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Interval package

12,662 bytes added, 12 August
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The {{OctaveForge| name = interval| logo = [[File:Interval.png|100px]]| short description = Real-valued interval arithmetic.| version = 3.2.0| date = 2018-07-01| author 1 name = Oliver Heimlich| author 1 email = <oheim@posteo.de>| maintainer 1 name = Oliver Heimlich| maintainer 1 email = <oheim@posteo.de>| license = GPL-3.0+| group = Community package| doc 1 = [https://octave.sourceforge.netio/interval/overview.html Function reference]| doc 2 = [https:/p/octave.sourceforge.io/interval/ package_doc/ User manual]| download 1 = [https://octave.sourceforge.io/download.php?package=interval package] provides data types and fundamental operations for real valued -3.2.0.tar.gz interval arithmetic based on the common floating-point format “binary64” a3.2. k0. atar. doublegz]| repository 1 = https://octave.sourceforge.io/pkg-precisionrepository/interval/| dependency 1 = octave &ge; 3. 8.0| dependency 2 = '''Interval arithmeticRuntime:''' produces mathematically proven numerical resultsmpfr (&ge; 3. It aims to be standard compliant with the (upcoming1.0) [httphttps://standardspackages.ieeedebian.org/develop/project/1788.html IEEE 1788search?keywords=libmpfr4 libmpfr4 for Debian] and therefore implements the | dependency 3 = '''Build:'set-based'' interval arithmetic flavormpfr (&ge; 3.1.0) [https://packages.debian.org/search?keywords=libmpfr-dev libmpfr-dev for Debian]}}
WarningThe GNU Octave interval package for real-valued [https://en.wikipedia.org/wiki/Interval_arithmetic interval arithmetic].* Intervals are closed, connected subsets of the real numbers. Intervals may be unbound (in either or both directions) or empty. In special cases <code>+inf</code> and <code>-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 has not yet been releasedmanage to produce a very accurate such enclosure.* The result of an interval arithmetic function 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].
[[File:Interval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]<div style== Motivation =="clear:left"></div>
{{quote|Give a digital computer a problem in arithmetic== Distribution ==* [https://tracker.debian.org/pkg/octave-interval Debian GNU/Linux], and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer[https://launchpad. … An net/ubuntu/+source/octave-interval computation yields a pair of numbers, an upper and a lower bound, which are guaranteed to enclose the exact answerLaunchpad Ubuntu]* [https://aur. Maybe you still don’t know the truth, but at least you know how much you don’t knowarchlinux.|Brian Hayes|org/packages/octave-interval/ archlinux user repository]* Included in [httphttps://dxftp.doignu.org/10gnu/octave/windows/ official Windows installer] and installed automatically with Octave (since version 4.0.15111)* [https://github.com/macports/macports-ports/tree/master/math/octave-interval/2003MacPorts] for Mac OS X* [https://www.6freshports.484 DOIorg/math/octave-forge-interval/ FreshPorts] for FreeBSD* [https: 10//cygwin.1511com/2003cgi-bin2/package-grep.6cgi?grep=octave-interval Cygwin] for Windows* [https://build.484opensuse.org/package/show/science/octave-forge-interval openSUSE build service]}}
{| class="wikitable" style="margin: auto"Development status ==!Standard floating point arithmetic* Completeness!Interval ** All required functions from [https://standards.ieee.org/findstds/standard/1788-2015.html IEEE Std 1788-2015], IEEE standard for interval arithmetic|, are implemented. The standard was approved by IEEE-SA on June 11, 2015. It will remain active for ten years. The standard was approved by ANSI in 2016.| style = "vertical-align: top" | octave** Also, the minimalistic standard [https://standards.ieee.org/findstds/standard/1788.1> 19 * 0-2017.html IEEE Std 1788.1 - 2 + 02017], IEEE standard for interval arithmetic (simplified) is fully implemented. The standard was approved by IEEE-SA on December 6, 2017 (and published in January 2018).** In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.1 ans = 1* Quality** Most arithmetic operations produce tight, correctly-rounded results.3878eThat is, the smallest possible interval with double-16precision (binary64) endpoints, which encloses the exact result.| style = "vertical-align** Includes [https: top" |//github.com/oheim/ITF1788 large test suite] for arithmetic functions octave** For open bugs please refer to the [https://savannah.gnu.org/search/?words=forge+interval&type_of_search=bugs&only_group_id=1925&exact=1> x = infsup ("0bug tracker].1"); * Performance** All elementary functions have been [https://octave:2> 19 * x .org/doc/interpreter/Vectorization-and-Faster-Code- 2 + xExecution.html vectorized] and run fast on large input data. ans = ** Arithmetic is performed with the [-3http://www.mpfr.org/ GNU MPFR] library internally.1918911957973251e-16Where possible, +1the optimized [http://web.archive.org/web/20170128033523/http://lipforge.3877787807814457eens-16lyon.fr/www/crlibm/ CRlibm]library is used.* Portability** Runs in GNU Octave ≥ 3.8.2|}** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD
