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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]
* Free software licensed under the terms of the GNU General Public License* Many interval arithmetic functions [] with high, system-independent accuracy* Conforming to IEEE 1788 []* Support for interval vectors and interval matrices** very accurate vector sum, vector dot and matrix multiplication (< 1 ULP)** fast matrix multiplication and fast solver for dense linear systems (BLAS routines)** vectorized function evaluation** broadcasting* Easy usage: Convenient interval constructors and GNU Octave function names === Limitations ===* No complex numbers* No sparse matrices (maybe in the future, if requested by users)* No multidimensional arrays (maybe in the future, if requested by users) === Dependencies ===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. == Development Status status ==
* Completeness
** All required functions from [ IEEE Std 1788 functions -2015], IEEE standard for interval arithmetic, are implemented. The standard was approved by IEEE-SA on June 11, the 2015. It will remain active for ten years. The standard is currently was approved by ANSI in recirculation ballot phase and quite stable2016.** PlannedAlso, the minimalistic standard [https: [[GSoC_Project_Ideas#Interval_package|more solvers; plotting functions]// IEEE Std 1788.1-2017], IEEE standard for interval arithmetic (simplified) is fully implemented. The standard was approved by IEEE-SA on December 6, 2017 (and published in January 2018).** Planned: User documentation included in the packageIn 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 tests [ large test suite] for all functions, many tests for basic arithmetic functions** No known For open bugs please refer to the [ bug tracker].* Performance** All elementary functions have been [ vectorized] and run fast on large input data. The package ** Arithmetic is quite new and still has a small user baseperformed with the [ GNU MPFR] library internally. Where possible, so there might be hidden bugsthe optimized [ Also some advanced functions need more testingfr/www/crlibm/ CRlibm] library is used.
* Portability
** Runs in GNU Octave 3.8.2 and 4.0 release candidates** Runs Known to run under GNU/Linux, Microsoft Windows (install from [[Octave-Forge#Installing_packages|Octave Forge]]), macOS, and Mac OS X ([[Create_a_MacOS_X_App_Bundle_Using_MacPorts|MacPorts]])FreeBSD
==Project ideas (TODOs) = What =* 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 expect 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 [] []The * Implement user-controllable output from the interval arithmetic provided 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 this output_precision (...) / `format long` / `format short`) d) It should be possible to output the bounds of an interval package is '''slow'''without punctuation, e.g., 1.234 2.345 instead of [1.234, 2.345]. For instance, but '''accurate'''this might be a convenient way to write intervals to a file for use by another application.
''Why is the == Compatibility ==The interval package slow?''All arithmetic interval operations are simulated in high-level octave language using C99 or multi-precision floating-point routines, which s main goal is a lot slower than a hardware implementation []. Building interval arithmetic operations from floatingto be compliant with IEEE Std 1788-point routines is easy for simple monotonic functions2015, e. g., addition and subtraction, but so it is complex for others, e. g., [ with other standard-power-2011conforming implementations (on the set of operations described by the standard document).htm interval power function]Other implementations, atan2, or [[#Reverse_arithmetic_operations|reverse functions]]. For some interval operations it is not even possible which are known to rely on floating-point routines, since not all required routines aim for standard conformance are available in C99 or BLAS.:
For example* [ IntervalArithmetic.jl package] (Julia)* [ JInterval library] (Java)* [ ieeep1788 library] (C++) created by Marco Nehmeier, multiplication of matrices with many elements becomes unfeasible as it takes a lot of time.later forked by Dmitry Nadezhin
