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Interval package

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The GNU Octave interval package for real-valued [ interval/ interval packagearithmetic] provides data types and fundamental operations for .* Intervals are closed, connected subsets of the real valued interval arithmetic based on the common floating-point format “binary64” anumbers. kIntervals may be unbound (in either or both directions) or empty. a. doubleIn special cases <code>+inf</code> and <code>-precisioninf</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 '''Interval arithmeticenclosure of all possible values''' produces mathematically proven numerical resultsof f over x where the function is defined. Most interval arithmetic functions in this package manage to produce a very accurate such enclosure.* The result of an interval arithmetic function is an interval in general. It aims to might happen, that the mathematical range of a function consist of several intervals, but their union will be standard compliant with the (upcoming) [http://standardsreturned,, 1 /develop/project/1788.html IEEE 1788[-1, 1] = [Entire] and therefore implements the ''set-based'' interval arithmetic flavor.
Warning__TOC__[[File: The package has not yet been releasedInterval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]<div style="clear:left"></div>
== Motivation 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]
{{quote|Give a digital computer a problem in == Development status ==* Completeness** All required functions from [ IEEE Std 1788-2015], IEEE standard for interval arithmetic, and it are implemented. The standard was approved by IEEE-SA on June 11, 2015. It will grind away methodicallyremain active for ten years. The standard was approved by ANSI in 2016.** Also, tirelesslythe minimalistic standard [ IEEE Std 1788.1-2017], at gigahertz speedIEEE standard for interval arithmetic (simplified) is fully implemented. The standard was approved by IEEE-SA on December 6, until ultimately it produces the wrong answer2017 (and published in January 2018). … An ** In addition there are functions for interval computation yields a pair of numbersmatrix arithmetic, N-dimensional interval arrays, plotting, an upper and a lower boundsolvers.* Quality** Most arithmetic operations produce tight, correctly-rounded results. That is, the smallest possible interval with double-precision (binary64) endpoints, which are guaranteed encloses the exact result.** Includes [ large test suite] for arithmetic functions** For open bugs please refer to enclose the [ answer=1 bug tracker].* Performance** All elementary functions have been [https://octave. Maybe you still don’t know org/doc/interpreter/Vectorization-and-Faster-Code-Execution.html vectorized] and run fast on large input data.** Arithmetic is performed with the truth, but at least you know how much you don’t know.|Brian Hayes|[ MPFR] library internally.1511Where possible, the optimized [http://2003web.6archive.484 DOIorg/web/20170128033523/http: 10//lipforge.ens-lyon.1511fr/www/crlibm/2003CRlibm] library is used.* Portability** Runs in GNU Octave ≥ 3.68.484]}}2** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD
{| class="wikitable" style="marginProject ideas (TODOs) ==* To be considered in the future: Algorithms can be migrated from the C-XSC Toolbox (C++ code) from [http: auto"!Standard floating point //] (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.!* Interval arithmeticversion of <code>interp1</code>|-* Extend <code>subsasgn</code> to allow direct manipulation of inf and sup (and dec) properties.| style >> A = infsup ("vertical-align: top[2, 4]" |); octave:1> 19 * 0.1 - 2 + 0> A.inf = infsup ("[1, 3]") ans A = [1, 4] 1>> A.3878e-16inf = 5| style A = "vertical-align[Empty]: top" |* While at it, also allow multiple subscripts in <code>subsasgn</code> octave>> A(:1> x )(2:4)(2) = infsup 42; # equivalent to A("03) = 42 >> A.1"inf(3)= 42;# also A(3).inf = 42 >> A.inf.inf = 42 # should produce error? octave>> A.inf.sup = 42 # should produce error?* Tight Enclosure of Matrix Multiplication with Level 3 BLAS [] []* Verified Convex Hull for Inexact Data [] [http:2> 19 //]* x Implement user- 2 + xcontrollable 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). ans = 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 [-3Inf, 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.1918911957973251e-16, +1.3877787807814457e-16234 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.
