Interval package: Difference between revisions

From Octave
Jump to navigation Jump to search
(Reverse arithmetic operations explained)
(Use OctaveForge template.)
 
(148 intermediate revisions by 4 users not shown)
Line 1: Line 1:
The [https://sourceforge.net/p/octave/interval/ interval package] provides data types and fundamental operations for real valued interval arithmetic based on the common floating-point format “binary64” a. k. a. double-precision. It aims to be standard compliant with the (upcoming) [http://standards.ieee.org/develop/project/1788.html IEEE 1788] and therefore implements the ''set-based'' interval arithmetic flavor. '''Interval arithmetic''' produces mathematically proven numerical results.
{{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.io/interval/overview.html Function reference]
| doc 2 = [https://octave.sourceforge.io/interval/package_doc/ User manual]
| download 1 = [https://octave.sourceforge.io/download.php?package=interval-3.2.0.tar.gz interval-3.2.0.tar.gz]
| repository 1 = https://octave.sourceforge.io/pkg-repository/interval/
| dependency 1 = octave &ge; 3.8.0
| dependency 2 = '''Runtime:''' mpfr (&ge; 3.1.0) [https://packages.debian.org/search?keywords=libmpfr4 libmpfr4 for Debian]
| dependency 3 = '''Build:''' mpfr (&ge; 3.1.0) [https://packages.debian.org/search?keywords=libmpfr-dev libmpfr-dev for Debian]
}}


Warning: The package has not yet been released. If you want to experience the development version, you may (1) install the (currently deprecated) [http://octave.sourceforge.net/fenv/ fenv package], (2) download a [https://sourceforge.net/p/octave/interval/ci/default/tarball snapshot version of the interval package], (3) navigate to the <code>inst/</code> subfolder and run octave.
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].  


== Motivation ==
[[File:Interval-sombrero.png|280px|thumb|left|Example: Plotting the interval enclosure of a function]]
<div style="clear:left"></div>


{{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]}}
== Distribution ==
* [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]


{| class="wikitable" style="margin: auto"
== Development status ==
!Standard floating point arithmetic
* Completeness
!Interval arithmetic
** 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.
|-
** 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).
| style = "vertical-align: top" |
** In addition there are functions for interval matrix arithmetic, N-dimensional interval arrays, plotting, and solvers.
octave:1> 19 * 0.1 - 2 + 0.1
* Quality
ans =  1.3878e-16
** Most arithmetic operations produce tight, correctly-rounded results. That is, the smallest possible interval with double-precision (binary64) endpoints, which encloses the exact result.
| style = "vertical-align: top" |
** Includes [https://github.com/oheim/ITF1788 large test suite] for arithmetic functions
octave:1> x = infsup ("0.1");
** 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].
octave:2> 19 * x - 2 + x
* Performance
ans ⊂ [-3.1918911957973251e-16, +1.3877787807814457e-16]
** 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
** Runs in GNU Octave ≥ 3.8.2
** Known to run under GNU/Linux, Microsoft Windows, macOS, and FreeBSD


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 extended 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.
== 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, 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.


However, there are some downsides of floating point arithmetic in practice, which will eventually produce errors in computations.
== Compatibility ==
* 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]
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:
* 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]
* 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, … [http://www.cs.berkeley.edu/~wkahan/JAVAhurt.pdf]
* Results are hardly predictable. [https://hal.archives-ouvertes.fr/hal-00128124/en/] All operations produce the best possible accuracy ''at runtime'', this is how 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''.
* 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.


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.
* [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


== 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 interval arithmetic ===
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.
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 ==
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 interval arithmetic, interval objects must be created, typically from non-interval data. There are interval constants <code>empty</code> and <code>entire</code> and the class constructors <code>infsup</code> for bare intervals and <code>infsupdec</code> for decorated intervals. The class constructors are very sophisticated and can be used with several kinds of parameters: Interval boundaries can be given by numeric values or string values with decimal numbers. Also it is possible to use so called interval literals with square brackets.
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 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


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</code> and <code>sup</code>. The default text representation of intervals can be created with <code>intervaltotext</code>. The default text representation is not guaranteed to be exact (see function <code>intervaltoexact</code>), 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 and is accurate enough to separate different boundaries.
<div style="display:flex; align-items: flex-start">
<div style="margin-right: 2em">
{{Code|Computation with this interval package|<syntaxhighlight lang="octave">
pkg load interval
A1 = infsup (2, 3);
B1 = hull (-4, A2);
C1 = midrad (0, 2);


octave:7> infsup (1, 1 + eps)
A1 + B1 * C1
ans ⊂ [1, 1.0000000000000003]
</syntaxhighlight>
octave:8> infsup (1, 1 + 2 * eps)
}}
ans ⊂ [1, 1.0000000000000005]
</div><div>
{{Code|Computation with INTLAB|<syntaxhighlight lang="octave">
startintlab
A2 = infsup (2, 3);
B2 = hull (-4, A2);
C2 = midrad (0, 2);