Floating point == Project ideas (TODOs) ==* To be considered in the future: Algorithms can be migrated from the C-XSC Toolbox (C++ code) from [http://www2.math.uni-wuppertal.de/wrswt/xsc/cxsc_new.html] (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.* Interval version of <code>interp1</code>* Extend <code>subsasgn</code> to allow direct manipulation of inf and sup (and dec) properties. >> A = infsup ("[2, 4]"); >> A.inf = infsup ("[1, 3]") A = [1, 4] >> A.inf = 5 A = [Empty]:* While at it, as specified by also allow multiple subscripts in <code>subsasgn</code> >> A(:)(2:4)(2) = 42; # equivalent to A(3) = 42 >> A.inf(3) = 42; # also A(3).inf = 42 >> A.inf.inf = 42 # should produce error? >> A.inf.sup = 42 # should produce error?* Tight Enclosure of Matrix Multiplication with Level 3 BLAS [http://kam.mff.cuni.cz/conferences/swim2015/abstracts/ozaki.pdf] [http://kam.mff.cuni.cz/conferences/swim2015/slides/ozaki.pdf]* Verified Convex Hull for Inexact Data [http://kam.mff.cuni.cz/conferences/swim2015/abstracts/ohta.pdf] [http://enkam.wikipediamff.orgcuni.cz/conferences/swim2015/wikislides/IEEE_floating_point IEEE 754ohta.pdf]* Implement user-controllable output from the interval standard (e. g. via printf functions): a) It should be possible to specify the preferred overall field width (the length of s). b) It should be possible to specify how Empty, is available in almost every computer system todayEntire and NaI are output, e.g. It is wide, whether lower or upper case, and whether Entire becomes [Entire] or [-spreadInf, implemented in common hardware and integral part in programming languagesInf]. c) For examplel and u, it should be possible to specify the field width, and the extended precision number of digits after the point or the number of significant digits. (partly this is already implemented by output_precision (...) / `format is long` / `format short`) d) It should be possible to output the default numeric data type in GNU Octavebounds of an interval without punctuation, e.g., 1.234 2. Benefits are obvious: The performance 345 instead of arithmetic operations is well-defined[1.234, 2.345]. For instance, highly efficient and results are comparable between different systemsthis might be a convenient way to write intervals to a file for use by another application.
However, there are some downsides of floating point arithmetic in practice, which will eventually produce errors in computations.== Compatibility ==* Floating point arithmetic The interval package's main goal is often used mindlessly by developers. [http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html]* 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 to be proficient, most developing environments / technologies limit floating point arithmetic capabilities to a very limited subset of compliant with IEEE 754: Only one or two data typesStd 1788-2015, no rounding modes, …* Results are hardly predictable. All operations produce the best possible accuracy ''at runtime'', this so it is how floating point works. Contrariwise, financial computer systems typically use a [http://en.wikipedia.org/wiki/Fixedcompatible with other standard-point_arithmetic fixed-point arithmetic] conforming implementations (COBOL, PL/I, …), where overflow and rounding can be precisely predicted ''at compile-time''.* If you do not know on the technical details, cf. first bullet, you ignore set of operations described by the fact that the computer lies to you in many situationsstandard document). For exampleOther implementations, when looking at numerical output and the computer says “<code>ans = 0.1</code>,” this is not absolutely correct. In fact, the value is only ''close enough'' which are known to the value 0.1.aim for standard conformance are:
Interval arithmetic addresses above problems in its very special way and introduces new possibilities for algorithms* [https://github.com/JuliaIntervals/IntervalArithmetic. For example, the jl IntervalArithmetic.jl package] (Julia)* [httphttps://engithub.wikipediacom/jinterval/jinterval JInterval library] (Java)* [https://github.orgcom/wikinadezhin/Interval_arithmetic#Interval_Newton_method interval newton methodlibieeep1788 ieeep1788 library] is able to find ''all'' zeros of a particular function.(C++) created by Marco Nehmeier, later forked by Dmitry Nadezhin
== Theory =Octave Forge simp package ===In 2008/2009 there was a Single Interval Mathematics Package (SIMP) for Octave, which has eventually become unmaintained at Octave Forge.
