Difference between revisions of "Interval package"

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The GNU Octave [http://octave.sourceforge.net/interval/ interval package] for real-valued interval arithmetic.
+
The 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 manage 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].  
  
The interval package provides data types and operations for
+
__TOC__
verified computing. It can be used to handle uncertainties, estimate
+
[[File:Interval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]
arithmetic errors and produce reliable results.  Also it can help find
+
<div style="clear:left"></div>
guaranteed solutions to numerical problems.  Thanks to GNU MPFR many
 
arithmetic operations are provided with best possible accuracy.  The
 
implementation of intervals in inf-sup format is based on interval boundaries
 
represented by binary64 numbers and is standard conforming to the  
 
(upcoming) standard for interval arithmetic
 
[http://standards.ieee.org/develop/project/1788.html IEEE 1788].
 
  
== Features ==
+
== Distribution ==
 +
* [https://octave.sourceforge.io/interval/ Latest version at Octave Forge]
 +
** <code>pkg install -forge interval</code>
 +
** [https://octave.sourceforge.io/interval/overview.html function reference]
 +
** [https://octave.sourceforge.io/interval/package_doc/index.html package documentation] (user manual)
 +
'''Third-party'''
 +
* [https://tracker.debian.org/pkg/octave-interval Debian GNU/Linux], [https://launchpad.net/ubuntu/+source/octave-interval Launchpad Ubuntu]
 +
* [https://aur.archlinux.org/packages/octave-interval/ archlinux user repository]
 +
* Included in [https://ftp.gnu.org/gnu/octave/windows/ official Windows installer] and installed automatically with Octave (since version 4.0.1)
 +
* [https://github.com/macports/macports-ports/tree/master/math/octave-interval/ MacPorts] for Mac OS X
 +
* [https://www.freshports.org/math/octave-forge-interval/ FreshPorts] for FreeBSD
 +
* [https://cygwin.com/cgi-bin2/package-grep.cgi?grep=octave-interval Cygwin] for Windows
 +
* [https://build.opensuse.org/package/show/science/octave-forge-interval openSUSE build service]
  
* Free software licensed under the terms of the GNU General Public License
+
== Development status ==
* Many interval arithmetic functions [http://octave.sourceforge.net/interval/overview.html] with high, system-independent accuracy
 
* Conforming to IEEE 1788 [http://sourceforge.net/p/octave/interval/ci/default/tree/IEEE%201788%20CONFORMANCE]
 
* 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 ==
 
 
* Completeness
 
* Completeness
** All required IEEE 1788 functions are implemented, the standard is currently in recirculation ballot phase and quite stable.
+
** 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.
** Planned: [[GSoC_Project_Ideas#Interval_package|more solvers; plotting functions]]
+
** Also, the minimalistic standard [https://standards.ieee.org/findstds/standard/1788.1-2017.html 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 package
+
** In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.
 
* Quality
 
* Quality
** Includes tests for all functions, many tests for basic functions
+
** Most arithmetic operations produce tight, correctly-rounded results. That is, the smallest possible interval with double-precision (binary64) endpoints, which encloses the exact result.
** No known bugs. The package is quite new and still has a small user base, so there might be hidden bugs. Also some advanced functions need more testing.
+
** Includes [https://github.com/oheim/ITF1788 large test suite] for arithmetic functions
 +
** 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 bug tracker].
 +
* Performance
 +
** All elementary functions have been [https://octave.org/doc/interpreter/Vectorization-and-Faster-Code-Execution.html vectorized] and run fast on large input data.
 +
** Arithmetic is performed with the [http://www.mpfr.org/ GNU MPFR] library internally. Where possible, the optimized [http://web.archive.org/web/20170128033523/http://lipforge.ens-lyon.fr/www/crlibm/ CRlibm] library is used.
 
* Portability
 
* Portability
** Runs in GNU Octave 3.8.2 and 4.0 release candidates
+
** Runs in GNU Octave 3.8.2
** Runs under GNU/Linux, Microsoft Windows (install from [[Octave-Forge#Installing_packages|Octave Forge]]), and Mac OS X ([[Create_a_MacOS_X_App_Bundle_Using_MacPorts|MacPorts]])
+
** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD
  
=== What to expect ===
+
== Project ideas (TODOs) ==
The interval arithmetic provided by this interval package is '''slow''', but '''accurate'''.
+
* 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, 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://kam.mff.cuni.cz/conferences/swim2015/slides/ohta.pdf]
 +
* Implement user-controllable output from the interval standard (e. g. via printf functions):
 +
a) It should be possible to specify the preferred overall field width (the length of s).
 +
b) It should be possible to specify how Empty, Entire and NaI are output,
 +
    e.g., whether lower or upper case, and whether Entire becomes [Entire] or [-Inf, Inf].
 +
c) For l and u, it should be possible to specify the field width,
 +
    and the number of digits after the point or the number of significant digits.
 +
    (partly this is already implemented by output_precision (...) / `format long` / `format short`)
 +
d) It should be possible to output the bounds of an interval without punctuation,
 +
    e.g., 1.234 2.345 instead of [1.234, 2.345]. For instance, this might be a
 +
    convenient way to write intervals to a file for use by another application.
  