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.
A2 + B2 * C2
</syntaxhighlight>
}}
</div>
</div>


octave:9> infsup (<span style = "color:red">0.2</span>)
==== Known differences ====
ans ⊂ [.20000000000000001, .20000000000000002]
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.
octave:10> infsup (<span style = "color:green">"0.2"</span>)
{|
ans ⊂ [.19999999999999998, .20000000000000002]
! interval package
 
! INTLAB
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.
|-
 
| infsup (x)
octave:11> infsup ("17.7") + 1
| intval (x)
ans ⊂ [18.699999999999999, 18.700000000000003]
|-
octave:12> ans + "[0, 2]"
| wid (x)
ans ⊂ [18.699999999999999, 20.700000000000003]
| diam (x)
 
|-
==== Specialized interval constructors ====
| subset (a, b)
Above mentioned interval construction with decimal numbers or numeric data is straightforward. Beyond that, there are more ways to define intervals or interval boundaries.
| in (a, b)
* 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.
| interior (a, b)
* 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
| in0 (a, b)
** <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),
| isempty (x)
** <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),
| isnan (x)
** <code>''E ''</code> is either empty or an exponent field comprising the character <code>e</code> followed by a decimal integer exponent (base 10).
 
octave:13> infsup ("0x1.999999999999Ap-4")
ans ⊂ [.1, .10000000000000001]
octave:14> infsup ("1/3", "7/9")
ans ⊂ [.33333333333333331, .7777777777777778]
octave:15> infsup ("121.2?")
ans ⊂ [121.14999999999999, 121.25]
octave:16> infsup ("5?32e2")
ans = [-2700, +3700]
octave:17> infsup ("-42??u")
ans = [-42, +Inf]
 
=== Decorations ===
With the subclass <code>infsupdec</code> it is possible to extend interval arithmetic with a decoration system. Every interval and intermediate result will additionally carry a decoration, which may provide additional information about the final result. The following decorations are available:
 
{| class="wikitable" style="margin: auto"
!Decoration
!Bounded
!Continuous
!Defined
!Definition
|-
|-
| com<br/>(common)
| disjoint (a, b)
| style="text-align: center" | ✓
| emptyintersect (a, b)
| style="text-align: center" | ✓
| 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)
| hdist (a, b)
|
| qdist (a, b)
| 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)
| disp (x)
|
| disp2str (x)
|
| style="text-align: center" | ✓
| '''''x''''' is a nonempty subset of Dom(''f'')
|-
|-
| trv<br/>(trivial)
| infsup (s)
|
| str2intval (s)
|
|
| always true (so gives no information)
|-
|-
| ill<br/>(ill-formed)
| isa (x, "infsup")
|
| isintval (x)
|
|
| Not an interval, at least one interval constructor failed during the course of computation
|}
|}


In the following example, all decoration information is lost when the interval is possibly divided by zero, i. e., the overall function is not guaranteed to be defined for all possible inputs.
== Developer Information ==
=== Source Code Repository ===
https://sourceforge.net/p/octave/interval/ci/default/tree/


octave:1> infsupdec (3, 4)
=== Dependencies ===
ans = [3, 4]_com
  apt-get install liboctave-dev mercurial make automake libmpfr-dev
octave:2> ans + 12
ans = [15, 16]_com
  octave:3> ans / "[0, 2]"
ans = [7.5, Inf]_trv


=== Arithmetic operations ===
=== Build ===
The interval packages comprises many interval arithmetic operations. Function names match GNU Octave standard functions where applicable, and follow recommendations by IEEE 1788 otherwise. It is possible to look up all functions by their corresponding IEEE 1788 name in the index {{Citation needed}}.
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>


One operation that should be noted is the <code>fma</code> function (fused multiply and add). It computes '''''x''''' × '''''y''''' + '''''z''''' in a single step and is much slower than multiplication followed by addition. However, it is more accurate and therefore preferred in some situations.
=== Architecture ===


octave:1> sin (infsup (0.5))
In a nutshell the package provides two new data types to users: bare intervals and decorated intervals. The data types are implemented as:
ans ⊂ [.47942553860420294, .47942553860420307]
* class <code>infsup</code> (bare interval) with attributes <code>inf</code> (lower interval boundary) and <code>sup</code> (upper interval boundary)
octave:2> pow (infsup (2), infsup (3, 4))
* class <code>infsupdec</code> (decorated interval) which extends the former and adds attribute <code>dec</code> (interval decoration).
ans = [8, 16]
octave:3> atan2 (infsup (1), infsup (1))
ans ⊂ [.785398163397448, .7853981633974487]


=== Reverse arithmetic 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:
[[File:Reverse-power-functions.png|400px|thumb|right|Reverse power operations. The relevant subset of the function's domain, where ''x''<sup>''y''</sup> ∈ [2, 3], is outlined and hatched.]]