=== Moore's fundamental theroem of The simp package contains a few basic interval arithmetic ===Let '''''y''''' = ''f''('''''x''''') be the result ofinterval-evaluation of ''f'' over a box '''''x''''' = (''x''<sub>1</sub>, … , ''x''<sub>''n''</sub>)using any interval versions of its component library functions. Then# In all cases, '''''y''''' contains the range of ''f'' over '''''x''''', that is, the set of ''f''('''''x''''') at points of '''''x''''' where it is defined: '''''y''''' ⊇ Rge(''f'' | '''''x''''') = {''f''(''x'') | ''x'' ∈ '''''x''''' ∩ Dom(''f'') }# If also each library operation in ''f'' is everywhere defined on its inputs, while evaluating '''''y''''', then ''f'' is everywhere defined operations on '''''x''''', that is Dom(''f'') ⊇ '''''x'''''scalar or vector intervals.# If It does not consider inaccurate built-in additionarithmetic functions, each library operation in ''f'' is everywhere continuous on its inputs, while evaluating '''''y'''''round-off, then ''f'' is everywhere continuous on '''''x'''''conversion and representational errors.# If some library operation in ''f'' is nowhere defined on As a result its inputs, while evaluating '''''y''''', then ''f'' syntax is nowhere defined on '''''x'''''very easy, that is Dom(''f'') ∩ '''''x''''' = Øbut the arithmetic fails to produce guaranteed enclosures.
== Quick start introduction ==It is recommended to use the interval package as a replacement for simp. However, function names and interval constructors are not compatible between the packages.
=== Input and output INTLAB ===Before exercising This interval arithmeticpackage is ''not'' meant to be a replacement for INTLAB and any compatibility with it is pure coincidence. Since both are compatible with GNU Octave, they happen to agree on many function names and programs written for INTLAB may possibly run with this interval objects must be created, typically from package as well. Some fundamental differences that I am currently aware of:* INTLAB is non-interval datafree software, it grants none of the [http://www.gnu. There are interval constants <code>empty<org/code> and <code>entire<philosophy/code> free-sw.html four essential freedoms] of free software* INTLAB is not conforming to IEEE Std 1788-2015 and the class constructors <code>infsup</code> parsing of intervals from strings uses a different format—especially for bare the uncertain form* INTLAB supports intervals with complex numbers and <code>infsupdec</code> sparse interval matrices, but no empty intervals* INTLAB uses inferior accuracy for decorated intervals. The class constructors are very sophisticated and most arithmetic operations, because it focuses on speed* Basic operations can be used with several kinds found in both packages, but the availability of parameters: Interval boundaries can be given by numeric values or string values with decimal numbers. Also it is possible to use so called interval literals with square brackets.special functions depends
octave:1> infsup (1)
ans = [1]
octave:2> infsup (1, 2)
ans = [1, 2]
octave:3> infsup ("3", "4")
ans = [3, 4]
octave:4> infsup ("1.1")
ans = [1.0999999999999998, 1.1000000000000001]
octave:5> infsup ("[5, 6.5]")
ans = [5, 6.5]
octave:6> infsup ("[5.8e-17]")
ans = [5.799999999999999e-17, 5.800000000000001e-17]
It is possible to access the exact numeric interval boundaries with the functions <code>inf</codediv style="display:flex; align-items: flex-start"> and <codediv style="margin-right: 2em">sup</code>. The default text representation of intervals can be created {{Code|Computation with this interval package|<codesyntaxhighlight lang="octave">intervaltotext</code>. The default text representation is not guaranteed to be exact pkg load intervalA1 = infsup (see function <code>intervaltoexact</code>2, 3);B1 = hull (-4, because this would massively spam console output. For exampleA2);C1 = midrad (0, the exact text representation of <code>realmin</code> would be over 700 decimal places long! However, the default text representation is correct as it guarantees to contain the actual boundaries and is accurate enough to separate different boundaries.2);
A1 + B1 * C1</syntaxhighlight>}}</div><div>{{Code|Computation with INTLAB|<syntaxhighlight lang="octave:7"> startintlabA2 = infsup (12, 1 + eps3); ans B2 = [1hull (-4, 1.0000000000000003]A2); octave:8> infsup C2 = midrad (10, 1 + 2 * eps) ans = [1, 1.0000000000000005];
Warning: Decimal fractions 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.A2 + B2 * C2</syntaxhighlight>}}</div></div>
