240

edits

Jump to navigation
Jump to search
##
Interval package
(edit)

###
Revision as of 14:18, 14 January 2015

455 bytes removed
, 14:18, 14 January 2015
m

Removed dependency to fenv package

The [https://sourceforge.net/p/octave/interval/ interval package] provides data types and fundamental operations for real valued interval arithmetic based on the common floating-point format “binary64” a. k. a. double-precision. It aims to be standard compliant with the (upcoming) [http://standards.ieee.org/develop/project/1788.html IEEE 1788] and therefore implements the ''set-based'' interval arithmetic flavor. '''Interval arithmetic''' produces mathematically proven numerical results.

Warning: The package has not yet been released. If you want to experience the development version, you may (1~~) install the (currently deprecated) [http://octave.sourceforge.net/fenv/ fenv package], (2~~) download a [https://sourceforge.net/p/octave/interval/ci/default/tarball snapshot version of the interval package], (~~3~~2) install the [http://www.mpfr.org/mpfr-current/#download development library of MPFR] for your system, (~~4~~3) execute <code>make install</code> in the <code>src/</code> subfolder, (5) navigate to the <code>inst/</code> subfolder and run octave.

== Motivation ==

''Why is the interval package slow?''

All arithmetic interval operations are simulated in high-level octave language using C99 floating-point routines or multi-precision floating-point routines, which is a lot slower than hardware implementations [https://books.google.de/books?id=JTc4XdXFnQIC&pg=PA61]. Building interval arithmetic operations from floating-point routines is easy for simple monotonic functions, e. g., addition and subtraction, but is complex for others, e. g., [http://exp.ln0.de/heimlich-power-2011.htm interval power function], atan2, or [[#Reverse_arithmetic_operations|reverse functions]]. For some interval operations it is not even possible to rely on floating-point routines, since not all required routines are available in C99 or BLAS~~.~~ ~~For example, for some tightly rounded results of vector and matrix operations the interval package has to simulate a [http://books.google.de/books?hl=de&id=I7X9EVfeV5EC&q=accumulator Kulisch accumulator], which introduces a computational overhead of factor 10. However, the Kulisch accumulator could be implemented in hardware and then outperform floating-point routines~~.

Great algorithms and optimizations exist for matrix arithmetic in GNU octave. Good interval versions of these still have to be found and implemented.

Some basic operations are provided by the C library on common platforms with directed rounding and correctly rounded result: plus, minus, division, multiplication and square root. All other GNU Octave arithmetic functions are not guaranteed to produce accurate results, because they are based on C99 floating-point routines [http://www.gnu.org/software/libc/manual/html_node/Errors-in-Math-Functions.html#Errors-in-Math-Functions]. Their results depend on hardware, system libraries and compilation options.

The interval package handles all arithmetic functions with the help of the [http://www.mpfr.org/ GNU MPFR library]. With MPFR it is possible to compute system-independent, valid and tight enclosures of the correct results.

== Quick start introduction ==

octave:1> sin (infsup (0.5))

ans ⊂ [.47942553860420294, .~~47942553860420307~~47942553860420301]

octave:2> pow (infsup (2), infsup (3, 4))

ans = [8, 16]

octave:3> atan2 (infsup (1), infsup (1))

ans ⊂ [.~~785398163397448~~7853981633974482, .~~7853981633974487~~7853981633974484]

=== Reverse arithmetic operations ===

In the following example, we compute the constraints for base and exponent of the power function <code>pow</code> as shown in the figure.

octave:1> x = powrev1 (infsup ("[1.1, 1.45]"), infsup (2, 3))

x ⊂ [1.~~6128979635153644~~6128979635153646, 2.~~7148547265657923~~7148547265657915]

octave:2> y = powrev2 (infsup ("[2.14, 2.5]"), infsup (2, 3))

y ⊂ [.~~7564707973660297~~7564707973660299, 1.~~4440113978403293~~4440113978403284]

=== Numerical operations ===

Above mentioned operations can also be applied element-wise to interval vectors and matrices. Many operations use [http://www.gnu.org/software/octave/doc/interpreter/Vectorization-and-Faster-Code-Execution.html#Vectorization-and-Faster-Code-Execution vectorization techniques].

In addition, there are matrix operations on interval matrices. These operations comprise: tight ~~dot product, fast ~~dot product, tight ~~matrix multiplication, fast ~~matrix multiplication, tight vector sums, matrix inversion, matrix powers, and solving linear systems. As a result of missing hardware / low-level library support and missing optimizations, these operations are quite slow compared to familiar operations in floating-point arithmetic.

octave:1> A = infsup ([1, 2, 3; 4, 0, 0; 0, 0, 1]); A (2, 3) = "[0, 6]"

240

edits

Retrieved from "https://wiki.octave.org/Special:MobileDiff/5271"

- Not logged in
- Talk
- Contributions
- Create account
- Log in