Numerics, or, Why can’t you just tell the computer to do it?

When most people think of math, they think of the math they did in school: repeated arithmetic until your brain goes numb, followed by basic algebra and trig. You weren’t allowed to use calculators on most tests for the simple reason that almost everything you did could be done by a calculator in a fraction of the time.

Real math isn’t like that. Mathematicians handle proofs and abstract concepts, definitions and constructions and functions and generally not a single actual number in sight. That much, at least, shouldn’t be surprising.

What might be surprising is that even tasks which seem very much like things computers could do easily take a fair bit of human ingenuity.

In physics, I do a lot of integrals. For those of you unfamiliar with calculus, integrals can be thought of as the area between a curve and the x-axis.

Areas seem like the sort of thing it would be easy for a computer to find. Chop the space into little rectangles, add up all the rectangles under the curve, and if your rectangles are small enough you should get the right answer. Broadly, this is the method of numerical integration. Since computers can do billions of calculations per second, you can chop things up into billions of rectangles and get as close as you’d like, right?

Heck, ten is a lot. Can we just do ten?

Heck, ten is a lot. Can we just do ten?

For some curves, this works fine. For others, though…

Ten might not be enough for this one.

Ten might not be enough for this one.

See how the left side of that plot goes off the chart? That curve goes to infinity. No matter how many rectangles you put on that side, you still won’t have any that are infinitely tall, so you’ll still miss that part of the curve.

Surprisingly enough, the area under this curve isn’t infinite. Do the integral correctly, and you get a result of 2. Set a computer to calculate this integral via the sort of naïve numerical integration discussed above though, and you’ll never find that answer. You need smarter methods: smart humans doing the math, or smart humans programming the computer.

Another way this can come up is if you’re adding up two parts of something that go to infinity in opposite directions. Try to integrate each part by itself and you’ll be stuck.

firstplot

secondplot

But add them together, and you get something quite a bit more tractable.

Yeah, definitely a ten-rectangle job.

Yeah, definitely a ten-rectangle job.

Numerical integration, and computers in general, are a very important tool in a scientist’s arsenal. But in order to use them, we have to be smart, and know what we’re doing. Knowing how to use our tools right can take almost as much expertise and care as working without tools.

So no, I can’t just tell the computer to do it.

Advertisements

One thought on “Numerics, or, Why can’t you just tell the computer to do it?

  1. Wyrd Smythe

    Well, yeah. 🙂 Computers, after all, are nothing more than over-grown calculators.

    Lately, due I think to how heavily they’re used in popular media, computers have become a kind of modern “magic” people believe capable of all sorts of (ridiculous) things. We have a kind of modern myth that any (good enough) hacker can break into any computer and make it do anything. And they often play the role of modern Frankenstein’s monster.

    Like

    Reply

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s