Floating point arithmetic, as specified by [ == Compatibility ==The interval package's main goal is to be compliant with IEEE 754]Std 1788-2015, so it is available in almost every computer system today. It is widecompatible with other standard-spread, implemented in common hardware and integral part in programming languages. For example, conforming implementations (on the extended precision format is set of operations described by the default numeric data type in GNU Octavestandard document). Benefits Other implementations, which are known to aim for standard conformance are obvious: The performance of arithmetic operations is well-defined, highly efficient and results are comparable between different systems.
However, there are some downsides of floating point arithmetic in practice, which will eventually produce errors in computations.* Floating point arithmetic is often used mindlessly by developers. [http IntervalArithmetic.htmljl package](Julia)* The binary data types categorically are not suitable for doing financial computations[https://github. Very often representational errors are introduced when using “real world” decimal numbers.* Even if the developer would be proficient, most developing environments com/jinterval/ technologies limit floating point arithmetic capabilities to a very limited subset of IEEE 754: Only one or two data types, no rounding modes, …jinterval JInterval library] (Java)* Results are hardly predictable. All operations produce the best possible accuracy ''at runtime'', this is how floating point works. Contrariwise, financial computer systems typically use a [httphttps://engithub.wikipedia.orgcom/wikinadezhin/Fixed-point_arithmetic fixed-point arithmeticlibieeep1788 ieeep1788 library] (COBOL, PL/I, …C++)created by Marco Nehmeier, where overflow and rounding can be precisely predicted ''at compile-time''.* If you do not know the technical details, cf. first bullet, you ignore the fact that the computer lies to you in many situations. For example, when looking at numerical output and the computer says “<code>ans = 0.1</code>,” this is not absolutely correct. In fact, the value is only ''close enough'' to the value 0.1.later forked by Dmitry Nadezhin
=== Octave Forge simp package ===In 2008/2009 there was a Single Interval arithmetic addresses above problems in its very special way and introduces new possibilities Mathematics Package (SIMP) for algorithms. For exampleOctave, the [ interval newton method] is able to find ''all'' zeros of a particular functionwhich has eventually become unmaintained at Octave Forge.
== Theory ==The simp package contains a few basic interval arithmetic operations on scalar or vector intervals. It does not consider inaccurate built-in arithmetic functions, round-off, conversion and representational errors. As a result its syntax is very easy, but the arithmetic fails to produce guaranteed enclosures.
=== Moore's fundamental theroem of interval arithmetic ===Let '''''y''''' = ''f''('''''x''''') be It is recommended to use the result ofinterval-evaluation of ''f'' over package as a box '''''x''''' = (''x''<sub>1</sub>, … replacement for simp. However, ''x''<sub>''n''</sub>)using any function names and interval versions of its component library functions. Then# In all cases, '''''y''''' contains constructors are not compatible between 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''''' = Øpackages.
== Quick start introduction =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 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 Std 1788-2015 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
=== Input and output ===
Before exercising interval arithmetic, interval objects must be created, typically from non-interval data. There are interval constants <code>empty</code> and <code>entire</code> and the class constructors <code>infsup</code> for bare intervals and <code>infsupdec</code> for decorated intervals. The class constructors are very sophisticated and can be used with several kinds of parameters: Interval boundaries can be given by numeric values or string values with decimal numbers. Also it is possible to use so called interval literals with square brackets.
