Numerical linear algebra is principally concerned with solving two
types of problems: A x = b and A v = lambda v. In both cases, it
can be surprisingly hard to produce accurate values for solutions
when inputs to the problem may be off by very small amounts. In
applications, the problems being solved are often intended to
approximate operators in infinite dimensional settings. The problems
with the first type of calculation are well understood, and very
satisfactory theorems about approximation of solutions of high (or
even infinite) dimensional operators are known. In the case of the
eigenvalue problem the situation is more interesting. For normal
operators a great deal is known, however, for non-normal operators
the situation is not as well understood. Operators may converge
without spectra (eigenvalues) converging. We will explore the
concept of the $\epsilon$-pseudospectrum which "repairs" some of the
problems that the spectrum has, and we will look at some of the tools
that are available for computing the pseudospectrum. In addition we
will discuss some of the applications of these ideas to analysis of
the stability of solutions of PDEs.