Interval package

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

• Latest version at Octave Forge
  • pkg install -forge interval
  • function reference
  • package documentation (user manual)

Third-party

• 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 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)

• 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

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:

• 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 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: 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

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

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 apt-get install libmpfr-dev 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 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

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

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
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