octave<div style="display:flex; align-items:1flex-start"> infsup (1) ans <div style= [1] octave"margin-right:22em"> infsup (1, 2) ans {{Code|Computation with this interval package|<syntaxhighlight lang= [1, 2] "octave:3"> infsup ("3", "4") ans = [3, 4]pkg load interval octave:4> infsup ("1.1") ans A1 = [1.0999999999999998, 1.1000000000000001] octave:5> infsup ("[52, 6.5]"3); ans B1 = [5, 6.5] octave:6> infsup hull ("[5.8e-17]"4, A2); ans C1 = [5.799999999999999e-17midrad (0, 5.800000000000001e-17]2);
It is possible to access the exact numeric interval boundaries with the functions A1 + B1 * C1<code/syntaxhighlight>inf}}</code> and <codediv>sup</codediv>. The default text representation of intervals can be created {{Code|Computation with INTLAB|<code>intervaltotext</codesyntaxhighlight lang="octave">. The default text representation is not guaranteed to be exact startintlabA2 = infsup (see function <code>intervaltoexact</code>2, 3);B2 = hull (-4, because this would massively spam console output. For exampleA2);C2 = midrad (0, the exact text representation of <code>realmin</code> would be over 700 decimal places long! However, the default text representation is correct as it guarantees to contain the actual boundaries and is accurate enough to separate different boundaries.2);
octave:7A2 + B2 * C2</syntaxhighlight> infsup (1, 1 + eps) ans = [1, 1.0000000000000003]}} octave:8</div> infsup (1, 1 + 2 * eps) ans = [1, 1.0000000000000005]</div>
Warning==== Known differences ====Simple programs written for INTLAB should run without modification with this interval package. The following table lists common functions that use a different name in INTLAB.{|! 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)|} == Developer Information ===== Source Code Repository ===https: Decimal fractions should always be passed as // === Dependencies === apt-get install liboctave-dev mercurial make automake libmpfr-dev === Build ===The repository contains a string to Makefile which controls the constructorbuild process. Otherwise it is possible, that GNU Some common targets are:* <code>make release</code> Create a release tarball and the HTML documentation for [[Octave introduces conversion errors when Forge]] (takes a while).* <code>make check</code> Run the numeric literal is converted into floatingfull test-point format suite to verify that code changes didn'''before''' it is passed to t break anything (takes a while).* <code>make run</code> Quickly start Octave with minimal recompilation and functions loaded from the constructorworkspace (for interactive testing of code changes).
octave:9> infsup (<span style = "color:red">0.2</span>) ans = [.20000000000000001, .20000000000000002]'''Build dependencies''' octave:10> infsup (<span style = "color:green"code>"0.2"apt-get install libmpfr-dev autoconf automake inkscape zopfli</spancode>) ans = [.19999999999999998, .20000000000000002]
For convenience it is possible to implicitly call the interval constructor during all interval operations if at least one input already is an interval object.=== Architecture ===
octaveIn a nutshell the package provides two new data types to users:11bare intervals and decorated intervals. The data types are implemented as:* class <code> infsup </code> ("17.7"bare interval) with attributes <code>inf</code> (lower interval boundary) and <code>sup</code> (upper interval boundary) + 1 ans = [18.699999999999999, 18.700000000000003] octave:12* class <code>infsupdec</code> (decorated interval) which extends the former and adds attribute <code>dec</code> ans + "[0, 2]" ans = [18.699999999999999, 20(interval decoration).700000000000003]
=== Decorations ===Almost all functions in the package are implemented as methods of these classes, e. g. <code>@infsup/sin</code> implements the sine function for bare intervals. Most code is kept in m-files. Arithmetic operations that require correctly-rounded results are implemented in oct-files (C++ code), these are used internally by the m-files of the package. The source code is organized as follows:
=== Arithmetic operations === +- doc/ – package manual +- inst/ | +- @infsup/ | | +- infsup.m – class constructor for bare intervals | | +- sin.m – sine function for bare intervals (uses mpfr_function_d internally) | | `- ... – further functions on bare intervals | +- @infsupdec/ | | +- infsupdec.m – class constructor for decorated intervals | | +- sin.m – sine function for decorated intervals (uses @infsup/sin internally) | | `- ... – further functions on decorated intervals | `- ... – a few global functions that don't operate on intervals `- src/ +- – computes various arithmetic functions correctly rounded (using MPFR) `- ... – other oct-file sources