''Why is the interval package slow?''
+
== Compatibility ==
All arithmetic interval operations are simulated in high-level octave language using C99 or multi-precision floating-point routines, which is a lot slower than a hardware implementation [https://books.google.de/books?id=JTc4XdXFnQIC&pg=PA61]. Building interval arithmetic operations from floating-point routines is easy for simple monotonic functions, e. g., addition and subtraction, but is complex for others, e. g., [http://exp.ln0.de/heimlich-power-2011.htm interval power function], atan2, or [[#Reverse_arithmetic_operations|reverse functions]]. For some interval operations it is not even possible to rely on floating-point routines, since not all required routines are available in C99 or BLAS.
+
The interval package's main goal is to be compliant with IEEE Std 1788-2015, so it is compatible with other standard-conforming implementations (on the set of operations described by the standard document). Other implementations, which are known to aim for standard conformance are:
  
For example, multiplication of matrices with many elements becomes unfeasible as it takes a lot of time.
+
* [https://github.com/JuliaIntervals/IntervalArithmetic.jl IntervalArithmetic.jl package] (Julia)
 +
* [https://github.com/jinterval/jinterval JInterval library] (Java)
 +
* [https://github.com/nadezhin/libieeep1788 ieeep1788 library] (C++) created by Marco Nehmeier, later forked by Dmitry Nadezhin
  
{| class="wikitable" style="margin: auto; align:right"
+
=== Octave Forge simp package ===
|+ Approximate runtime (wall clock time in seconds) <span style="font-weight: normal">— Results have been produced with GNU Octave 3.8.2 and Interval package 0.1.4 on an Intel® Core™ i5-4340M CPU (2.9–3.6 GHz)</span>
+
In 2008/2009 there was a Single Interval Mathematics Package (SIMP) for Octave, which has eventually become unmaintained at Octave Forge.
!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</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"| 0.002
 
|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 package accurate?''
+
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.
The GNU Octave built-in floating-point routines are not useful for interval arithmetic: Their results depend on hardware, system libraries and compilation options.  
 
The interval package handles all arithmetic functions with the help of the [http://www.mpfr.org/ GNU MPFR library]. With MPFR it is possible to compute system-independent, valid and tight enclosures of the correct results for most functions. However, it should be noted that some reverse operations and matrix operations do not exists in GNU MPFR and therefore cannot be computed with the same accuracy.
 
  
== 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 a problem in arithmetic, and it will grind away methodically, tirelessly, at gigahertz speed, until ultimately it produces the wrong answer. … An interval computation yields a pair 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|[http://dx.doi.org/10.1511/2003.6.484 DOI: 10.1511/2003.6.484]}}
+
=== 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:
{| class="wikitable" style="margin: auto"
+
* INTLAB is non-free software, it grants none of the [http://www.gnu.org/philosophy/free-sw.html four essential freedoms] of free software
!Standard floating point 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
!Interval arithmetic
+
* 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
| style = "vertical-align: top" |
+
* Basic operations can be found in both packages, but the availability of special functions depends
<span style="opacity:.5">octave:1> </span>19 * 0.1 - 2 + 0.1
 
ans =  1.3878e-16
 
| style = "vertical-align: top" |
 
<span style="opacity:.5">octave:1> </span>x = infsup ("0.1");
 
<span style="opacity:.5">octave:2> </span>19 * x - 2 + x
 
ans ⊂ [-3.191891195797326e-16, +1.3877787807814457e-16]
 
|}
 
  
Floating-point arithmetic, as specified by [http://en.wikipedia.org/wiki/IEEE_floating_point 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.
+
<div style="display:flex; align-items: flex-start">
* Floating-point arithmetic is often used mindlessly by developers. [http://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html] [http://www.cs.berkeley.edu/~wkahan/Mindless.pdf] [http://www.ima.umn.edu/~arnold/disasters/]
+
<div style="margin-right: 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. [http://en.wikipedia.org/wiki/Decimal_computer]
+
{{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 … [http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf]
+
pkg load interval
* Results are hardly predictable. [https://hal.archives-ouvertes.fr/hal-00128124/en/] All operations produce the best possible accuracy ''at runtime'', this is how a floating point works. Contrariwise, financial computer systems typically use a [http://en.wikipedia.org/wiki/Fixed-point_arithmetic fixed-point arithmetic] (COBOL, PL/I, ), where overflow and rounding can be precisely predicted ''at compile-time''.
+
A1 = infsup (2, 3);
* 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. [http://www.gnu.org/software/libc/manual/html_node/Errors-in-Math-Functions.html#Errors-in-Math-Functions] [http://developer.amd.com/tools-and-sdks/cpu-development/libm/]
+
B1 = hull (-4, A2);
* 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. Additionally, many functions produce limit values (∞ × −∞ = −∞, ∞ ÷ 0 = ∞, ∞ ÷ −0 = −∞, log (0) = −∞), which is sometimes (but not always!) useful when overflow and underflow occur.
+
C1 = midrad (0, 2);
  
