Interval package
The GNU Octave interval package for realvalued interval arithmetic.
Contents
Distribution
 Latest version at Octave Forge
pkg install forge interval
 function reference
 package documentation (user manual)
 MacPorts for Mac OS X
 FreshPorts for FreeBSD
Development status
 Completeness
 All required functions from IEEE Std 17882015, IEEE standard for interval arithmetic, are implemented. The standard was approved on June 11, 2015. It will remain active for ten years.
 In addition there are functions for interval matrix arithmetic, plotting and solvers.
 Quality
 Includes tests for all functions, many tests for basic functions
 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 will always need more testing.
 Portability
 Runs in GNU Octave 3.8.2 and 4.0.0
 Known to run under GNU/Linux, Microsoft Windows, Mac OS X and FreeBSD
 Possible TODOs
 To be considered in the future: Algorithms can be migrated from the CXSC Toolbox (C++ code) from [1] (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.
 Interval version of
interp1
 The CAPD package contains an algorithm for QR factorization. Analyze. Migrate.
 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
 While at it, also allow multiple subscripts in
>> 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?
Compatibility
The interval package's main goal is to be compliant with IEEE Std 17882015, so it is compatible with other standardconforming implementations (on the set of operations described by the standard document).
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 builtin arithmetic functions, roundoff, 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 nonfree software, it grants none of the four essential freedoms of free software
 INTLAB is not conforming to IEEE Std 17882015 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. 
# 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

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) 
Similar software
For C++ there is an interval library libieeep1788 by Marco Nehmeier (member of IEEE P1788). It aims to be standard compliant with IEEE Std 17882015 and is designed in a modular way, supporting several interval data types and different flavors of interval arithmetic [2]. The GNU Octave interval package shares several unit tests with libieeep1788.
For Julia there is an evolving interval library ValidatedNumerics.jl by Luis Benet and David P. Sanders. It is planned to become conforming to IEEE Std 17882015 (or to the basic standard 1788.1) in the future.
Developer Information
Source Code Repository
https://sourceforge.net/p/octave/interval/ci/default/tree/
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 testsuite 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
 Octave
 Version 3.8.0 or greater
 No need to compile from source, but you need development files e.g. package
liboctavedev
in Debian.
 Mercurial
 Texinfo
 MPFR
 Version 3.1.0 or greater
 No need to compile from source, but you need development files e.g. package
libmpfrdev
in Debian.
 Octave package: doctest
 Purpose: Verification of the examples in the manual and in the function documentation
 Installation: Use
pkg install forge doctest
inside Octave
 Octave package: generate_html
 Purpose: Generate HTML documenation for publication on Octave Forge (only needed for release)
 Installation: Use
pkg install forge generate_html
inside Octave
 ITF1788
 Purpose: Compilation of unittest
 Installation:
 Clone the git repository from https://github.com/oheim/ITF1788
 Install python 3 and the dependencies described by ITF1788's
setup.py
file  Set up an environment variable ITF1788_HOME to point to your local git workspace, for example put the line
export IFT1788_HOME=/home/user/ITF1788
into your.bashrc
.
 LilyPond, Inkscape, Poppler
 Purpose: Generate / convert images for the manual
 Installation: Use your distribution's package manager (look for packages called
lilypond
inkscape
popplerutils
)
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 attributesinf
(lower interval boundary) andsup
(upper interval boundary)  class
infsupdec
(decorated interval) which extends the former and adds attributedec
(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 mfiles. Arithmetic operations that require correctlyrounded results are implemented in octfiles (C++ code), these are used internally by the mfiles 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 octfile sources ` test/ – interval arithmetic unit tests
Best practices
Parameter checking
 All methods must check
nargin
and callprint_usage
if the number of parameters is wrong. This prevents simple errors by the user.  Methods with more than 1 parameter must convert noninterval parameters to intervals using the class constructor. This allows the user to mix noninterval 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
 Methods of class
infsupdec
as well as methods of classinfsup
that are not overridden byinfsupdec
must check parameters using theisnai
function and return the NAI value if it is present. This will propagate NAI values through any function evaluations.
if (isnai (x)) result = x; return 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 result = ceil (x) ... parameter checking ... result = infsup (ceil (x.inf), ceil (x.sup)); endfunction
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 result = 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; result = infsup (l, u); endfunction