27 Feb 2021

Testing 4x4 matrix inversion precision

It is extremely rare that a hobby software project of mine gets completed, but now it has happened. Behold! Fourbyfour!

Have you ever had to implement a mathematical algorithm, say, matrix inversion? You want it to be fast and measuring the speed is fairly simple, right. But what about correctness? Or precision? Behavior around inputs that are on the edge? You can hand-pick a few example inputs, put those into your test suite, and verify the result is what you expect. If you do not pick only trivial inputs, this is usually enough to guarantee your algorithm does not have fundamental mistakes. But what about those almost invalid inputs, can you trust your algorithm to not go haywire on them? How close to invalid can your inputs be before things break down? Does your algorithm know when it stops working and tell you?

Inverting a square matrix requires that the inverse matrix exists to begin with. Matrices that do not mathematically have an inverse matrix are called singular. Can your matrix inversion algorithm tell you when you are trying to invert a matrix that cannot be inverted, or does it just give you a bad result pretending it is ok?

Working with computers often means working with floating-point numbers. With floating-point, the usual mathematics is not enough, it can actually break down. You calculate something and the result a computer gives you is total nonsense, like 1+2=2 in spirit. In the case of matrix inversion, it's not enough that the input matrix is not singular mathematically, it needs to be "nice enough" numerically as well. How do you test your matrix inversion algorithm with this in mind?

These questions I tried to answer with Fourbyfour. The README has the links to the sub-pages discussing how I solved this, so I will not repeat it here. However, as the TL;DR, if there is one thing you should remember, it is this:

    Do not use the matrix determinant to test if a matrix is invertible!

Yes, the determinant is zero for a singular matrix. No, close to zero determinant does not tell you how close to singular the matrix is. There are better ways.