{| class="wikitable" style="margin: auto; align:right"|+ Approximate runtime (wall clock time in seconds) <span style="font-weight: normal">— Results have been produced with GNU Octave 3.8.2 and Interval Forge simp package 0.1.4 on an Intel® Core™ i5-4340M CPU (2.9–3.6 GHz)</span>!rowspan="2"|Interval matrix size!valign="top" style="min-width:6em"|<code>plus</code>!valign="top" style="min-width:6em"|<code>times</code>!valign="top" style="min-width:6em"|<code>log<In 2008/code>!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>!valign="top" style="min-width:6em"|<code>inv</code>|-!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"|tightest<br/>accuracy!style="font-weight:normal"|valid<br/>accuracy!style="font-weight:normal"|tightest<br/>accuracy!style="font-weight:normal"|valid<br/>accuracy|-| 10 × 10|align="right"| < 0.001|align="right"| < 0.001|align="right"| 0.001|align="right"| 0.008|align="right"| 0.001|align="right"| 02009 there was a Single Interval Mathematics Package (SIMP) for Octave, which has eventually become unmaintained at Octave Forge.002|align="right"| 0.025|-| 100 × 100|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"| 0.30|-| 500 × 500|align="right"| 0.060|align="right"| 0.24|align="right"| 1.3|align="right"| 15|align="right"| 0.30|align="right"| 63|align="right"| 4.2|-|}
''Why is the interval The simp package accurate?''The GNU Octave built-in floating-point routines are not useful for contains a few basic interval arithmetic: Their results depend operations on hardware, system libraries and compilation optionsscalar or vector intervals. The interval package handles all It does not consider inaccurate built-in arithmetic functions with the help of the [ GNU MPFR library]. With MPFR it is possible to compute system, round-independentoff, valid conversion and tight enclosures of the correct results for most functionsrepresentational errors. HoweverAs a result its syntax is very easy, it should be noted that some reverse operations and matrix operations do not exists in GNU MPFR and therefore cannot be computed with but the same accuracyarithmetic fails to produce guaranteed enclosures.
== Motivation ==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.
{{quote|Give a digital computer === INTLAB ===This interval package is ''not'' meant to be a problem in arithmetic, replacement for INTLAB and any compatibility with it will grind away methodicallyis pure coincidence. Since both are compatible with GNU Octave, tirelesslythey 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, at gigahertz speed, until ultimately it produces the wrong answer. … An interval computation yields a pair grants none of numbers, an upper and a lower bound, which are guaranteed to enclose the exact answer. Maybe you still don’t know the truth, but at least you know how much you don’t know.|Brian Hayes|[ DOI: 10free-sw.1511/2003.6.484html four essential freedoms]}}of free software {| class="wikitable" style="margin: auto"!Standard floating point arithmetic!Interval arithmetic|* INTLAB is not conforming to IEEE Std 1788-2015 and the parsing of intervals from strings uses a different format—especially for the uncertain form| style = "vertical-align: top" |* INTLAB supports intervals with complex numbers and sparse interval matrices, but no empty intervals <span style="opacity:.5">octave:1> </span>19 * 0.1 - 2 + 0.1INTLAB uses inferior accuracy for most arithmetic operations, because it focuses on speed ans = 1.3878e-16| style = "vertical-align: top" | <span style="opacity:.5">octave:1> </span>x = infsup ("0.1"); <span style="opacity:.5">octave:2> </span>19 * x - 2 + x ans ⊂ [-3.191891195797326e-16Basic operations can be found in both packages, +1.3877787807814457e-16]|}but the availability of special functions depends
Floating-point arithmetic, as specified by [ IEEE 754], is available in almost every computer system today. It is wide-spread, implemented in common hardware and integral part in programming languages. For example, the double-precision format is the default numeric data type in GNU Octave. Benefits are obvious: The results of arithmetic operations are well-defined and comparable between different systems and computation is highly efficient.
