An implementation of an algebraically closed interval system of the extended real number set
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:
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
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
division
when both intervals contain zero creates a whole intervalcosine, tangent
works with positive/negative values out of the boxDue 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
:
const 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
const compile = require('interval-arithmetic-eval');
compile('sin(exp(x)) + tan(x) - 1/cos(PI) * [1, 3]^2').eval({ x: [0, 1] })
$ npm install --save interval-arithmetic
import IOps, { Interval } from 'interval-arithmetic'
IOps.add(Interval(1), Interval(2))
See the homepage
npm test
Deployment steps
// after the working directory is clean
npm test
npm run build
npm version (major|minor|patch)
git push origin master
// if everything went well
npm publish
npm run deploy
2015-2020 © Mauricio Poppe