Some fast algorithms sequentially semiseparable representations

An extended sequentially semiseparable (SSS) representation derived from time-varying system theory is used to capture, on the one hand, the low-rank of the off-diagonal blocks of a matrix for the purposes of efficient computations and, on the other, to provide for sufficient descriptive richness to allow for backward stability in the computations. We present (i) a fast algorithm (linear in the number of equations) to solve least squares problems in which the coefficient matrix is in SSS form, (ii) a fast algorithm to find the SSS form of $X$ such that $AX=B$, where $A$ and $B$ are in SSS form, and (iii) a fast model reduction technique to improve the SSS form.