Minghao W. Rostami
Lyapunov inverse iteration for computing pseudospectral abscissa

Department of Mathematical Sciences
Stratton Hall
100 Institute Road
Worcester
MA 01609-2280
mwu@wpi.edu

$\varepsilon$ -pseudospectral abscissa $\alpha_\varepsilon$ refers to the maximum real part of points in the $\varepsilon$ -pseudospectrum of a matrix $A$ . It is a robust measure of the stability of the linear dynamical system $\dot{u}=Au$ . If $\alpha_\varepsilon>0$ , the steady state of this system can be unstable (i.e., sensitive to small perturbations) in reality even if all the eigenvalues of $A$ have negative real parts.

In [SIAM J. Matrix Anal. Appl, Vol. 32, No. 4, pp. 1166-1192], an algorithm for computing $\alpha_\varepsilon$ for large, sparse matrices was proposed, which requires computing the rightmost eigenvalues for a sequence of (complex) matrices.

Finding the rightmost eigenvalues for large, sparse and nonsymmetric matrices is known to be challenging. In this talk, we first present a modified version of the above algorithm that computes the real $\varepsilon$ -pseudospectral abscissa $\alpha_\varepsilon^{\mathbb{R}}$ of $A$ . It requires computing the rightmost eigenvalues for a sequence of real matrices, which can be done in a robust and efficient manner by applying a recently developed eigenvalue solver called the Lyapunov inverse iteration [SIAM. J. Matrix Anal. Appl. Vol. 34, No. 4, pp. 1685-1707]. We then show that a good approximation of $\alpha_\varepsilon$ can be obtained from $\alpha_\varepsilon^{\mathbb{R}}$ . Numerical results for $A$ arising from spacial discretization of incompressible Navier-Stokes equations will be presented.