However, there are some downsides of floating-point arithmetic in practice, which will eventually produce errors in computations.* Floating<div style="display:flex; align-point arithmetic is often used mindlessly by developers. [httpitems://"><div style="margin-3568/ncg_goldberg.html] [httpright://] []2em">* The binary data types categorically are not suitable for doing financial computations. Very often representational errors are introduced when using “real world” decimal numbers. []{{Code|Computation with this interval package|<syntaxhighlight lang="octave">* Even if the developer would be proficient, most developing environments / technologies limit floating-point arithmetic capabilities to a very limited subset of IEEE 754: Only one or two data types, no rounding modes, missing functions … []pkg load interval* Results are hardly predictable. [] All operations produce the best possible accuracy ''at runtime'', this is how a floating point works. Contrariwise, financial computer systems typically use a [ fixed-point arithmetic] A1 = infsup (COBOL, PL/I2, 3), where overflow and rounding can be precisely predicted ''at compile-time''.;* Results are system dependent. All but the most basic floating-point operations are not guaranteed to be accurate and produce different results depending on low level libraries and hardware. [] []* 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 B1 = 0.1</code>,” this is not absolutely correct. In fact, the value is only ''close enough'' to the value 0.1. Additionally, many functions produce limit values hull (∞ × −∞ = −∞-4, ∞ ÷ 0 A2);C1 = ∞, ∞ ÷ −0 = −∞, log midrad (0) = −∞), which is sometimes (but not always!2) useful when overflow and underflow occur.;
Interval arithmetic addresses above problems in its very special way and introduces new possibilities for algorithms. For example, the [ interval newton method] is able to find ''all'' zeros of a particular function. == Theory == === Online Introductions === [ Interval analysis in MATLAB] Note: The INTLAB toolbox for Matlab is not entirely compatible with this interval package for GNU Octave, cf. [[#Compatibility]]. However, basic operations can be compared and should be compatible for common intervals. === Moore's fundamental theroem of 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 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. == Quick start introduction == === Input and output ===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 intervals for performing interval arithmetic|<syntaxhighlight lang="octave">infsupdec (1) # [1]_cominfsupdec (1, 2) # [1, 2]_cominfsupdec ("3", "4") # [3, 4]_cominfsupdec ("1.1") # [1.0999999999999998, 1.100000000000001]_cominfsupdec ("5.8e-17") # [5.799999999999999e-17, 5.800000000000001e-17]_commidrad (12, 3) # [9, 15]_commidrad ("4.2", "1e-7") # [4.199999899999999, 4.200000100000001]_comhull (3, 42, "19.3", "-2.3") # [-2.300000000000001, A1 +42]_comhull ("pi", "e") # [2.718281828459045, 3.1415926535897936]_comB1 * C1
{{Code|Computation with INTLAB|<syntaxhighlight lang="octave">
A2 = infsup (2, 3);
B2 = hull (-4, A2);
C2 = midrad (0, 2);
The default text representation of intervals is not guaranteed to be exact, because this would massively spam console output. For example, the exact text representation of <code>realmin</code> would be over 700 decimal places long! However, the default text representation is correct as it guarantees to contain the actual boundaries. {{Warning|Decimal fractions as well as numbers of high magnitude (> 2<sup>53</sup>) should always be passed as a string to the constructor. Otherwise it is possible, that GNU Octave introduces conversion errors when the numeric literal is converted into floating-point format '''before''' it is passed to the constructor.}} {{Code|Beware of the conversion pitfall|<syntaxhighlight lang="octave">## The numeric constant “0.3” is an approximation of the## decimal number 0.3. An interval around this approximation## will not contain the decimal number 0.3.infsupdec (0.3) # [0.29999999999999998, 0.29999999999999999]_com## However, passing the decimal number 0.3 as a string## to the interval constructor will create an interval which## actually encloses the decimal number.infsupdec ("0.3") # [0.29999999999999998, 0.3000000000000001]_comA2 + B2 * C2
==== Interval vectors and matrices Known differences ====Vectors and matrices of intervals can be created by passing numerical matrices, char vectors or cell arrays to the Simple programs written for INTLAB should run without modification with this interval constructorspackage. With cell arrays it is also possible to mix several types of boundariesThe following table lists common functions that use a different name in INTLAB.{|! interval package! INTLAB|-| infsup (x)| intval (x)|-| wid (x)| diam (x)|-| subset (a, b)| in (a, b)|-| interior (a, b)| in0 (a, b)|-| isempty (x)| isnan (x)|-| disjoint (a, b)| emptyintersect (a, b)|-| hdist (a, b)| qdist (a, b)|-| disp (x)| disp2str (x)|-| infsup (s)| str2intval (s)|-| isa (x, "infsup")| isintval (x)|}
Interval matrices behave like normal matrices in GNU Octave and can be used for broadcasting and vectorized function evaluation== Developer Information ===== Source Code Repository ===
{{Code|Create interval matrices|<syntaxhighlight lang="octave">M = infsup (magic (3))= Dependencies === # [8] [1] [6] # [3] [5] [7] # [4] [9] [2]infsup (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; "4/7", "pi"}, {2, 1; "e", "0xff"}) # [1, 2] [2.220446049250313e apt-get install liboctave-dev mercurial make automake libmpfr-16, 1] # [0.5714285714285713, 2.7182818284590456] [3.141592653589793, 255]</syntaxhighlight>}}dev
=== Arithmetic operations Build ===The interval packages comprises many interval arithmetic operationsrepository contains a Makefile which controls the build process. A complete list can be found in its [httpSome common targets are:* <code>make release</code> Create a release tarball and the HTML documentation for [[Octave Forge]] (takes a while).* <code>make check</octavecode> Run the full test-suite to verify that code changes didn't break anything (takes a while)* <code>make run</interval/overview.html function reference]. Function names match GNU code> Quickly start Octave standard with minimal recompilation and functions where applicable and follow recommendations by IEEE 1788 otherwise, cf. [[#IEEE_1788_index|IEEE 1788 index]]loaded from the workspace (for interactive testing of code changes).
