Interval package
The GNU Octave interval package for realvalued 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
andinf
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].
Distribution
 Latest version at Octave Forge
pkg install forge interval
 function reference
 package documentation (user manual)
Thirdparty
 Debian GNU/Linux, Launchpad Ubuntu
 archlinux user repository
 Included in official Windows installer and installed automatically with Octave (since version 4.0.1)
 MacPorts for Mac OS X
 FreshPorts for FreeBSD
 Cygwin for Windows
 openSUSE build service
Development status
 Completeness
 All required functions from IEEE Std 17882015, IEEE standard for interval arithmetic, are implemented. The standard was approved by IEEESA 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 1788.12017, IEEE standard for interval arithmetic (simplified) is fully implemented. The standard was approved by IEEESA on December 6, 2017 (and published in January 2018).
 In addition there are functions for interval matrix arithmetic, Ndimensional interval arrays, plotting, and solvers.
 Quality
 Most arithmetic operations produce tight, correctlyrounded results. That is, the smallest possible interval with doubleprecision (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 CXSC 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
 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?
 Tight Enclosure of Matrix Multiplication with Level 3 BLAS [2] [3]
 Verified Convex Hull for Inexact Data [4] [5]
 Implement usercontrollable 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 17882015, so it is compatible with other standardconforming implementations (on the set of operations described by the standard document). Other implementations, which are known to aim for standard conformance are:
 IntervalArithmetic.jl package (Julia)
 JInterval library (Java)
 ieeep1788 library (C++) created by Marco Nehmeier, later forked by Dmitry Nadezhin
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: 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
aptget install liboctavedev mercurial make automake libmpfrdev
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
aptget install libmpfrdev 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 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
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
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

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)  
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

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

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

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 GramSchmidt process, which seems to be more accurate and faster than the Cholsky decomposition or Householder reflections used in verqr. No migration needed.

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