Interval arithmetic addresses above problems in its very special way and introduces new possibilities for algorithms. For example, the [http://en.wikipedia.org/wiki/Interval_arithmetic#Interval_Newton_method interval newton method] is able to find ''all'' zeros of a particular function.
+
A1 + B1 * C1
 
 
== Theory ==
 
 
 
=== Online Introductions ===
 
 
 
[http://www.maths.manchester.ac.uk/~higham/narep/narep416.pdf  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 of
 
interval-evaluation of ''f'' over a box '''''x''''' = (''x''<sub>1</sub>, … , ''x''<sub>''n''</sub>)
 
using any interval versions of its component library functions. Then
 
# In all cases, '''''y''''' contains the range of ''f'' over '''''x''''', that is, the set of ''f''('''''x''''') at points of '''''x''''' where it is defined: '''''y''''' ⊇ Rge(''f'' | '''''x''''') = {''f''(''x'') | ''x'' ∈ '''''x''''' ∩ Dom(''f'') }
 
# If also each library operation in ''f'' is everywhere defined on its inputs, while evaluating '''''y''''', then ''f'' is everywhere defined 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]_com
 
infsupdec (1, 2) # [1, 2]_com
 
infsupdec ("3", "4") # [3, 4]_com
 
infsupdec ("1.1") # [1.0999999999999998, 1.100000000000001]_com
 
infsupdec ("5.8e-17")
 
  # [5.799999999999999e-17, 5.800000000000001e-17]_com
 
midrad (12, 3) # [9, 15]_com
 
midrad ("4.2", "1e-7") # [4.199999899999999, 4.200000100000001]_com
 
hull (3, 42, "19.3", "-2.3") # [-2.300000000000001, +42]_com
 
hull ("pi", "e") # [2.718281828459045, 3.1415926535897936]_com
 
 
</syntaxhighlight>
 
</syntaxhighlight>
 
}}
 
}}
 +
</div><div>
 +
{{Code|Computation with INTLAB|<syntaxhighlight lang="octave">
 +
startintlab
 +
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.
+
A2 + B2 * C2
 
 
{{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]_com
 
 
</syntaxhighlight>
 
</syntaxhighlight>
 
}}
 
}}
 +
</div>
 +
</div>
  
==== 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 interval constructors. With cell arrays it is also possible to mix several types of boundaries.
+
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)
 +
|}
  
Interval matrices behave like normal matrices in GNU Octave and can be used for broadcasting and vectorized function evaluation.
+
== Developer Information ==
 +
=== Source Code Repository ===
 +
https://sourceforge.net/p/octave/interval/ci/default/tree/
  
{{Code|Create interval matrices|<syntaxhighlight lang="octave">
+
=== Dependencies ===
M = infsup (magic (3))
+
apt-get install liboctave-dev mercurial make automake libmpfr-dev
  # [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]_com
 
infsup ({1, eps; "4/7", "pi"}, {2, 1; "e", "0xff"})
 
  #                                  [1, 2]  [2.220446049250313e-16, 1]
 
  # [0.5714285714285713, 2.7182818284590456]    [3.141592653589793, 255]
 
</syntaxhighlight>
 
}}
 
  
=== Arithmetic operations ===
+
=== Build ===
The interval packages comprises many interval arithmetic operations. A complete list can be found in its [http://octave.sourceforge.net/interval/overview.html function reference]. Function names match GNU Octave standard functions where applicable and follow recommendations by IEEE 1788 otherwise, cf. [[#IEEE_1788_index|IEEE 1788 index]].
+
The repository contains a Makefile which controls the build process. 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 with minimal recompilation and functions 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'') is essentially the natural extension to sets of the corresponding point-wise function on real numbers. That is, the function is evaluated for each number in the interval where the function is defined and the result must be an enclosure of all possible values that may occur.
+
'''Build dependencies'''
 +
<code>apt-get install libmpfr-dev 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|<syntaxhighlight lang="octave">
+
In a nutshell the package provides two new data types to users: bare intervals and decorated intervals. The data types are implemented as:
sin (infsupdec (0.5)) # [.4794255386042029, .4794255386042031]_com
+
* class <code>infsup</code> (bare interval) with attributes <code>inf</code> (lower interval boundary) and <code>sup</code> (upper interval boundary)
pow (infsupdec (2), infsupdec (3, 4)) # [8, 16]_com
+
* class <code>infsupdec</code> (decorated interval) which extends the former and adds attribute <code>dec</code> (interval decoration).
atan2 (infsupdec (1), infsupdec (1))
 
  # [.7853981633974482, .7853981633974484]_com
 
midrad (magic (3), 0.5) * 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 ===
+
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:
Some operations on intervals do not return an interval enclosure, but a single number (in double-precision). Most important are <code>inf</code> and <code>sup</code>, which return the lower and upper interval boundaries.
 