octave:9> infsup (<span style = "color:red">0.2</span>) ans = [.20000000000000001, .20000000000000002] octave:10> infsup (<span style = "color:green">"0.2"</span>)= Known differences ==== ans = [Simple programs written for INTLAB should run without modification with this interval package.19999999999999998, The following table lists common functions that use a different name in INTLAB.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.package! INTLAB octave:11> |-| infsup ("17.7"x) + 1 ans = [18.699999999999999, 18.700000000000003]| intval (x) octave:12> ans + "[0, 2]"|- ans = [18.699999999999999, 20.700000000000003]| wid (x)| diam (x)=== Decorations ===|-With the subclass <code>infsupdec</code> it is possible to extend interval arithmetic with | subset (a decoration system. Every interval and intermediate result will additionally carry , b)| in (a decoration, which may provide additional information about the final result. The following decorations are available:b)|-{| class="wikitable" style="margin: auto"interior (a, b)!Decoration| in0 (a, b)!Bounded|-!Continuous| isempty (x)!Defined!Definition| isnan (x)
|-
| com<br/>disjoint (commona, b)| | ✓| ✓| '''''x''''' is emptyintersect (a bounded, nonempty subset of Dom(''f''); ''f'' is continuous at each point of '''''x'''''; and the computed interval ''f''('''''x'''''b) is bounded
|-
| dac<br/>hdist (defined &amp; continuousa, b)|| ✓| ✓| '''''x''''' is qdist (a nonempty subset of Dom(''f'', b); and the restriction of ''f'' to '''''x''''' is continuous
|-
| def<br/>disp (definedx)||| ✓| '''''disp2str (x''''' is a nonempty subset of Dom(''f'')
|-
| trv<br/>infsup (trivials)|||| always true str2intval (so gives no informations)
|-
| ill<br/>isa (ill-formedx, "infsup")|||| Not an interval, at least one interval constructor failed during the course of computationisintval (x)
|}
In the following example, all decoration information is lost when the == Developer Information ===== Source Code Repository ===https://sourceforge.net/p/octave/interval is possibly divided by zero, i. e., the overall function is not guaranteed to be defined for all possible inputs./ci/default/tree/
octave:1> infsupdec(3, 4) ans = [3, 4]_com octave:2> ans + 12 ans = [15, 16]_com= Dependencies === octave:3> ans / "[0, 2]" ans = [7.5, Inf]_trvapt-get install liboctave-dev mercurial make automake libmpfr-dev
=== Arithmetic operations Build ===The interval packages comprises many interval arithmetic operationsrepository contains a Makefile which controls the build process. Function names match GNU Some common targets are:* <code>make release</code> Create a release tarball and the HTML documentation for [[Octave Forge]] (takes a while).* <code>make check</code> Run the full test-suite to verify that code changes didn't break anything (takes a while).* <code>make run</code> Quickly start Octave standard with minimal recompilation and functions where applicableloaded 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 functions are evaluated for each number in the interval where the function is defined and the result must be an enclosure of all possible values that may occur.dev autoconf automake inkscape zopfli</code>
=== Reverse arithmetic operations Architecture ===
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</code> (bare interval) with attributes <code>inf</code> (lower interval boundary) and <code>sup</code> (upper interval boundary)* class <code>infsupdec</code> (decorated interval) which extends the former and adds attribute <code>dec</code> (interval decoration). 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:  +- 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/ +- mpfr_function_d.cc – computes various arithmetic functions correctly rounded (using MPFR) `- ... – other oct-file sources === 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 ==== 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 [http://uivtx.cs.cas.cz/~rohn/matlab/ 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 [http://www.ti3.tuhh.de/paper/rump/Ru06c.pdf])|-|colspan="3"|Real (or complex) data only: Polynomials|-|verroots|style="color:red"| trapped| main part implemented in <code>vereig</code>|-|colspan="3"|Interval (or real) data: Matrices|-|verhurwstab|style="color:red"| trapped| depends on <code style="color:red">verposdef</code>|-|verinverse|style="color:green"| free| depends on <code style="color:green">verintervalhull</code>, to be migrated|-|<s>verinvnonneg</s>|style="color:green"| free, migrated|-|verposdef|style="color:red"| trapped| depends on <code>isspd</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 [http://uivtx.cs.cas.cz/~rohn/publist/singreg.pdf]|-|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= Numerical operations "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 [http://uivtx.cs.cas.cz/~rohn/publist/absvaleqnall.pdf], to be migrated|-| verbasintnpprob|style="color:red"| trapped| depends on <code style="color:red">verregsing</code>|-|}
=== Boolean operations ===
[[Category:Octave-Forge]]

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