Publications related to Implicit QR algorithms for palindromic and even eigenvalue problems

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Singular quadratic eigenvalue problems: linearization and weak condition numbers

Daniel Kressner, Ivana Sain Glibic

The numerical solution of singular eigenvalue problems is complicated by the fact that small perturbations of the coefficients may have an arbitrarily bad effect on eigenvalue accuracy. However, it has been known for a long time that such perturbations are ...

SPRINGER2023

A mixed precision LOBPCG algorithm

Daniel Kressner, Meiyue Shao, Yuxin Ma

The locally optimal block preconditioned conjugate gradient (LOBPCG) algorithm is a popular approach for computing a few smallest eigenvalues and the corresponding eigenvectors of a large Hermitian positive definite matrix A. In this work, we propose a mix ...

SPRINGER2023

Iterative refinement of Schur decompositions

Daniel Kressner, Zvonimir Bujanovic

The Schur decomposition of a square matrix A is an important intermediate step of state-of-the-art numerical algorithms for addressing eigenvalue problems, matrix functions, and matrix equations. This work is concerned with the following task: Compute a (m ...

SPRINGER2022

Robust Differentiable SVD

Pascal Fua, Mathieu Salzmann, Zheng Dang, Wei Wang, Yinlin Hu

Eigendecomposition of symmetric matrices is at the heart of many computer vision algorithms. However, the derivatives of the eigenvectors tend to be numerically unstable, whether using the SVD to compute them analytically or using the Power Iteration (PI) ...

2021

Complementary Asymptotically Sharp Estimates for Eigenvalue Means of Laplacians

Joachim Stubbe, Luigi Provenzano

We present asymptotically sharp inequalities, containing a 2nd term, for the Dirichlet and Neumann eigenvalues of the Laplacian on a domain, which are complementary to the familiar Berezin-Li-Yau and Kroger inequalities in the limit as the eigenvalues tend ...

OXFORD UNIV PRESS2021

Compress-and-restart block Krylov subspace methods for Sylvester matrix equations

Daniel Kressner, Stefano Massei, Kathryn Dianne Lund

Block Krylov subspace methods (KSMs) comprise building blocks in many state-of-the-art solvers for large-scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov ...

2020

A Householder-Based Algorithm For Hessenberg-Triangular Reduction

Daniel Kressner, Zvonimir Bujanovic

The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil A - lambda B requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations re ...

SIAM PUBLICATIONS2018

MATHICSE Technical Report : Fast QR decomposition of HODLR matrices

Daniel Kressner, Ana Susnjara

The efficient and accurate QR decomposition for matrices with hierarchical low-rank structures, such as HODLR and hierarchical matrices, has been challenging. Existing structure-exploiting algorithms are prone to numerical instability as they proceed indi- ...

MATHICSE2018

Optimally Packed Chains of Bulges in Multishift QR Algorithms

Daniel Kressner

The QR algorithm is the method of choice for computing all eigenvalues of a dense nonsymmetric matrix A. After an initial reduction to Hessenberg form, a QR iteration can be viewed as chasing a small bulge from the top left to the bottom right corner along ...

Association for Computing Machinery2014

Implicit QR algorithms for palindromic and even eigenvalue problems

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Singular quadratic eigenvalue problems: linearization and weak condition numbers

A mixed precision LOBPCG algorithm

Iterative refinement of Schur decompositions

Robust Differentiable SVD

Complementary Asymptotically Sharp Estimates for Eigenvalue Means of Laplacians

Compress-and-restart block Krylov subspace methods for Sylvester matrix equations

Fast hierarchical solvers for symmetric eigenvalue problems

A Householder-Based Algorithm For Hessenberg-Triangular Reduction

MATHICSE Technical Report : Fast QR decomposition of HODLR matrices

Optimally Packed Chains of Bulges in Multishift QR Algorithms

Singular quadratic eigenvalue problems: linearization and weak condition numbers

Iterative refinement of Schur decompositions

Robust Differentiable SVD

Fast hierarchical solvers for symmetric eigenvalue problems

MATHICSE Technical Report : Fast QR decomposition of HODLR matrices

A mixed precision LOBPCG algorithm

Complementary Asymptotically Sharp Estimates for Eigenvalue Means of Laplacians

Compress-and-restart block Krylov subspace methods for Sylvester matrix equations

A Householder-Based Algorithm For Hessenberg-Triangular Reduction

Optimally Packed Chains of Bulges in Multishift QR Algorithms