  
More such operations are <code>mid</code> (approximation of the interval's midpoint), <code>wid</code> (approximation of the interval's width), <code>rad</code> (approximation of the interval's radius), <code>mag</code> (interval's magnitude) and <code>mig</code> (interval's mignitude).
+
+- 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
  
=== 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 ∀<sub>''a''</sub> ∃<sub>''b''</sub> ''a'' ≤ ''b'' ∧ ∀<sub>''b''</sub> ∃<sub>''a''</sub> ''a'' ≤ ''b''.
+
==== 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
  
  <span style="opacity:.5">octave:1> </span>infsup (1, 3) <= infsup (2, 4)
+
  if (not (isa (x, "infsupdec")))
  ans =  1
+
  x = infsupdec (x);
 +
endif
 +
if (not (isa (y, "infsupdec")))
 +
  y = infsupdec (y);
 +
  endif
  
=== Matrix operations ===
+
==== Use of Octave functions ====
Above mentioned operations can also be applied element-wise to interval vectors and matrices. Many operations use [http://www.gnu.org/software/octave/doc/interpreter/Vectorization-and-Faster-Code-Execution.html#Vectorization-and-Faster-Code-Execution vectorization techniques].
+
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
  
In addition, there are matrix operations on interval matrices. These operations comprise: dot product, matrix multiplication, vector sums (all with tightest accuracy), matrix inversion, matrix powers, and solving linear systems (the latter are less accurate). As a result of missing hardware / low-level library support and missing optimizations, these operations are relatively slow compared to familiar operations in floating-point arithmetic.
+
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.
  
{{Code|Examples of using interval matrix functions|<syntaxhighlight lang="octave">
+
==== Vectorization & Indexing ====
A = infsup ([1, 2, 3; 4, 0, 0; 0, 0, 1]); A (2, 3) = "[0, 6]"
+
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.
   # [1]  [2]      [3]
+
function x = plus (x, y)
   # [4]  [0]  [0, 6]
+
   ... parameter checking ...
  # [0]  [0]      [1]
+
   l = mpfr_function_d ('plus', -inf, x.inf, y.inf);
B = inv (A)
+
   u = mpfr_function_d ('plus', +inf, x.sup, y.sup);
  #  [0]    [0.25]      [-1.5, 0]
+
   # [0.5]  [-0.125]  [-1.5, -0.75]
+
   emptyresult = isempty (x) | isempty (y);
   #  [0]        [0]            [1]
+
  l(emptyresult) = inf;
A * B
+
   u(emptyresult) = -inf;
   # [1]  [0]  [-1.5, +1.5]
+
  
   # [0]  [1]      [-6, +6]
+
endfunction
  # [0]  [0]            [1]
 
  
A = infsup (magic (3))
+
== VERSOFT ==
  # [8]  [1]  [6]
+
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.
  # [3]  [5]  [7]
 
  # [4]  [9]  [2]
 
c = A \ [3; 4; 5]
 
  # [0.18333333333333315, 0.18333333333333358]
 
  # [0.43333333333333318, 0.43333333333333346]
 
  # [0.18333333333333307, 0.18333333333333355]
 
A * c
 
  # [2.9999999999999964, 3.0000000000000036]
 
  #  [3.9999999999999964, 4.000000000000004]
 
  #  [4.999999999999997, 5.000000000000003]
 
</syntaxhighlight>
 
}}
 
  
==== Notes on linear systems ====
+
{|
A system of linear equations in the form A''x'' = 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 (possibly several) linear systems. Some characteristics should definitely be known when linear interval systems are solved:
+
! Function
* If the linear system is underdetermined 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.
+
! Status
* If the interval result is empty in at least one of its coordinates, the linear system is guaranteed to be underdetermined and has no solutions. Contrariwise, from a non-empty result it can not be concluded whether all or some of the systems have solutions or not.
+
! Information
* Wide intervals within the matrix A can easily lead to a superposition of cases, where the rank of A is no longer unique. If the linear interval system contains cases of linear independent equations as well as linear dependent equations, the resulting enclosure of solutions will inevitably be very broad.
+
|-
 
+
|colspan="3"|Real (or complex) data only: Matrices
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.
+
|-
 
