The QR Factorization for Banded-Plus-Semiseparable Matrices Is Computable in Linear Complexity

We show that the $QR$ factorization of a banded-plus-semiseparable (BPS) matrix is computable in optimal linear complexity with respect to the discretization size by showing that the intermediate stages of a $QR$ factorization as computed using Householder reflection maintain a specific structure which has optimal storage. This theoretical result enables the design of stable, linear-complexity algorithms for solving the associated linear systems. For symmetric BPS matrices, we further show that the $RQ$ product – central to eigenvalue computations via the $QR$ algorithm – also preserves the BPS structure, leading to a linear-complexity algorithm for each iteration. Numerical experiments validate the optimal linear complexity, confirm high numerical accuracy, and demonstrate substantial speedups compared with existing hierarchical approaches. The algorithms have been implemented in an open-source Julia package, providing an efficient and accessible platform for practical use.