# Difference between revisions of "Interval package"

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

Plotting the interval enclosure of a function

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

### 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`