9 papers
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…
Linear convergence of iterative contour integral-based eigensolvers for nonlinear eigenvalue problems
Daniel Kressner, Yuqi Liu, Jose E. Roman +2
Solving nonlinear eigenvalue problems is an important and challenging task in scientific computing. Contour integral-based approaches are attractive for such eigenvalue problems be…
Stabilizing the Rayleigh--Ritz procedure by randomization
Nian Shao
Extracting approximate eigenpairs from a prescribed subspace is of fundamental importance in eigenvalue computation. While projecting the target eigenvector onto the subspace yield…
Lanczos with compression for symmetric eigenvalue problems
Angelo A. Casulli, Daniel Kressner, Nian Shao
The Lanczos method with implicit restarting is one of the most popular methods for finding a few exterior eigenpairs of a large symmetric matrix . Usually based on polynomial fi…
Restoring similarity in randomized Krylov methods with applications to eigenvalue problems and matrix functions
Laura Grigori, Daniel Kressner, Nian Shao +1
The randomized Arnoldi process has been used in large-scale scientific computing because it produces a well-conditioned basis for the Krylov subspace more quickly than the standard…