interval arithmetic, which gives additional information about an interval result in exchange for a tiny overhead. The following decorations are available: {| class="wikitable" style="margin: auto"!Decoration!Bounded!Continuous!Defined!Definition
| com<br/>(common)vereigval| style="text-aligncolor: centerred" | trapped| depends on <code style="text-aligncolor: centerred" | ✓| 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>verregsing</code>
| dac<brs>vereigvec</s>(defined &amp; continuous)|| style="text-align: center" | ✓| style="text-aligncolor: centergreen" | | '''''x''''' is a nonempty subset of Dom(''f''); and the restriction of ''f'' to '''''x''''' is continuousfree, migrated
| def<br/>(defined)||verperrvec| style="text-aligncolor: centergreen" | free| '''''x''''' the function is just a nonempty subset of Dom(''f'')wrapper around <code style="color:green">vereigvec</code>?!?
| trv<br/>(trivial)versingval|style="color:red"|trapped|| always true (so gives no information)depends on <code style="color:red">vereigsym</code>
| ill<br/>(ill-formed)|||| Not an interval, at least one interval constructor failed during the course of computation|} The decoration information is especially useful after a very long and complicated function evaluation. For example, when the “def” decoration survives until the final result, it is proven that the overall function is actually defined for all values covered by the input intervals. {{Code|Examples of using the decoration system|<syntaxhighlight langcolspan="octave">x = infsupdec (3, 4) # [3, 4]_comy = x - 3.5 # [-0.5, +0.5]_com## The square root function ignores any negative part of the input,## but the decoration indicates whether this has or has not happened.sqrt (x) # [1.732050807568877, 2]_comsqrt (y) # [0, 0.7071067811865476]_trv</syntaxhighlight>}} === Specialized interval constructors ===Above mentioned interval construction with decimal numbers or numeric data is straightforward. Beyond 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 a point) and an exponent field with an integral power of two as defined by the C99 standard ([ ISO/IEC9899, N1256, §]). 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.* 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 [ “units in last place” (ULPs)] as an uncertainty. The format is <code>''m''?''ruE''</code>, where** <code>''m ''</code> is a mantissa in decimal,** <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 (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). {{Code|Examples of different formats during interval construction|<syntaxhighlight lang="octave">infsup ("0x1.999999999999Ap-4") # hex-form # [0.1, 0.1000000000000001]infsup ("1/3", "7/9") # rational form # [0.3333333333333333, 0.7777777777777778]infsup ("121.2?") # uncertain form # [121.14999999999999, 121.25]infsup ("5?32e2") # uncertain form with ulp count # [-2700, +3700]infsup ("-42??u") # unbound uncertain form # [-42, +Inf]</syntaxhighlight>}} === 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 Interval (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. <span style="opacity:.5">octave:1> </span>x = powrev1 (infsup ("[1.1, 1.45]"), infsup (2, 3)) x ⊂ [1.6128979635153646, 2.714854726565793] <span style="opacity:.5">octave:2> </span>y = powrev2 (infsup ("[2.14, 2.5]"), infsup (2, 3)) y ⊂ [0.7564707973660299, 1.4440113978403289] === Tips & Tricks ===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.  <span style="opacity:.5">octave:1> </span>infsupdec ("17.7") + 1 ans ⊂ [18.699999999999999, 18.70000000000001]_com <span style="opacity:.5">octave:2> </span>ans + "[0, 2]" ans ⊂ [18.699999999999999, 20.70000000000001]_com Interval functions with only one argument can be called by using property syntax, e. g. <code>x.inf</code>, <code>x.sup</code> or even <code>x.sqr</code>. When matrices are resized using subscripted assignment, any implicit new matrix elements will carry an empty interval. <span style="opacity:.5">octave:1> </span>M = infsup (magic (3real)); M (4, 4) = 42 M = 4×4 interval matrix [8] [1] [6] [Empty] [3] [5] [7] [Empty] [4] [9] [2] [Empty] [Empty] [Empty] [Empty] [42] 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. <span style="opacity:.5">octaveMatrices:1> </span>builtin ("isempty", empty ()), isempty (empty ()) ans = 0 ans = 1 == IEEE 1788 index == {| class="wikitable" style="margin: auto; min-width: 50%"|+ In terms of a better integration into the GNU Octave language, several operations use a function name which is different from the name proposed in the standard document.!IEEE 1788!GNU OctaveDecompositions
| numsToIntervalverqr (experimental)| bare versionstyle="color: infsup (''l'', ''u'') []green"| free| <code>qr<br/code>decorated version: infsupdec (''l''has already been implemented using the Gram-Schmidt process, ''u'') [http://octave.sourceforgewhich seems to be more accurate and faster than the Cholsky decomposition or Householder reflections used in migration needed.html]
