interval-arithmetic

An implementation of an algebraically closed interval system of the extended real number set

Table of Contents generated with DocToc

• Description
  • floating point operations
  • Interval arithmetic
  • Notable modifications
• Interval arithmetic evaluator
• Installation
• API
• Development

Description

An interval is a pair of numbers which represents all the numbers between them, closed means that the bounds are also included in the representation, extended real because the real number system is extended with two elements: -∞ and +∞ representing negative infinity and positive infinity respectively.

The implementation is a modified port of the Boost's interval arithmetic library, the modifications are based on some guidelines from the following papers/presentations:

• Interval Arithmetic: from Principles to Implementation - T. Hickey, Q. Ju, M.H. van Emden
• Interval Arithmetic: Python Implementation and Applications - Stefano Taschini
• The Boost interval arithmetic library - Hervé Brönnimann, Guillaume Melquiond, Sylvain Pion
• Graphing equations with generalized interval arithmetic - Jeffrey Allen Tupper

floating point operations

Floating point is a way to represent a real number in an approximate way (due to the finite space existing on a computer), most calculations with real numbers will produce quantities that cannot be exactly represented with the space allocated and therefore this operation needs to be rounded in order to fit back into its finite representation, such errors are described in more detail here

Interval arithmetic

Instead of using a single floating point number as an approximation of a real number, interval arithmetic represents the approximated value as a set of possible values (considering the numbers that floating point cannot represent), let's say we want to represent the number 1 / 3, as a single floating point number it's approximated as 0.333333333333..., in the end there will be some 333... decimals that will be lost due to the nature of floating point, instead we can represent this number with the interval [0.2, 0.4], with this interval we're completely sure that 1 / 3 is within the interval (although the interval is also representing many more numbers), to improve the scope of the interval we have to understand that numbers in JavaScript are represented with 64 bits, therefore to get the next floating point number of a single precision number the last bit needs to be incremented to get the upper bound, and the last bit also needs to be decremented to get the lower point

Notable modifications

• next/previous IEEE754 floating point number implementation based on Typed Arrays
• division when both intervals contain zero creates a whole interval
• cosine, tangent works with positive/negative values out of the box

Interval arithmetic evaluator

Due to the expressive nature of the way the methods interact with intervals it's sad that even the simplest operation needs a lot of characters to be typed, let's consider evaluating the result of 1 + 2 expressed with intervals

Interval.add(new Interval(1, 1), new Interval(2, 2))

This gets worse when the expression to be evaluated becomes complex like sin(exp(x)) + tan(x) - 1/cos(PI) * [1, 3]^2:

var x = Interval(0, 1);
Interval.add(
  Interval.sin(Interval.exp(x)),
  Interval.sub(
    Interval.tan(x),
    Interval.mul(
      Interval.div(Interval.ONE, Interval.cos(Interval.PI)),
      Interval.pow(Interval(1, 3), 2)
    )
  )
);

To avoid this 'expressiveness' mess there's an interval arithmetic evaluator module which I've created to deal with all the work of parsing/evaluating expressions like the one above

var compile = require('interval-arithmetic-eval');
compile('sin(exp(x)) + tan(x) - 1/cos(PI) * [1, 3]^2').eval({ x: [0, 1] })

Installation

$ npm install --save interval-arithmetic

API

See the homepage

Development

npm start

Deploy steps

// after the working directory is clean
(optional) npm version (major|minor|patch)
npm run deploy
// if everything went well
npm publish

2015 © Mauricio Poppe