+
|verbasis
{{Code|Standard floating point arithmetic versus interval arithmetic on ill-conditioned linear systems|<syntaxhighlight lang="octave">
+
|style="color:red"| trapped
A = [1, 0; 2, 0];
+
| depends on <code style="color:red">verfullcolrank</code>
## This linear system has no solutions
+
|-
A \ [3; 0]
+
|vercondnum
  # warning: matrix singular to machine precision, rcond = 0
+
|style="color:red"| trapped
  # 0.60000
+
| depends on <code style="color:red">versingval</code>
  # 0.00000
+
|-
## This linear system has many solutions
+
|verdet
A \ [4; 8]
+
|style="color:red"| trapped
  # 4
+
| depends on <code>vereig</code>
  # 0
+
|-
 
+
|verdistsing
## The empty interval vector proves that there is no solution
+
|style="color:red"| trapped
infsup (A) \ [3; 0]
+
| depends on <code style="color:red">versingval</code>
  # [Empty]
+
|-
  # [Empty]
+
|verfullcolrank
## The unbound interval vector indicates that there may be many solutions
+
|style="color:red"| trapped
infsup (A) \ [4; 8]
+
| depends on <code>verpinv</code>
  #      [4]
+
|-
  # [Entire]
+
|vernorm2
</syntaxhighlight>
+
|style="color:red"| trapped
}}
+
| depends on <code style="color:red">versingval</code>
 
+
|-
== Advanced topics ==
+
|vernull (experimental)
 
+
| unknown
=== Error handling ===
+
| depends on <code style="color:red">verlsq</code>; todo: compare with local function inside <code style="color:green">verintlinineqs</code>
 
+
|-
Due to the nature of set-based interval arithmetic, one should not observe errors (in the sense of raised GNU Octave error messages) during computation unless operations are evaluated for incompatible data types. Arithmetic operations which are not defined for (parts of) their input, simply ignore anything that is outside of their domain.
+
|verorth
 
+
|style="color:red"| trapped
However, the interval constructors can produce errors depending on the input. The <code>infsup</code> constructor will fail if the interval boundaries are invalid. Contrariwise, the (preferred) <code>infsupdec</code>, <code>midrad</code> and <code>hull</code> constructors will only issue a warning and return a [NaI] object, which will propagate and survive through computations. NaI stands for “not an interval”.
+
| depends on <code style="color:red">verbasis</code> and <code style="color:red">verthinsvd</code>
 
+
|-
{{Code|Effects of set-based interval arithmetic on partial functions and the NaI object|<syntaxhighlight lang="octave">
+
|verorthproj
## Evaluation of a function outside of its domain returns an empty interval
+
|style="color:red"| trapped
infsupdec (2) / 0 # [Empty]_trv
+
| depends on <code style="color:red">verpinv</code> and <code style="color:red">verfullcolrank</code>
infsupdec (0) ^ infsupdec (0) # [Empty]_trv
+
|-
 
+
|verpd
## Illegal interval construction creates a NaI
+
|style="color:red"| trapped
infsupdec (3, 2) # [NaI]
+
| depends on <code>isspd</code> (by Rump, to be checked) and <code style="color:red">vereig</code>
## NaI even survives through computations
+
|-
ans + 1 # [NaI]
+
|verpinv
</syntaxhighlight>
+
|style="color:red"| trapped
}}
+
| dependency <code>verifylss</code> is implemented as <code>mldivide</code>; depends on <code style="color:red">verthinsvd</code>
 
+
|-
There 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 <code>help warning</code> for instructions.
+
|verpmat
 
+
|style="color:red"| trapped
{| class="wikitable" style="margin: auto"
+
| depends on <code style="color:red">verregsing</code>
!Warning ID
+
|-
!Reason
+
|verrank
!Possible consequences
+
|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>
 
|-
 
|-
| interval:PossiblyUndefined
+
|verregsing
|style="vertical-align:top" | Interval construction with boundaries in decimal format, and the constructor can't decide whether the lower boundary is smaller than the upper boundary. Both boundaries are very close and lie between subsequent binary64 numbers.
+
|style="color:red"| trapped
|style="vertical-align:top" | The constructed interval is a valid and tight enclosure of both numbers. If the lower boundary was actually greater than the upper boundary, this illegal interval is not considered an error.
+
| 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]
 
|-
 
|-
| interval:ImplicitPromote
+
|colspan="3"|Interval (or real) data: Matrices: Eigenvalues and singular values
|style="vertical-align:top" | 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 (1, 2) + infsup (3, 4)</code> does. A bare interval can be explicitly promoted with the newdec [http://octave.sourceforge.net/interval/function/@infsup/newdec.html] function.
 
|style="vertical-align: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 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.
 
 
|-
 
|-
| interval:NaI
+
|vereigsym
|style="vertical-align:top" | 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 [http://octave.sourceforge.net/interval/function/nai.html] function.
+
|style="color:red"| 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.
+
| main part implemented in <code>vereig</code>, depends on <code style="color:red">verspectrad</code>
|}
 
 
 