| textToInterval<s>verchol (experimental)</s>|style="color:green"| free, migrated| bare migrated version: infsup (''s'')has been named after the standard Octave function <code>chol<br/code>decorated version: infsupdec (''s'')
| setDeccolspan="3"| infsupdec Interval (''x'', ''dx''or real) data: Linear systems (square)
| exp10verenclinthull| pow10 [httpstyle="color://]green"| free| to be migrated
| exp2verhullparam| pow2 [httpstyle="color:green"| free| depends on <code>verintervalhull<//]code>, to be migrated
| recipverhullpatt| inv [httpstyle="color:green"| free| depends on <code>verhullparam<//]code>, to be migrated
| sqrtverintervalhull| realsqrt [httpstyle="color://]green"| free| to be migrated
| rootncolspan="3"| nthroot [httpInterval (or real) data://]Linear systems (rectangular)
| logp1verintlinineqs| log1p [httpstyle="color:green"| free| depends on <code style="color:green">verlinineqnn<//]code>
| roundTiesToAwayveroettprag| round [httpstyle="color://]green"| free
| roundTiesToEvenvertolsol| roundb [httpstyle="color:green"| free| depends on <code style="color:green">verlinineqnn<//]code>
| trunccolspan="3"| fix [httpInterval (or real) data://]Matrix equations (rectangular)
| sumvermatreqn| on intervalsstyle="color: sum []<br/>on numbers: mpfr_vector_sum_d []green"| free
| dotcolspan="3"| on intervalsReal data only: dot []<br/>on numbers: mpfr_vector_dot_d []Uncommon problems
| sumAbsplusminusoneset| on intervalsstyle="color: sumabs []<br/>on numbers: mpfr_vector_sum_d (''rounding mode'', abs (''x''))green"| free
| sumSquareverabsvaleqn| on intervalsstyle="color: sumsq []<br/>on numbers: mpfr_vector_dot_d (''rounding mode'', abs (''x''), abs (''x''))green"| free| to be migrated
| intersectionverabsvaleqnall| intersect style="color:green"| free| depends on <code>verabsvaleqn</code>, see also [], to be migrated
| convexHullverbasintnpprob| union [httpstyle="color:red"| trapped| depends on <code style="color:red">verregsing<//]code>
| mulRevToPair
| mulrev [] with two output parameters
== Compatibility ==
The interval package's main goal is to be compliant with IEEE 1788, so it is compatible with other standard-conforming implementations (on the set of operations described by the standard document).
This interval package is ''not'' meant to be a replacement for INTLAB and any compatibility with it is pure coincidence. Since both are compatible with GNU Octave, they happen to agree on many function names and programs written for INTLAB may possibly run with this interval package as well. Some fundamental differences that I am currently aware of:
* INTLAB is non-free software, it grants none of the [ four essential freedoms] of free software
* INTLAB is not conforming to IEEE 1788 and the parsing of intervals from strings uses a different format—especially for the uncertain form
* INTLAB supports intervals with complex numbers and sparse interval matrices, but no empty intervals
* INTLAB uses inferior accuracy for most arithmetic operations, because it focuses on speed
* Basic operations can be found in both packages, but the availability of special functions depends
{{Code|In GNU Octave the interval package can also be run alongside INTLAB.|<syntaxhighlight lang="octave">
# INTLAB intervals
A1 = infsup (2, 3);
B1 = hull (-4, A1);
C1 = midrad (0, 2);
# Interval package intervals
pkg load interval
A2 = infsup (2, 3);
B2 = hull (-4, A2);
C2 = midrad (0, 2);
pkg unload interval
# Computation with INTLAB
A1 + B1 * C1
# Computation without INTLAB
A2 + B2 * C2
== Related work ==
For C++ there is an open source interval library [ libieeep1788] by Marco Nehmeier (member of IEEE P1788). It aims to be standard compliant with IEEE 1788 and is designed in a modular way, supporting several interval data types and different flavors of interval arithmetic []. The GNU Octave interval package shares several unit tests with libieeep1788.
For C++, Pascal and Fortran there is a free interval library [ XSC]. It is not standard compliant with IEEE 1788. Some parts of the GNU Octave interval package have been derived from C-XSC.
For Java there is a library [ jinterval] by Dmitry Nadezhin (member of IEEE P1788). It aims to be standard compliant with IEEE 1788, but is not complete yet.
For MATLAB there is a popular, nonfree interval arithmetic toolbox [ INTLAB] by Siegfried Rump. It had been free of charge for academic use in the past, but no longer is. Its origin dates back to 1999, so it is well tested and comprises a lot of functionality, especially for vector / matrix operations. INTLAB is compatible with GNU Octave since Version 9 [].

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