|Real-valued interval arithmetic.|
|Author(s):||Oliver Heimlich <firstname.lastname@example.org>|
|Maintainer(s):||Oliver Heimlich <email@example.com>|
|Documentation:||Function reference |
|Dependencies:||octave ≥ 3.8.0 |
Runtime: mpfr (≥ 3.1.0) libmpfr4 for Debian
Build: mpfr (≥ 3.1.0) libmpfr-dev for Debian
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
-infare 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].
- 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
- 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.
- 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  (nlinsys.cpp and cpzero.cpp), however these would need gradient arithmetic and complex arithmetic.
- Interval version of
subsasgnto 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
- 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  
- Verified Convex Hull for Inexact Data  
- 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.
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.
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
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.
|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)|
Source Code Repository
apt-get install liboctave-dev mercurial make automake libmpfr-dev
The repository contains a Makefile which controls the build process. Some common targets are:
make releaseCreate a release tarball and the HTML documentation for Octave Forge (takes a while).
make checkRun the full test-suite to verify that code changes didn't break anything (takes a while).
make runQuickly start Octave with minimal recompilation and functions loaded from the workspace (for interactive testing of code changes).
apt-get install libmpfr-dev autoconf automake inkscape zopfli
In a nutshell the package provides two new data types to users: bare intervals and decorated intervals. The data types are implemented as:
infsup(bare interval) with attributes
inf(lower interval boundary) and
sup(upper interval boundary)
infsupdec(decorated interval) which extends the former and adds attribute
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
- All methods must check
print_usageif 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
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.
|Real (or complex) data only: Matrices|
|verbasis||trapped||depends on |
|vercondnum||trapped||depends on |
|verdet||trapped||depends on |
|verdistsing||trapped||depends on |
|verfullcolrank||trapped||depends on |
|vernorm2||trapped||depends on |
|vernull (experimental)||unknown||depends on |
|verorth||trapped||depends on |
|verorthproj||trapped||depends on |
|verpd||trapped||depends on |
|verpmat||trapped||depends on |
|verrank||trapped||depends on |
|verrref||trapped||depends on |
|Real (or complex) data only: Matrices: Eigenvalues and singular values|
|vereig||trapped||depends on proprietary |
|free, migrated (for real eigenvalues)||dependency |
|verspectrad||trapped||main part implemented in |
|Real (or complex) data only: Matrices: Decompositions|
|verpoldec||trapped||depends on |
|verrankdec||trapped||depends on |
|verspectdec||trapped||main part implemented in |
|verthinsvd||trapped||implemented in |
|Real (or complex) data only: Matrix functions|
|vermatfun||trapped||main part implemented in |
|Real data only: Linear systems (rectangular)|
|free, migrated||use |
|verlsq||trapped||depends on |
|Real data only: Optimization|
|verlcpall||free||depends on |
|free, migrated||use |
|verlinprogg||trapped||depends on |
|Real (or complex) data only: Polynomials|
|verroots||trapped||main part implemented in |
|Interval (or real) data: Matrices|
|verhurwstab||trapped||depends on |
|verinverse||free||depends on |
|verposdef||trapped||depends on |
|Interval (or real) data: Matrices: Eigenvalues and singular values|
|vereigsym||trapped||main part implemented in |
|vereigval||trapped||depends on |
|verperrvec||free||the function is just a wrapper around |
|versingval||trapped||depends on |
|Interval (or real) data: Matrices: Decompositions|
|free, migrated||migrated version has been named after the standard Octave function |
|Interval (or real) data: Linear systems (square)|
|verenclinthull||free||to be migrated|
|verhullparam||free||depends on |
|verhullpatt||free||depends on |
|verintervalhull||free||to be migrated|
|Interval (or real) data: Linear systems (rectangular)|
|verintlinineqs||free||depends on |
|vertolsol||free||depends on |
|Interval (or real) data: Matrix equations (rectangular)|
|Real data only: Uncommon problems|
|verabsvaleqn||free||to be migrated|
|verabsvaleqnall||free||depends on |
|verbasintnpprob||trapped||depends on |