An Algorithm to Compute Sepλ

The following problem is addressed: given square matrices A and B, compute the smallest e such that A + E and B + F have a common eigenvalue for some E, F with max(∥E∥ 2 , ∥F∥ 2 ) < ∈. An algorithm to compute this quantity to any prescribed accuracy is presented, assuming that eigenvalues can be computed exactly.

[1]  G. Stewart.  Error and Perturbation Bounds for Subspaces Associated with Certain Eigenvalue Problems , 1973 .

[2]  J. Varah.  On the Separation of Two Matrices , 1979 .

[3]  J. Demmel.  A Numerical Analyst's Jordan Canonical Form. , 1983 .

[4]  J. Demmel.  Computing stable eigendecompositions of matrices , 1986 .

[5]  J. Demmel.  A counterexample for two conjectures about stability , 1987 .

[6]  R. Byers.  A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices , 1988 .

[7]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[8]  Ming Gu,et al.  New Methods for Estimating the Distance to Uncontrollability , 2000, SIAM J. Matrix Anal. Appl..

[9]  Volker Mehrmann,et al.  Structure-Preserving Methods for Computing Eigenpairs of Large Sparse Skew-Hamiltonian/Hamiltonian Pencils , 2001, SIAM J. Sci. Comput..

[10]  Adrian S. Lewis,et al.  Optimization and Pseudospectra, with Applications to Robust Stability , 2003, SIAM J. Matrix Anal. Appl..

[11]  Adrian S. Lewis,et al.  Pseudospectral Components and the Distance to Uncontrollability , 2005, SIAM J. Matrix Anal. Appl..

[12]  Adrian S. Lewis,et al.  A Robust Gradient Sampling Algorithm for Nonsmooth, Nonconvex Optimization , 2005, SIAM J. Optim..

[13]  R. Alam,et al.  On sensitivity of eigenvalues and eigendecompositions of matrices , 2005 .

[14]  V. Mehrmann,et al.  Skew-Hamiltonian and Hamiltonian Eigenvalue Problems: Theory, Algorithms and Applications , 2005 .

[15]  Ming Gu,et al.  Fast Methods for Estimating the Distance to Uncontrollability , 2006, SIAM J. Matrix Anal. Appl..

[16]  L. Trefethen,et al.  Spectra and Pseudospectra , 2020 .