Interval package

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The GNU Octave interval package for real-valued interval arithmetic.

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 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 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]

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).

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: 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 1788-2015 and is designed in a modular way, supporting several interval data types and different flavors of interval arithmetic [6]. 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 1788-2015 (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 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

  • Octave
    • Version 3.8.0 or greater
    • No need to compile from source, but you need development files e.g. package liboctave-dev 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 libmpfr-dev 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 unit-test
    • Installation:
      1. Clone the git repository from https://github.com/oheim/ITF1788
      2. Install python 3 and the dependencies described by ITF1788's setup.py file
      3. 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 poppler-utils)

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
`- test/                       – interval arithmetic unit tests

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
  • Methods of class infsupdec as well as methods of class infsup that are not overridden by infsupdec must check parameters using the isnai 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

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 p-coded ol
verdistsing trapped depends on versingval
verfullcolrank encrypted implemented in p-coded zd
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 encrypted implemented in p-coded ol
vereigback free dependency norm is already implemented
verspectrad trapped main part implemented in p-coded ol
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 p-coded ol
verthinsvd encrypted implemented in p-coded ol
Real (or complex) data only: Matrix functions
vermatfun trapped main part implemented in p-coded ol
Real data only: Linear systems (rectangular)
verlinineqnn free 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 trapped depends on verabsvaleqnall
verlinprog free use glpk as a replacement for linprog; dependency verifylss is implemented as mldivide
verlinprogg encrypted implemented in p-coded at
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 [7])
Real (or complex) data only: Polynomials
verroots trapped main part implemented in p-coded ol
Interval (or real) data: Matrices
verhurwstab trapped depends on verposdef
verinverse trapped depends on verintervalhull
verinvnonneg free
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 [8]
Interval (or real) data: Matrices: Eigenvalues and singular values
vereigsym trapped main part implemented in p-coded ol, depends on verspectrad
vereigval trapped depends on verregsing
vereigvec free
verperrvec free depends on vereigvec
versingval trapped depends on vereigsym
Interval (or real) data: Matrices: Decompositions
verqr (experimental) free 2-norm not implemented yet, qr has already been implemented. todo: check if this function is better; depends on verchol
verchol (experimental) free todo: check if already implemented as part of qr
Interval (or real) data: Linear systems (square)
verenclinthull trapped main part implemented in p-coded ea
verhullparam encrypted implemented in p-coded jz
verhullpatt trapped main part implemented in p-coded jz
verintervalhull encrypted implemented in p-coded intervalhull (find algorithm in [9])
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 trapped main part implemented in p-coded ek (find algorithm in [10], improved version in [11])
verabsvaleqnall trapped depends on verabsvaleqn, see also [12]
verbasintnpprob trapped depends on verregsing
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