=== Reverse arithmetic operations Best practices ======= Parameter checking ====* All methods must check <code>nargin</code> and call <code>print_usage</code> if the number of parameters is wrong. This prevents simple errors by the user.* 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
if (not (isa (x, "infsupdec"))) x =infsupdec (x); endif if (not (isa (y, "infsupdec"))) y =infsupdec (y); endif = Numerical operations ===Use of Octave functions ====Octave functions may be used as long as they don't introduce arithmetic errors. For example, the ceil function can be used safely since it is exact on binary64 numbers. function x = ceil (x) ... parameter checking ... x.inf = ceil (x.inf); x.sup = ceil (x.sup); endfunction If Octave functions would introduce arithmetic/rounding errors, there are interfaces to MPFR (<code>mpfr_function_d</code>) and crlibm (<code>crlibm_function</code>), which can produce guaranteed boundaries. ==== Vectorization & Indexing ====All functions should be implemented using vectorization and indexing. This is very important for performance on large data. For example, consider the plus function. It computes lower and upper boundaries of the result (x.inf, y.inf, x.sup, y.sup may be vectors or matrices) and then uses an indexing expression to adjust values where empty intervals would have produces problematic values. function x = plus (x, y) ... parameter checking ... l = mpfr_function_d ('plus', -inf, x.inf, y.inf); u = mpfr_function_d ('plus', +inf, x.sup, y.sup); emptyresult = isempty (x) | isempty (y); l(emptyresult) = inf; u(emptyresult) = -inf; endfunction == VERSOFT ==The [ VERSOFT] software package (by Jiří Rohn) has been released under a free software license (Expat license) and algorithms may be migrated into the interval package. {|! Function! Status! Information|-|colspan="3"|Real (or complex) data only: Matrices|-|verbasis|style="color:red"| trapped| depends on <code style="color:red">verfullcolrank</code>|-|vercondnum|style="color:red"| trapped| depends on <code style="color:red">versingval</code>|-|verdet|style="color:red"| trapped| depends on <code>vereig</code>|-|verdistsing|style="color:red"| trapped| depends on <code style="color:red">versingval</code>|-|verfullcolrank|style="color:red"| trapped| depends on <code>verpinv</code>|-|vernorm2|style="color:red"| trapped| depends on <code style="color:red">versingval</code>|-|vernull (experimental)| unknown| depends on <code style="color:red">verlsq</code>; todo: compare with local function inside <code style="color:green">verintlinineqs</code>|-|verorth|style="color:red"| trapped| depends on <code style="color:red">verbasis</code> and <code style="color:red">verthinsvd</code>|-|verorthproj|style="color:red"| trapped| depends on <code style="color:red">verpinv</code> and <code style="color:red">verfullcolrank</code>|-|verpd|style="color:red"| trapped| depends on <code>isspd</code> (by Rump, to be checked) and <code style="color:red">vereig</code>|-|verpinv|style="color:red"| trapped| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code style="color:red">verthinsvd</code>|-|verpmat|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>|-|verrank|style="color:red"| trapped| depends on <code style="color:red">versingval</code> and <code style="color:red">verbasis</code>|-|verrref|style="color:red"| trapped| depends on <code style="color:red">verfullcolrank</code> and <code style="color:red">verpinv</code>|-|colspan="3"|Real (or complex) data only: Matrices: Eigenvalues and singular values|-|vereig|style="color:red"| trapped| depends on proprietary <code>verifyeig</code> function from INTLAB, depends on complex interval arithmetic|-|<s>vereigback</s>|style="color:green"| free, migrated (for real eigenvalues)| dependency <code>norm</code> is already implemented|-|verspectrad|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3"|Real (or complex) data only: Matrices: Decompositions|-|verpoldec|style="color:red"| trapped| depends on <code style="color:red">verthinsvd</code>|-|verrankdec|style="color:red"| trapped| depends on <code style="color:red">verfullcolrank</code> and <code style="color:red">verpinv</code>|-|verspectdec|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|verthinsvd|style="color:red"| trapped| implemented in <code>vereig</code>|-|colspan="3"|Real (or complex) data only: Matrix functions|-|vermatfun|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3"|Real data only: Linear systems (rectangular) |-|<s>verlinineqnn</s>|style="color:green"| free, migrated| use <code>glpk</code> as a replacement for <code>linprog</code>|-|verlinsys|style="color:red"| trapped| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code style="color:red">verpinv</code>, <code style="color:red">verfullcolrank</code>, and <code style="color:red">verbasis</code>|-|verlsq|style="color:red"| trapped| depends on <code style="color:red">verpinv</code> and <code style="color:red">verfullcolrank</code>|-|colspan="3"|Real data only: Optimization|-|verlcpall|style="color:green"| free| depends on <code>verabsvaleqnall</code>|-|<s>verlinprog</s>|style="color:green"| free, migrated| use <code>glpk</code> as a replacement for <code>linprog</code>; dependency <code>verifylss</code> is implemented as <code>mldivide</code>|-|verlinprogg|style="color:red"| trapped| depends on <code>verfullcolrank</code>|-|verquadprog| unknown| use <code>quadprog</code> from the optim package; use <code>glpk</code> as a replacement for <code>linprog</code>; dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code>isspd</code> (by Rump, to be checked, algorithm in [])|-|colspan="3"|Real (or complex) data only: Polynomials|-|verroots|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3"|Interval (or real) data: Matrices|-|verhurwstab|style="color:red"| trapped| depends on <code style="color:red">verposdef</code>|-|verinverse|style="color:green"| free| depends on <code style="color:green">verintervalhull</code>, to be migrated|-|<s>verinvnonneg</s>|style="color:green"| free, migrated|-|verposdef|style="color:red"| trapped| depends on <code>isspd</code> (by Rump, to be checked) and <code style="color:red">verregsing</code>|-|verregsing|style="color:red"| trapped| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code>isspd</code> (by Rump, to be checked) and <code>verintervalhull</code>; see also []|-|colspan="3"|Interval (or real) data: Matrices: Eigenvalues and singular values|-|vereigsym|style="color:red"| trapped| main part implemented in <code>vereig</code>, depends on <code style="color:red">verspectrad</code>|-|vereigval|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>|-|<s>vereigvec</s>|style="color:green"| free, migrated|-|verperrvec|style="color:green"| free| the function is just a wrapper around <code style="color:green">vereigvec</code>?!?|-|versingval|style="color:red"| trapped| depends on <code style="color:red">vereigsym</code>|-|colspan="3"|Interval (or real) data: Matrices: Decompositions|-|verqr (experimental)|style="color:green"| free| <code>qr</code> has already been implemented using the Gram-Schmidt process, which seems to be more accurate and faster than the Cholsky decomposition or Householder reflections used in verqr. No migration needed.|-|<s>verchol (experimental)</s>|style="color:green"| free, migrated| migrated version has been named after the standard Octave function <code>chol</code>|-|colspan="3"|Interval (or real) data: Linear systems (square)|-|verenclinthull|style="color:green"| free| to be migrated|-|verhullparam|style="color:green"| free| depends on <code>verintervalhull</code>, to be migrated|-|verhullpatt|style="color:green"| free| depends on <code>verhullparam</code>, to be migrated|-|verintervalhull|style="color:green"| free| to be migrated|-|colspan="3"|Interval (or real) data: Linear systems (rectangular)|-|verintlinineqs|style="color:green"| free| depends on <code style="color:green">verlinineqnn</code>|-|veroettprag|style="color:green"| free|-|vertolsol|style="color:green"| free| depends on <code style="color:green">verlinineqnn</code>|-|colspan="3"|Interval (or real) data: Matrix equations (rectangular)|-|vermatreqn|style="color:green"| free|-|colspan="3"|Real data only: Uncommon problems|-| plusminusoneset|style="color:green"| free|-| verabsvaleqn|style="color:green"| free| to be migrated|-| verabsvaleqnall|style="color:green"| free| depends on <code>verabsvaleqn</code>, see also [], to be migrated|-| verbasintnpprob|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>|-|}
=== Boolean operations ===