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>X = sqrrev (C, X)</code> and will compute the enclosure of all numbers ''x'' ∈ X that fulfill the constraint ''x''² ∈ C.
+- 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


In the following example, we compute the constraints for base and exponent of the power function <code>pow</code> as shown in the figure.
=== Best practices ===
  octave:1> x = powrev1 (infsup ("[1.1, 1.45]"), infsup (2, 3))
==== Parameter checking ====
  x ⊂ [1.6128979635153644, 2.7148547265657923]
* 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.
  octave:2> y = powrev2 (infsup ("[2.14, 2.5]"), infsup (2, 3))
* 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.
  y ⊂ [.7564707973660297, 1.4440113978403293]
  if (not (isa (x, "infsup")))
  x = infsup (x);
  endif
  if (not (isa (y, "infsup")))
  y = infsup (y);
  endif


=== Numerical operations ===
if (not (isa (x, "infsupdec")))
  x = infsupdec (x);
endif
if (not (isa (y, "infsupdec")))
  y = infsupdec (y);
endif


=== Boolean operations ===
==== Use of Octave functions ====
Octave functions may be used as long as they don't introduce arithmetic errors. For example, the ceil function can be used safely since it is exact on binary64 numbers.
function x = ceil (x)
  ... parameter checking ...
  x.inf = ceil (x.inf);
  x.sup = ceil (x.sup);
endfunction


=== Error handling ===
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.
Due to the nature of set-based interval arithmetic, you should never observe errors during computation. If you do, there either is a bug in the code or there are unsupported data types.


  octave:1> infsup (2, 3) / 0
==== Vectorization & Indexing ====
  ans = [Empty]
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.
octave:2> infsup (0) ^ infsup (0)
  function x = plus (x, y)
  ans = [Empty]
  ... 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


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 <code>infsupdec</code> constructor will only issue a warning and return a [NaI], which will propagate and survive through computations.
== 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.


octave:3> infsup (3, 2) + 1
{|
error: illegal interval boundaries: infimum greater than supremum
! Function
''… (call stack) …''
! Status
octave:3> infsupdec (3, 2) + 1
! Information
warning: illegal interval boundaries: infimum greater than supremum
|-
ans = [NaI]
|colspan="3"|Real (or complex) data only: Matrices
 
|-
== Related work ==
|verbasis
 
|style="color:red"| trapped
For MATLAB there is a popular interval arithmetic toolbox [http://www.ti3.tu-harburg.de/rump/intlab/ INTLAB] by Siegfried Rump (member of IEEE P1788). It had been free (as in free beer) 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 not compatible with GNU Octave. I don't know if INTLAB is or will be compliant with IEEE 1788.
| depends on <code style="color:red">verfullcolrank</code>
 
|-
For C++ there is an interval library [https://github.com/nehmeier/libieeep1788/ libIEEE1788] by Marco Nehmeier (member of IEEE P1788). It aims to be standard compliant with IEEE 1788, but is not complete yet.
|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="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>
|-
|}


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.


[[Category:Octave-Forge]]
[[Category:Octave Forge]]

Latest revision as of 04:55, 13 August 2019

Octave Forge
interval
Interval.png
Real-valued interval arithmetic.
pkg install -forge interval
Version: 3.2.0 (2018-07-01)
Author(s): Oliver Heimlich <oheim@posteo.de>
Maintainer(s): Oliver Heimlich <oheim@posteo.de>
License: GPL-3.0+
Group: Community package
Documentation: Function reference
User manual
Download: interval-3.2.0.tar.gz
Repository: https://octave.sourceforge.io/pkg-repository/interval/
Dependencies: octave ≥ 3.8.0
Runtime: mpfr (≥ 3.1.0) libmpfr4 for Debian
Build: mpfr (≥ 3.1.0) libmpfr-dev for Debian


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[edit]

Development status[edit]

  • 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)[edit]

  • 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[edit]

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[edit]

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[edit]

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[edit]

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[edit]

Source Code Repository[edit]

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

Dependencies[edit]

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

Build[edit]

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[edit]

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[edit]

Parameter checking[edit]

  • 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[edit]

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[edit]

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[edit]

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
Retrieved from "https://wiki.octave.org/wiki/index.php?title=Interval_package&oldid=12214"