Here are ten things about numerical linear algebra that you may find surprising if you’re not familiar with the field.
- Numerical linear algebra applies very advanced mathematics to solve problems that can be stated with high school mathematics.
- Practical applications often require solving enormous systems of equations, millions or even billions of variables.
- The heart of Google is an enormous linear algebra problem. PageRank is essentially an eigenvalue problem.
- The efficiency of solving very large systems of equations has benefited at least as much from advances in algorithms as from Moore’s law.
- Many practical problems — optimization, differential equations, signal processing, etc. — boil down to solving linear systems, even when the original problems are non-linear. Finite element software, for example, spends nearly all its time solving linear equations.
- A system of a million equations can sometimes be solved on an ordinary PC in under a millisecond, depending on the structure of the equations.
- Iterative methods, methods that in theory require an infinite number of steps to solve a problem, are often faster and more accurate than direct methods, methods that in theory produce an exact answer in a finite number of steps.