Oxford course

Over and over again we see a pattern like this:

nonlinear -------—linearize & iterate--------—> LINEAR

PDE ---------—discretize----------—> ALGEBRA

Because of this, computers have brought linear algebra, and numerical linearalgebra, to the forefront of the mathematical sciences.

Standard algorithms to solve linear system $Ax = b$, i.e. matrix inversion, grow like $O(N^3)$. To improve this one can:

• use parallel computing, or
• algorithms to take advantage of sparsity of matrix (many entries are zero)

In recent years flop count is less and less important at the high end (i.e. for many processors) – communication is a bigger bottleneck.