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 autoconf automake inkscape zopfli</code>
By default arithmetic functions are computed with best possible accuracy (which is more than what is guaranteed by GNU Octave core functions). 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 command.=== Architecture ===
{{Code|Examples of using interval arithmetic functions|In a nutshell the package provides two new data types to users: bare intervals and decorated intervals. The data types are implemented as:* class <code>infsup<syntaxhighlight lang="octave"/code>sin (infsupdec (0.5bare interval)) # [.4794255386042029, .4794255386042031]_compow with attributes <code>inf</code> (infsupdec (2lower interval boundary), infsupdec and <code>sup</code> (3, 4upper interval boundary)) # [8, 16]_comatan2 (* class <code>infsupdec </code> (1decorated interval), infsupdec which extends the former and adds attribute <code>dec</code> (1)) # [.7853981633974482, .7853981633974484]_commidrad (magic (3), 0.5interval decoration) * pascal (3) # [13.5, 16.5]_com [25, 31]_com [42, 52]_com # [13.5, 16.5]_com [31, 37]_com [55, 65]_com # [13.5, 16.5]_com [25, 31]_com [38, 48]_com</syntaxhighlight>}}
=== Numerical operations ===Some operations on intervals do not return an interval enclosureAlmost all functions in the package are implemented as methods of these classes, but a single number (in double-precision)e. g. Most important are <code>inf@infsup/sin</code> and <implements the sine function for bare intervals. Most code>sup</is kept in m-files. Arithmetic operations that require correctly-rounded results are implemented in oct-files (C++ code>), which return these are used internally by the m-files of the lower and upper interval boundariespackage.The source code is organized as follows:
More such operations are <code>mid< +- doc/code> – package manual +- inst/ | +- @infsup/ | | +- infsup.m – class constructor for bare intervals | | +- sin.m – sine function for bare intervals (approximation of the interval's midpointuses mpfr_function_d internally), <code>wid< | | `- ... – further functions on bare intervals | +- @infsupdec/code> | | +- infsupdec.m – class constructor for decorated intervals | | +- sin.m – sine function for decorated intervals (approximation of the interval's width), <code>rad<uses @infsup/code> (approximation of the interval's radiussin internally), <code>mag</code> (interval | | `- ... – further functions on decorated intervals | `- ... – a few global functions that don's magnitude) and <code>mig<t operate on intervals `- src/code> +- – computes various arithmetic functions correctly rounded (interval's mignitudeusing MPFR) `- ... – other oct-file sources
=== Boolean operations Best practices ===Interval comparison operations produce boolean results. While some comparisons are especially for intervals (subset, interior, ismember, isempty, disjoint, …) others are extensions of simple numerical comparison. For example, the less-or-equal comparison is mathematically defined as ∀==== Parameter checking ====* All methods must check <subcode>''a''nargin</subcode> ∃and call <subcode>''b''print_usage</subcode> ''a'' ≤ ''b'' ∧ ∀<sub>''b''</sub> ∃<sub>''a''</sub> ''a'' ≤ ''b''if the number of parameters is wrong. This prevents simple errors by the user.* Methods with more than 1 parameter must convert non-interval parameters to intervals using the class constructor. This allows the user to mix non-interval parameters with interval parameters and the function treats any inputs as intervals. Invalid values will be handled by the class constructors. if (not (isa (x, "infsup"))) x = infsup (x); endif if (not (isa (y, "infsup"))) y = infsup (y); endif
<span style=if (not (isa (x, "opacity:.5infsupdec">octave:1> </span>infsup ))) x = infsupdec (x); endif if (1not (isa (y, 3"infsupdec") <)) y = infsup infsupdec (2, 4y); ans = 1endif
=== Matrix operations = Use of Octave functions ====Above mentioned operations Octave functions may be used as long as they don't introduce arithmetic errors. For example, the ceil function can also be applied element-wise to interval vectors and matricesused safely since it is exact on binary64 numbers. function x = ceil (x) ... Many operations use [http://wwwparameter checking vectorization techniques] x.inf = ceil (x.inf); x.sup = ceil (x.sup); endfunction