=== 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) <code>infsup</code> constructor creates bare, undecorated intervals and the <code>intervalpart</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-align: center" |
+
|style="color:red"| trapped
| style="text-align: center" | ✓
+
| depends on <code style="color:red">verregsing</code>
| 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
 
 
|-
 
|-
| dac<br/>(defined &amp; continuous)
+
|<s>vereigvec</s>
|
+
|style="color:green"| free, migrated
| style="text-align: center" | ✓
 
| style="text-align: center" |
 
| '''''x''''' is a nonempty subset of Dom(''f''); and the restriction of ''f'' to '''''x''''' is continuous
 
 
|-
 
|-
| def<br/>(defined)
+
|verperrvec
|
+
|style="color:green"| free
|
+
| the function is just a wrapper around <code style="color:green">vereigvec</code>?!?
| style="text-align: center" |
 
| '''''x''''' is a nonempty subset of Dom(''f'')
 
 
|-
 
|-
| trv<br/>(trivial)
+
|versingval
|
+
|style="color:red"| trapped
|
+
| depends on <code style="color:red">vereigsym</code>
|
 
| always true (so gives no information)
 
 
|-
 
|-
| ill<br/>(ill-formed)
+
|colspan="3"|Interval (or real) data: Matrices: Decompositions
|
 
|
 
|
 
| 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 lang="octave">
 
x = infsupdec (3, 4) # [3, 4]_com
 
y = 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]_com
 
sqrt (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 ([http://www.open-std.org/jtc1/sc22/WG14/www/docs/n1256.pdf ISO/IEC9899, N1256, §6.4.4.2]). This can be used as a convenient way to define interval boundaries 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 [http://en.wikipedia.org/wiki/Unit_in_the_last_place “units in last place” (ULPs)] as an uncertainty. The format is <code>''m''?''ruE''</code>, where
 
** <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 (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 (3)); 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">octave: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 Octave
 
 
|-
 
|-
| numsToInterval
+
|verqr (experimental)
| bare version: infsup (''l'', ''u'') [http://octave.sourceforge.net/interval/function/@infsup/infsup.html]<br/>decorated version: infsupdec (''l'', ''u'') [http://octave.sourceforge.net/interval/function/@infsupdec/infsupdec.html]
+
|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.
 
|-
 
|-
| textToInterval
+
|<s>verchol (experimental)</s>
| bare version: infsup (''s'')<br/>decorated version: infsupdec (''s'')
+
|style="color:green"| free, migrated
 +
| migrated version has been named after the standard Octave function <code>chol</code>
 
|-
 
|-
| setDec
+
|colspan="3"|Interval (or real) data: Linear systems (square)
| infsupdec (''x'', ''dx'')
 
 
|-
 
|-
| exp10
+
|verenclinthull
| pow10 [http://octave.sourceforge.net/interval/function/@infsup/pow10.html]
+
|style="color:green"| free
 +
| to be migrated
 
|-
 
|-
| exp2
+
|verhullparam
| pow2 [http://octave.sourceforge.net/interval/function/@infsup/pow2.html]
+
|style="color:green"| free
 +
| depends on <code>verintervalhull</code>, to be migrated
 
|-
 
|-
| recip
+
|verhullpatt
| inv [http://octave.sourceforge.net/interval/function/@infsup/inv.html]
+
|style="color:green"| free
 +
| depends on <code>verhullparam</code>, to be migrated
 
|-
 
|-
| sqrt
+
|verintervalhull
| realsqrt [http://octave.sourceforge.net/interval/function/@infsup/realsqrt.html]
+
|style="color:green"| free
 +
| to be migrated
 
|-
 
|-
| rootn
+
|colspan="3"|Interval (or real) data: Linear systems (rectangular)
| nthroot [http://octave.sourceforge.net/interval/function/@infsup/nthroot.html]
 
 
|-
 
|-
| logp1
+
|verintlinineqs
| log1p [http://octave.sourceforge.net/interval/function/@infsup/log1p.html]
+
|style="color:green"| free
 +
| depends on <code style="color:green">verlinineqnn</code>
 
|-
 
|-
| roundTiesToAway
+
|veroettprag
| round [http://octave.sourceforge.net/interval/function/@infsup/round.html]
+
|style="color:green"| free
 
|-
 
|-
| roundTiesToEven
+
|vertolsol
| roundb [http://octave.sourceforge.net/interval/function/@infsup/roundb.html]
+
|style="color:green"| free
 +
| depends on <code style="color:green">verlinineqnn</code>
 
|-
 
|-
| trunc
+
|colspan="3"|Interval (or real) data: Matrix equations (rectangular)
| fix [http://octave.sourceforge.net/interval/function/@infsup/fix.html]
 
 
|-
 
|-
| sum
+
|vermatreqn
| on intervals: sum [http://octave.sourceforge.net/interval/function/@infsup/sum.html]<br/>on numbers: mpfr_vector_sum_d [http://octave.sourceforge.net/interval/function/mpfr_vector_sum_d.html]
+
|style="color:green"| free
 
