Honest computer mistakes

Computers can make mistakes when they are doing calculations. How do you know how big these mistakes are? Your calculator is definitely not going to give an honest answer. So we better find a way to pry that information from him/her :)

For mathematicians computers are a bit of a mixed basket. Computers can do computations very quickly, however the solution that a computer gives is not necessarily correct. For example, if you ask your calculator to compute √2 it will probably give you 1.41421356237, but √2 is not equal to 1.41421356237. You could say that this number is an approximation of √2. Is it a good approximation? Well, let's look at two examples:

Example 1: Let's take a look at the following equation

y²-2

Suppose that y=√2 then the equation is zero. Well now consider y=1.41421356237. Then the solution is approximately -8.75 × 10^-12. That is very tiny! So for this example we can conclude that it is a good approximation.

Example 2: Let's take a look at the following equation

10¹² × (y²-2)

Again take y=√2 then it is easy to see that the above gives 0. Well now take y=1.41421356237 well then the above gives something which is approximately equal to -8.75. Wow so that is a big difference. So now you would conclude that it is a bad approximation.

Photo by PublicDomainPictures / 18042 - Pixabay- CC0 creative commons
(I didn't know what pictures to use but owls are always a hoot :D )

A contradiction?

We got two contradicting results from example 1 and 2. So is there anything we can conclude? In example 1 and 2 we were looking at different equations. So whether something is a good approximation might depend on the equation you are working with. But how can we determine if an equation gives a bad solution resulting from the approximation?

Some new math

And this is where a relative new field of mathematics comes in which is called interval arithmetic (actually it was already created around the 1950s but it recently became a hot topic because of the great improvement in processing power [1]). Instead of thinking about √2 as a number which can be approximated we are going to think about it as a number which is contained in between two numbers or equivalently, think of √2 as contained in an interval. For example, 1<√2<2 but with a bit more effort you can show that

1.41421356236<√2<1.41421356238

Returning to Example 1 suppose you know that

1.41421356236<y<1.41421356238

Then you know that the solution of the equation is inbetween -3.07 × 10^-11 and 1.96 × 10^-11. Now consider the equation in Example 2. Then you get that the solution of the equation is inbetween -30.7 and 19.6. So no wonder that you get a bad approximation in the second case.

Photo by TonW - Pixabay- CC0 creative commons

On the frontlines of mathematics

The case I considered is just very simple but you can apply these techniques to more complicated equations such as integrals, derivatives or even ordinary differential equations (which are equations governed by rates of change, for example the motion of a pendulum or the orbits of planets around the sun) [2]. You can pretty much implement it for all operations learned in a calculus course. So why not just apply the rules of calculus by hand? Well, sometimes you cannot evaluate an integral, like this one

So in those case you can use interval arithmetic to find you a precise approximation. That is pretty neat!

In conclusion: Computers might be inaccurate but they can tell you how inaccurate they are (if you put in the extra work). Which is a sweet deal for mathematicians.

Photo by TonW - Pixabay- CC0 creative commons

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Technical Disclaimer: In the case of ordinary differential equation it can be quite complicated to implement interval arithmetic. Most of the times you need to build an implementation for a specific ordinary differential equation. In addition it does not always work. So it is not like a miracle method. However, in certain settings it is able to show the existence of vastly complicated dynamics. For example, for the Lorenz system interval arithmetic was used to prove the existence (of a specific type of) chaos [3].

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References
[1] R. Moore, Interval Analysis, by Prentice-Hall, Englewood Cliffs, NJ, 1966.
[2] W. Tucker, Validated Numerics: A Short Introduction to Rigorous Computations, Princeton University Press, 2011
[3] W. Tucker, The Lorenz attractor exists, C. R. Acad. Sci. Paris, t. 328, Serie I, 1197–1202, 1999

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All images made with inkscape. It is free! (the top picture was actually made for my Turing post but I thought it would fit nicely here as well). The integral equation made using http://quicklatex.com/

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Merchandise :D

There is a MathOwl shop which sells my artsy fartsy stuff. If you got some spare money head over there. You can learn about the colors of pi over here here. I also have really cheap stuff available like these stickers They are an absolute hoot.

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