This talk discusses some fast numerical methods using certain rank
structured matrices: sequentially semiseparable (SSS) matrices
and hierarchically semiseparable (HSS) matrices. I will first
briefly show an example using semiseparable matrices: to find
polynomial roots and to estimate their conditions in $O(n^2)$ flops
instead of classical $O(n^3)$, where $n$ is the degree of the
polynomial. Then I will discuss in detail a superfast multifrontal
type direct method for large discretized linear systems. Rank
structures in the multifrontal method for certain discretized
matrices are exploited. Then semiseparable matrices are used
to approximate dense frontal and update matrices in the elimination.
A generalized semiseparable type factorization is obtained in linear
time. The overall sover has nearly linear complexity, linear storage,
and good potential for parallelization. It can also work as an
effective preconditioner.