|-
 
|-
| dot
+
|colspan="3"|Real data only: Uncommon problems
| on intervals: dot [http://octave.sourceforge.net/interval/function/@infsup/dot.html]<br/>on numbers: mpfr_vector_dot_d [http://octave.sourceforge.net/interval/function/mpfr_vector_dot_d.html]
 
 
|-
 
|-
| sumAbs
+
| plusminusoneset
| on intervals: sumabs [http://octave.sourceforge.net/interval/function/@infsup/sumabs.html]<br/>on numbers: mpfr_vector_sum_d (''rounding mode'', abs (''x''))
+
|style="color:green"| free
 
|-
 
|-
| sumSquare
+
| verabsvaleqn
| on intervals: sumsq [http://octave.sourceforge.net/interval/function/@infsup/sumsq.html]<br/>on numbers: mpfr_vector_dot_d (''rounding mode'', abs (''x''), abs (''x''))
+
|style="color:green"| free
 +
| to be migrated
 
|-
 
|-
| intersection
+
| verabsvaleqnall
| intersect [http://octave.sourceforge.net/interval/function/@infsup/intersect.html]
+
|style="color:green"| free
 +
| depends on <code>verabsvaleqn</code>, see also [http://uivtx.cs.cas.cz/~rohn/publist/absvaleqnall.pdf], to be migrated
 
|-
 
|-
| convexHull
+
| verbasintnpprob
| union [http://octave.sourceforge.net/interval/function/@infsup/union.html]
+
|style="color:red"| trapped
 +
| depends on <code style="color:red">verregsing</code>
 
|-
 
|-
| mulRevToPair
 
| mulrev [http://octave.sourceforge.net/interval/function/@infsup/mulrev.html] 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 [http://www.gnu.org/philosophy/free-sw.html 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
 
</syntaxhighlight>
 
}}
 
 
== Related work ==
 
 
For C++ there is an open source interval library [https://github.com/nehmeier/libieeep1788/ 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 [http://www.youtube.com/watch?v=GOa9aWAZO_Q]. The GNU Octave interval package shares several unit tests with libieeep1788.
 
 
For C++, Pascal and Fortran there is a free interval library [http://www2.math.uni-wuppertal.de/~xsc/xsc-sprachen.html 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 [https://java.net/projects/jinterval/ jinterval] by Dmitry Nadezhin (member of IEEE P1788). It aims to be standard compliant with IEEE 1788, but is not complete yet.
 
 
For MATLAB there is a popular, nonfree interval arithmetic toolbox [http://www.ti3.tu-harburg.de/rump/intlab/ 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 [http://www.ti3.tuhh.de/rump/intlab/Octave/octave.html].
 
  
[[Category:Octave-Forge]]
+
[[Category:Octave Forge]]

Revision as of 03:15, 10 June 2019

The GNU Octave interval package for real-valued 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 +inf and -inf 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 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].
Example: Plotting the interval enclosure of a function

Distribution

Third-party

Development status

  • Completeness
    • All required functions from 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.
    • Also, the minimalistic standard 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).
    • In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.
  • Quality
    • Most arithmetic operations produce tight, correctly-rounded results. That is, the smallest possible interval with double-precision (binary64) endpoints, which encloses the exact result.
    • Includes large test suite for arithmetic functions
    • For open bugs please refer to the bug tracker.
  • Performance
    • All elementary functions have been vectorized and run fast on large input data.
    • Arithmetic is performed with the GNU MPFR library internally. Where possible, the optimized CRlibm library is used.
  • Portability
    • Runs in GNU Octave ≥ 3.8.2
    • Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD

Project ideas (TODOs)

  • To be considered in the future: Algorithms can be migrated from the C-XSC Toolbox (C++ code) from [1] (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.
  • Interval version of interp1
  • Extend subsasgn 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 subsasgn
>> 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 [2] [3]
  • Verified Convex Hull for Inexact Data [4] [5]
  • Implement user-controllable output from the interval standard (e. g. via printf functions):
a) It should be possible to specify the preferred overall field width (the length of s).
b) It should be possible to specify how Empty, Entire and NaI are output,
   e.g., whether lower or upper case, and whether Entire becomes [Entire] or [-Inf, Inf].
c) For l and u, it should be possible to specify the field width,
   and the number of digits after the point or the number of significant digits.
   (partly this is already implemented by output_precision (...) / `format long` / `format short`)
d) It should be possible to output the bounds of an interval without punctuation,
   e.g., 1.234 2.345 instead of [1.234, 2.345]. For instance, this might be a
   convenient way to write intervals to a file for use by another application.

Compatibility

The interval package's main goal is to be compliant with IEEE Std 1788-2015, so it is compatible with other standard-conforming implementations (on the set of operations described by the standard document). Other implementations, which are known to aim for standard conformance are:

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.

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.

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.

