Interval package

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Octave-interval.png

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
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