Problems Programming Math
Posted on October 17, 2008 Comments (2)
Arithmetic Is Hard – To Get Right by Mark Sofroniou
The standard “schoolbook” algorithms are pretty easy. But they’re inefficient and often unnecessarily inaccurate. So people like me have done a huge amount of work to find algorithms that are more efficient and accurate. And the problem is that these algorithms are inevitably more complicated, and one has to be very careful to avoid insidious bugs.
Take multiplying integers, for example. The standard “schoolbook” long-multiplication algorithm uses n^2 multiplications to multiply two n-digit numbers. But many of these multiplications are actually redundant, and we now know clever algorithms that take n^1.58, n log n, or even fewer multiplications for large n. So this means that if one wants to do a million-digit multiplication, Mathematica can do it in a fraction of a second using these algorithms–while it would take at least a few minutes using standard long multiplication.
It’s not easy to get reliable numerical computation, and it’s not something one can “bolt on” after the fact. It’s something one has to build in from the beginning, as we’ve done in Mathematica for nearly 20 years.