16 citations · 40 across the 9 of their papers we have counts for
11 papers
Optimal Polynomial Approximation to Rational Matrix Functions Using the Arnoldi Algorithm
Tyler Chen, Anne Greenbaum, Natalie Wellen
Given an by matrix and an -vector , along with a rational function , we show how to find the optimal approximation to from the K…
GMRES, pseudospectra, and Crouzeix's conjecture for shifted and scaled Ginibre matrices
Tyler Chen, Anne Greenbaum, Thomas Trogdon
We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted matrix whose entries are independent complex Gaussians. When the righ…
Nearly Optimal Approximation of Matrix Functions by the Lanczos Method
Noah Amsel, Tyler Chen, Anne Greenbaum +2
Approximating the action of a matrix function on a vector is an increasingly important primitive in machine learning, data science, and statistics, wit…
K-Spectral Sets
Anne Greenbaum, Natalie Wellen
We use results in [M. Crouzeix and A. Greenbaum,Spectral sets: numerical range and beyond, SIAM Jour. Matrix Anal. Appl., 40 (2019), pp. 1087-1101] to derive a variety of K-spectra…
Low-memory Krylov subspace methods for optimal rational matrix function approximation
Tyler Chen, Anne Greenbaum, Cameron Musco +1
We describe a Lanczos-based algorithm for approximating the product of a rational matrix function with a vector. This algorithm, which we call the Lanczos method for optimal ration…
Error bounds for Lanczos-based matrix function approximation
Tyler Chen, Anne Greenbaum, Cameron Musco +1
We analyze the Lanczos method for matrix function approximation (Lanczos-FA), an iterative algorithm for computing when is a Hermitian matri…
Crouzeix's Conjecture and related problems
Kelly Bickel, Pamela Gorkin, Anne Greenbaum +3
In this paper, we establish several results related to Crouzeix's conjecture. We show that the conjecture holds for contractions with eigenvalues that are sufficiently well-separat…
On the Convergence Rate of Variants of the Conjugate Gradient Algorithm in Finite Precision Arithmetic
Anne Greenbaum, Hexuan Liu, Tyler Chen
We consider three mathematically equivalent variants of the conjugate gradient (CG) algorithm and how they perform in finite precision arithmetic. It was shown in [{\em Behavior of…
First-order Perturbation Theory for Eigenvalues and Eigenvectors
Anne Greenbaum, Ren-cang Li, Michael L. Overton
We present first-order perturbation analysis of a simple eigenvalue and the corresponding right and left eigenvectors of a general square matrix, not assumed to be Hermitian or nor…
Spectral Sets: Numerical Range and Beyond
Michel Crouzeix, Anne Greenbaum
We extend the proof in [M.~Crouzeix and C.~Palencia, {\em The numerical range is a -spectral set}, SIAM Jour.~Matrix Anal.~Appl., 38 (2017), pp.~649-655] to show th…
Some Extensions of the Crouzeix-Palencia Result
Trevor Caldwell, Anne Greenbaum, Kenan Li
In [{\em The Numerical Range is a -Spectral Set}, SIAM J. Matrix Anal. Appl. 38 (2017), pp.~649-655], Crouzeix and Palencia show that the numerical range of a squar…