In additionIf Octave functions would introduce arithmetic/rounding errors, there are matrix operations on interval matrices. These operations comprise: dot product, matrix multiplication, vector sums interfaces to MPFR (all with tightest accuracy<code>mpfr_function_d</code>), matrix inversion, matrix powers, and solving linear systems crlibm (the latter are less accurate<code>crlibm_function</code>). As a result of missing hardware / low-level library support and missing optimizations, these operations are relatively slow compared to familiar operations in floating-point arithmeticwhich can produce guaranteed boundaries.
{{Code|Examples of using interval matrix functions|<syntaxhighlight lang="octave">=== Vectorization & Indexing ====A = infsup 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 ([1x.inf, 2y.inf, 3; 4x.sup, 0, 0; 0, 0, 1]y.sup may be vectors or matrices); A and then uses an indexing expression to adjust values where empty intervals would have produces problematic values. function x = plus (2x, 3y) = "[0, 6]" # [1] [2] [3]... parameter checking ... # [4] [0] [0, 6] # [0] [0] [1]B l = inv mpfr_function_d (A) # [0] [0.25] ['plus', -1inf, x.5inf, 0]y.inf); # [0.5] [-0.125] [-1u = mpfr_function_d ('plus', +inf, x.5sup, -0y.75]sup); # [0] [0] [1]emptyresult = isempty (x) | isempty (y);A * B l(emptyresult) = inf; # [1] [0] [u(emptyresult) = -1.5, +1.5]inf; # [0] [1] [-6, +6] # [0] [0] [1] endfunction
A = infsup (magic (3)) # [8] [1] [6] # [3] [5] [7] # [4] [9] [2]c = A \ [3; 4; 5]VERSOFT == # The [0.18333333333333315, 0http://uivtx.18333333333333358] # [0cs.43333333333333318, 0cas.43333333333333346cz/~rohn/matlab/ VERSOFT] # [0.18333333333333307, 0software package (by Jiří Rohn) has been released under a free software license (Expat license) and algorithms may be migrated into the interval package.18333333333333355]A * c # [2.9999999999999964, 3.0000000000000036] # [3.9999999999999964, 4.000000000000004] # [4.999999999999997, 5.000000000000003]</syntaxhighlight>}}
{|! 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= Notes "color:red"| trapped| depends on linear systems <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"| trappedA system of linear equations in the form A''x'' | depends on <code style= 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 "color:red">versingval</code>|-|vernull (possibly severalexperimental) linear systems. Some characteristics should definitely be known when linear interval systems are solved| unknown| depends on <code style="color:red">verlsq</code>; todo: compare with local function inside <code style="color:green">verintlinineqs</code>|-|verorth|style="color:red"| trapped| depends on <code style="color:red">verbasis</code> and <code style="color:red">verthinsvd</code>|-|verorthproj|style="color:red"| trapped* If the linear system is underdetermined | depends on <code style="color:red">verpinv</code> and has infinitely many solutions, the interval solution will be unbound in at least one of its coordinates. Contrariwise, from an unbound result it can not be concluded whether the linear system is underdetermined or has solutions.<code style="color:red">verfullcolrank</code>|-|verpd|style="color:red"| trapped* If the interval result is empty in at least one of its coordinates| depends on <code>isspd</code> (by Rump, the linear system is guaranteed to be underdetermined checked) and has no solutions. Contrariwise, from a non<code style="color:red">vereig</code>|-empty result it can not be concluded whether all or some of the systems have solutions or not.* Wide intervals within the matrix A can easily lead to a superposition of cases, where the rank of A |verpinv|style="color:red"| trapped| dependency <code>verifylss</code> is no longer unique. If the linear interval system contains cases of linear independent equations implemented as well as linear dependent equations, the resulting enclosure of solutions will inevitably be very broad.