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


Code: Computation with this interval package
pkg load interval
A1 = infsup (2, 3);
B1 = hull (-4, A2);
C1 = midrad (0, 2);

A1 + B1 * C1
Code: Computation with INTLAB
startintlab
A2 = infsup (2, 3);
B2 = hull (-4, A2);
C2 = midrad (0, 2);

A2 + B2 * C2

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://sourceforge.net/p/octave/interval/ci/default/tree/

Dependencies

apt-get install liboctave-dev mercurial make automake libmpfr-dev

Build

The repository contains a Makefile which controls the build process. Some common targets are:

  • make release Create a release tarball and the HTML documentation for Octave Forge (takes a while).
  • make check Run the full test-suite to verify that code changes didn't break anything (takes a while).
  • make run Quickly start Octave with minimal recompilation and functions loaded from the workspace (for interactive testing of code changes).

Build dependencies apt-get install libmpfr-dev autoconf automake inkscape zopfli

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 infsup (bare interval) with attributes inf (lower interval boundary) and sup (upper interval boundary)
  • class infsupdec (decorated interval) which extends the former and adds attribute dec (interval decoration).

Almost all functions in the package are implemented as methods of these classes, e. g. @infsup/sin 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 nargin and call print_usage 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 (mpfr_function_d) and crlibm (crlibm_function), 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
Real (or complex) data only: Matrices
verbasis trapped depends on verfullcolrank
vercondnum trapped depends on versingval
verdet trapped depends on vereig
verdistsing trapped depends on versingval
verfullcolrank trapped depends on verpinv
vernorm2 trapped depends on versingval
vernull (experimental) unknown depends on verlsq; todo: compare with local function inside verintlinineqs
verorth trapped depends on verbasis and verthinsvd
verorthproj trapped depends on verpinv and verfullcolrank
verpd trapped depends on isspd (by Rump, to be checked) and vereig
verpinv trapped dependency verifylss is implemented as mldivide; depends on verthinsvd
verpmat trapped depends on verregsing
verrank trapped depends on versingval and verbasis
verrref trapped depends on verfullcolrank and verpinv
Real (or complex) data only: Matrices: Eigenvalues and singular values
vereig trapped depends on proprietary verifyeig function from INTLAB, depends on complex interval arithmetic
vereigback free, migrated (for real eigenvalues) dependency norm is already implemented
verspectrad trapped main part implemented in vereig
Real (or complex) data only: Matrices: Decompositions
verpoldec trapped depends on verthinsvd
verrankdec trapped depends on verfullcolrank and verpinv
verspectdec trapped main part implemented in vereig
verthinsvd trapped implemented in vereig
Real (or complex) data only: Matrix functions
vermatfun trapped main part implemented in vereig
Real data only: Linear systems (rectangular)
verlinineqnn free, migrated use glpk as a replacement for linprog
verlinsys trapped dependency verifylss is implemented as mldivide; depends on verpinv, verfullcolrank, and verbasis
verlsq trapped depends on verpinv and verfullcolrank
Real data only: Optimization
verlcpall free depends on verabsvaleqnall
verlinprog free, migrated use glpk as a replacement for linprog; dependency verifylss is implemented as mldivide
verlinprogg trapped depends on verfullcolrank
verquadprog unknown use quadprog from the optim package; use glpk as a replacement for linprog; dependency verifylss is implemented as mldivide; depends on isspd (by Rump, to be checked, algorithm in [6])
Real (or complex) data only: Polynomials
verroots trapped main part implemented in vereig
Interval (or real) data: Matrices
verhurwstab trapped depends on verposdef
verinverse free depends on verintervalhull, to be migrated
verinvnonneg free, migrated
verposdef trapped depends on isspd (by Rump, to be checked) and verregsing
verregsing trapped dependency verifylss is implemented as mldivide; depends on isspd (by Rump, to be checked) and verintervalhull; see also [7]
Interval (or real) data: Matrices: Eigenvalues and singular values
vereigsym trapped main part implemented in vereig, depends on verspectrad
vereigval trapped depends on verregsing
vereigvec free, migrated
verperrvec free the function is just a wrapper around vereigvec?!?
versingval trapped depends on vereigsym
Interval (or real) data: Matrices: Decompositions
verqr (experimental) free qr 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.
verchol (experimental) free, migrated migrated version has been named after the standard Octave function chol
Interval (or real) data: Linear systems (square)
verenclinthull free to be migrated
verhullparam free depends on verintervalhull, to be migrated
verhullpatt free depends on verhullparam, to be migrated
verintervalhull free to be migrated
Interval (or real) data: Linear systems (rectangular)
verintlinineqs free depends on verlinineqnn
veroettprag free
vertolsol free depends on verlinineqnn
Interval (or real) data: Matrix equations (rectangular)
vermatreqn free
Real data only: Uncommon problems
plusminusoneset free
verabsvaleqn free to be migrated
verabsvaleqnall free depends on verabsvaleqn, see also [8], to be migrated
verbasintnpprob trapped depends on verregsing