About

Anne Greenbaum works in the area of numerical analysis, especially numerical linear algebra, matrix theory and its applications. She is the author of the book Iterative Methods for Solving Linear Systems, published by SIAM, and the coauthor of the undergraduate textbook Numerical Methods: Design, Analysis, and Computer Implementation of Algorithms, published by Princeton University Press. She received her Bachelor's degree from the University of Michigan in 1974 and her PhD from the University of California at Berkeley in 1981. She worked as a mathematician at Lawrence Livermore National Laboratory from 1974 to 1986, then joined the Courant Institute at New York University, where she was a Research Professor from 1986 to 1997. She is now a Professor in the Applied Mathematics Department at the University of Washington. Her awards include the B. Bolzano Honorary Medal for Merit in the Mathematical Sciences, awarded in 1997 by the Academy of Sciences of the Czech Republic, and the SIAM Activity Group on Linear Algebra award for Outstanding Paper in Applicable Linear Algebra during 1991–1993. She was elected a Fellow of SIAM in 2015. Aside from mathematics, her interests include tennis and hiking.

Research topics

• Mathematics
• Applied mathematics

Selected publications

• Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

  Open MIND · 2026-02-05

  preprint

  This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was organized at the Simons Institute for the Theory of Computing program on Complexity and Linear Algebra in Fall 2025. The complexity and numerical solution of linear algebra problems is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions.

  DOI

• Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop

  ArXiv.org · 2026-02-05

  articleOpen access

  This document presents a series of open questions arising in matrix computations, i.e., the numerical solution of linear algebra problems. It is a result of working groups at the workshop Linear Systems and Eigenvalue Problems, which was organized at the Simons Institute for the Theory of Computing program on Complexity and Linear Algebra in Fall 2025. The complexity and numerical solution of linear algebra problems is a crosscutting area between theoretical computer science and numerical analysis. The value of the particular problem formulations here is that they were produced via discussions between researchers from both groups. The open questions are organized in five categories: iterative solvers for linear systems, eigenvalue computation, low-rank approximation, randomized sketching, and other areas including tensors, quantum systems, and matrix functions.

  PublisherOA PDF

• When is the Resolvent Like a Rank One Matrix?

  ArXiv.org · 2025-01-13

  preprintOpen access1st authorCorresponding

  For a square matrix $A$, the resolvent of $A$ at a point $z \in \mathbb{C}$ is defined as $(A-zI )^{-1}$. We consider the set of points $z \in \mathbb{C}$ where the relative difference in 2-norm between the resolvent and the nearest rank one matrix is less than a given number $ε\in (0,1)$. We establish a relationship between this set and the $ε$-pseudospectrum of $A$, and we derive specific results about this set for Jordan blocks and for a class of large Toeplitz matrices. We also derive disks about the eigenvalues of $A$ that are contained in this set, and this leads to some new results on disks about the eigenvalues that are contained in the $ε$-pseudospectrum of $A$. In addition, we consider the set of points $z \in \mathbb{C}$ where the absolute value of the inner product of the left and right singular vectors corresponding to the largest singular value of the resolvent is less than $ε$. We demonstrate numerically that this set can be almost as large as the one where the relative difference between the resolvent and the nearest rank one matrix is less than $ε$ and we give a partial explanation for this. Some possible applications are discussed.

  PublisherOA PDFDOI

• Stable algorithms for general linear systems by preconditioning the normal equations

  ArXiv.org · 2025-02-25

  preprintOpen access

  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 LSQR is seen to produce errors many orders of magnitude larger than classical direct methods; this paper demonstrates that the attainable accuracy of preconditioned LSQR can be greatly improved by applying iterative refinement or restarting when the accuracy stalls. This observation is supported by rigorous backward error analysis. This paper also provides a discussion of the relative merits of GMRES and LSQR for solving nonsymmetric linear systems, demonstrates stability for left-preconditioned LSQR without iterative refinement, and shows that iterative refinement can also improve the accuracy of preconditioned conjugate gradient.

  PublisherOA PDFDOI

• A New Proof that the Numerical Range is a Complete 2-Spectral Set for Weighted Shift Matrices

  ArXiv.org · 2025-08-18

  preprintOpen accessSenior author

  In this paper we give an alternative proof that the family of matrices studied by Daeshik Choi in A proof of Crouzeix's conjecture for a class of matrices, Linear Algebra and its Applications, 438, no. 8 (2013), pp. 3247-3257, satisfy Crouzeix's conjecture. We also show that they satisfy the completely bounded version of the conjecture.

  PublisherOA PDFDOI

• Optimal polynomial approximation to rational matrix functions using the Arnoldi algorithm

  Numerical Algorithms · 2025-05-06 · 1 citations

  article
  PublisherDOI

• GMRES, pseudospectra, and Crouzeix’s conjecture for shifted and scaled Ginibre matrices

  Mathematics of Computation · 2024-03-09 · 3 citations

  article

  We study the GMRES algorithm applied to linear systems of equations involving a scaled and shifted <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N times upper N"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>×</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\times N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> matrix whose entries are independent complex Gaussians. When the right-hand side of this linear system is independent of this random matrix, the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi mathvariant="normal">∞</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">N\to \infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> behavior of the GMRES residual error can be determined exactly. To handle cases where the right hand side depends on the random matrix, we study the pseudospectra and numerical range of Ginibre matrices and prove a restricted version of Crouzeix’s conjecture.

  PublisherDOI

• Nearly Optimal Approximation of Matrix Functions by the Lanczos Method

  2024-01-01 · 2 citations

  article
  PublisherDOI

• Numerical bounds on the Crouzeix ratio for a class of matrices

  CALCOLO · 2024-06-01 · 1 citations

  articleOpen access
  PublisherOA PDFDOI

• Comparison of K-spectral set bounds on norms of functions of a matrix or operator

  Linear Algebra and its Applications · 2024-04-12 · 2 citations

  article1st authorCorresponding
  PublisherDOI

Recent grants

• Beyond Eigenvalues - Describing the Behavior of Nonnormal Matrices and Linear Operators

  NSF · $267k · 2002–2008

• Applied Matrix Theory and Complex Approximation: Estimating Norms of Functions of Matrices

  NSF · $353k · 2012–2017

Frequent coauthors

• Garry Rodrigue

  12 shared

• Zdeněk Strakoš

  Charles University

  9 shared

• Tyler Chen

  9 shared

• Leslie Greengard

  Flatiron Health (United States)

  8 shared

• Cameron Musco

  University of Massachusetts Amherst

  7 shared

• Michael L. Overton

  7 shared

• Jack Dongarra

  7 shared

• Michel Crouzeix

  Centre National de la Recherche Scientifique

  6 shared

Education

• B.S.

  University of Michigan

  1974

• Ph.D.

  University of California at Berkeley

  1981

Awards & honors

• B. Bolzano Honorary Medal for Merit in the Mathematical Scie…
• SIAM Activity Group on Linear Algebra award for Outstanding…
• Fellow of SIAM (2015)
• 2022 Sonia Kovalesky Lecturer