A Bisection Method for Measuring the Distance of a Stable Matrix to the Unstable Matrices
Author:
- Ralph ByersRBRalph Byers
- 1 September 1988
- journal article
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Scientific and Statistical Computing
- Vol. 9 (5) , 875-881
Abstract
We describe a bisection method to determine the 2-norm and Frobenius norm distances from a given matrix A to the nearest matrix with an eigenvalue on the imaginary axis. If A is stable in the sense that its eigenvalues lie in the open-left half plane, then this distance measures how “nearly unstable“ A is. Each step provides either a rigorous upper bound or a rigorous lower bound on the distance. A few bisection steps can bracket the distance within an order of magnitude. Bisection avoids the difficulties associated with nonlinear minimization techniques and the occasional failures associated with heuristic estimates. We show how the method might be used to estimate the distance to the nearest matrix with an eigenvalue on the unit circle.Keywords
This publication has 7 references indexed in Scilit:
- A counterexample for two conjectures about stabilityIEEE Transactions on Automatic Control, 1987
- A Hamiltonian $QR$ AlgorithmSIAM Journal on Scientific and Statistical Computing, 1986
- How near is a stable matrix to an unstable matrix?1985
- A symplectic method for approximating all the eigenvalues of a Hamiltonian matrixLinear Algebra and its Applications, 1984
- Generalizing the LINPACK condition estimator1982
- Matrix Eigensystem Routines — EISPACK Guide1976
- Canonical forms for symplectic and Hamiltonian matricesCelestial Mechanics and Dynamical Astronomy, 1974