<code>mldivide</code>; depends on <code style="color:red">verthinsvd</code>|-|verpmat|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>However, solving linear systems with interval arithmetic can produce useful results in many cases and automatically carries a guaranty for error boundaries. Additionally, it can give better information than the floating|-point variants for some cases.|verrank|style="color:red"| trapped{{Code|Standard floating point arithmetic versus interval arithmetic depends on ill<code style="color:red">versingval</code> and <code style="color:red">verbasis</code>|-conditioned linear systems|verrref|style="color:red"| trapped|depends on <syntaxhighlight langcode style="octavecolor:red">A verfullcolrank</code> and <code style= [1, 0; 2, 0];"color:red">verpinv</code>## This linear system has no solutions|-A \ [|colspan="3; 0] # warning"|Real (or complex) data only: Matrices: matrix Eigenvalues and singular to machine precision, rcond = 0values # 0.60000|- # 0.00000|vereig## This linear system has many solutions|style="color:red"| trappedA \ [4; 8]| depends on proprietary <code>verifyeig</code> function from INTLAB, depends on complex interval arithmetic # 4|- # 0|<s>vereigback</s>|style="color:green"| free, migrated (for real eigenvalues)## The empty interval vector proves that there | dependency <code>norm</code> is no solutionalready implemented|-infsup (A) \ [3; 0]|verspectrad # [Empty]|style="color:red"| trapped # [Empty]| main part implemented in <code>vereig</code>## The unbound interval vector indicates that there may be many solutions|-infsup |colspan="3"|Real (Aor complex) \ [4; 8]data only: Matrices: Decompositions|- # [4]|verpoldec # [Entire]|style="color:red"| trapped| depends on <code style="color:red">verthinsvd</syntaxhighlightcode>}}|-|verrankdec|style="color:red"| trapped| depends on <code style= Advanced topics "color:red">verfullcolrank</code> and <code style="color:red">verpinv</code>|-|verspectdec|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|verthinsvd|style="color:red"| trapped| implemented in <code>vereig</code>|-|colspan="3"|Real (or complex) data only: Matrix functions|-|vermatfun|style= Error handling ==="color:red"| trapped| main part implemented in <code>vereig</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 |colspan="3"|Real data types. Arithmetic operations which are not defined for only: Linear systems (parts ofrectangular) their input|-|<s>verlinineqnn</s>|style="color:green"| free, simply ignore anything that is outside of their domain.migrated| use <code>glpk</code> as a replacement for <code>linprog</code>|-|verlinsysHowever, the interval constructors can produce errors depending |style="color:red"| trapped| 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 (preferred) <codestyle="color:red">infsupdecverfullcolrank</code>, and <code style="color:red">verbasis</code>|-|verlsq|style="color:red"| trapped| depends on <codestyle="color:red">midradverpinv</code> and <codestyle="color:red">hullverfullcolrank</code> constructors will |-|colspan="3"|Real data only issue : 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 warning and return a [NaI] object, which will propagate and survive through computations. NaI stands replacement for “not an interval”.<code>linprog</code>; dependency <code>verifylss</code> is implemented as <code>mldivide</code>|-{{Code|Effects of set-based interval arithmetic on partial functions and the NaI objectverlinprogg|<syntaxhighlight langstyle="octavecolor:red"| trapped| depends on <code>verfullcolrank</code>## Evaluation of |-|verquadprog| unknown| use <code>quadprog</code> from the optim package; use <code>glpk</code> as a function outside of its domain returns an empty intervalinfsupdec replacement for <code>linprog</code>; dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code>isspd</code> (2) by Rump, to be checked, algorithm in [ 0 # [EmptyRu06c.pdf]_trv)|-infsupdec |colspan="3"|Real (0or complex) ^ infsupdec (0) # [Empty]_trvdata only: Polynomials|-|verroots|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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