11 papers
Error bounds for the Sherman-Morrison formula and its modification with improved stability
Behnam Hashemi, Yuji Nakatsukasa
It is known that the Sherman--Morrison (SM) formula is not numerically stable. In recent work, we introduced SMIR, an algorithm that incorporates iterative refinement to enhance th…
Accelerating preconditioned Jacobi methods via perturbation-inspired pivoting
Nian Shao, Yuji Nakatsukasa
Perturbation theory for symmetric matrices shows that eigenvalues with small spectral gaps are more sensitive to off-diagonal perturbation, implying that different entries affect t…
Convergence analysis of a nonlinear eigensolver based on rational approximation of the resolvent
Nian Shao, Yuji Nakatsukasa
Given a holomorphic matrix-valued function, the poles of its sketched resolvent are generically its eigenvalues. Once a good rational approximation of the sketched resolvent is obt…
Randomized flexible Krylov methods for regularization
Malena Sabaté Landman, Yuji Nakatsukasa
The computation of sparse solutions of large-scale linear discrete ill-posed problems remains a computationally demanding task. A powerful framework in this context is the use of i…
SubApSnap: Solving parameter-dependent linear systems with a snapshot and subsampling
Eleanor Jones, Yuji Nakatsukasa
A growing number of problems in computational mathematics can be reduced to the solution of many linear systems that are related, often depending smoothly or slowly on a parameter…
Instability of the Sherman-Morrison formula and stabilization by iterative refinement
Behnam Hashemi, Yuji Nakatsukasa
Owing to its simplicity and efficiency, the Sherman-Morrison (SM) formula has seen widespread use across various scientific and engineering applications for solving rank-one pertur…