11 papers · 1 filter
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…
Computing the Zeros of a Holomorphic Function Using Quadrature-Based Subdivision and Rational Approximation of the Logarithmic Derivative
Jake Bowhay, Yuji Nakatsukasa, Irwin Zaid
We introduce a new method that uses AAA approximation to reliably compute all the zeros of a holomorphic function in a specified search region in the complex plane. Specifically, t…
Efficient Function Approximation Under Heteroskedastic Noise
Yuji Nakatsukasa, Yifu Zhang
Approximating a function on based on samples is a classical problem in numerical analysis. If the samples come with heteroskedastic noise depending on of…
A minimax method for the spectral fractional Laplacian and related evolution problems
José A. Carrillo, Stefano Fronzoni, Yuji Nakatsukasa +1
We present a numerical method for the approximation of the inverse of the fractional Laplacian , based on its spectral definition, using rational functions to approximate…
Stable algorithms for general linear systems by preconditioning the normal equations
Ethan N. Epperly, Anne Greenbaum, Yuji Nakatsukasa
This paper studies the solution of nonsymmetric linear systems by preconditioned Krylov methods based on the normal equations, LSQR in particular. On some examples, preconditioned…
Preconditioned normal equations for solving discretised partial differential equations
Lorenzo Lazzarino, Yuji Nakatsukasa, Umberto Zerbinati
This paper explores preconditioning the normal equation for non-symmetric square linear systems arising from PDE discretization, focusing on methods like CGNE and LSQR. The concept…