Decay bounds for the numerical quasiseparable preservation in matrix functions
For numerical linear algebra researchers, it offers theoretical guarantees for structure preservation in matrix function computations, though the results are incremental extensions of existing techniques.
The paper provides bounds guaranteeing that when a holomorphic function is applied to a matrix with off-diagonal singular value decay, the resulting matrix preserves a similar decay, enabling efficient computation with hierarchical matrices.
Given matrices $A$ and $B$ such that $B=f(A)$, where $f(z)$ is a holomorphic function, we analyze the relation between the singular values of the off-diagonal submatrices of $A$ and $B$. We provide family of bounds which depend on the interplay between the spectrum of the argument $A$ and the singularities of the function. In particular, these bounds guarantee the numerical preservation of quasiseparable structures under mild hypotheses. We extend the Dunford-Cauchy integral formula to the case in which some poles are contained inside the contour of integration. We use this tool together with the technology of hierarchical matrices ($\mathcal H$-matrices) for the effective computation of matrix functions with quasiseparable arguments.