Introduction to Numerical Analysis
Exact answers are a luxury, and your calculator has been politely rounding them for you all along.
Press on your calculator. It says 1.414213562, calm and confident. But never ends and never repeats, so that answer is wrong. Politely wrong, wrong by less than a billionth, but wrong. Now here is the puzzle: if every number a machine gives you is slightly off, how did anyone ever land a spacecraft on the Moon with one? This chapter is about the small, honest lies inside every calculation, and how to keep them small.
In secondary school, equations came with tidy answers. gives . The quadratic formula handles every . It is easy to grow up believing there is always a formula waiting somewhere.
There isn't. In 1824 the Norwegian mathematician Niels Abel proved that there is no general formula for equations of degree five or higher. And an equation as innocent-looking as has no answer you can write with ordinary functions at all. Yet engineers need these numbers every day.
So we change the question. Instead of asking "what is the exact answer?", we ask "can I get an answer that is right to 2, 4, or 10 decimal places, and can I prove how close it is?" That shift is the whole of this course.
Numerical computation trades exactness for two things that are far more useful: a number that is close enough, and an honest statement of how close.
Formally, numerical analysis is the study of algorithms that use arithmetic to obtain approximate solutions to mathematical problems, together with bounds on the errors of those approximations. Notice the second half. An answer without an error estimate is just a rumour.
Spotted a mistake?