  The class of quasiseparable matrices is defined by the property that any
submatrix entirely below or above the main diagonal has small rank, namely
below a bound called the order of quasiseparability. These matrices arise
naturally in solving PDE's for particle interaction with the Fast Multi-pole
Method (FMM), or computing generalized eigenvalues. From these application
fields, structured representations and algorithms have been designed in
numerical linear algebra to compute with these matrices in time linear in the
matrix dimension and either quadratic or cubic in the quasiseparability order.
Motivated by the design of the general purpose exact linear algebra library
LinBox, and by algorithmic applications in algebraic computing, we adapt
existing techniques introduce novel ones to use quasiseparable matrices in
exact linear algebra, where sub-cubic matrix arithmetic is available. In
particular, we will show, the connection between the notion of
quasiseparability and the rank profile matrix invariant, that we have
introduced in 2015. It results in two new structured representations, one being
a simpler variation on the hierarchically semiseparable storage, and the second
one exploiting the generalized Bruhat decomposition. As a consequence, most
basic operations, such as computing the quasiseparability orders, applying a
vector, a block vector, multiplying two quasiseparable matrices together,
inverting a quasiseparable matrix, can be at least as fast and often faster
than previous existing algorithms.
