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Robust stability and a criss-cross algorithm for pseudospectra. (English) Zbl 1042.65060

A dynamical system \(\dot{x}=Ax\) is robustly stable when all eigenvalues of complex matrices within a given distance of the square matrix \(A\) lie in the left half-plane. The ‘pseudospectral abscissa’, which is the largest real part of such an eigenvalue, measures the robust stability of \(A\). A criss-cross algorithm is presented for computing the pseudospectral abscissa; global and local quadratic convergence is proved, and a numerical implementation is discussed. As with analogous methods for calculating \(H_\infty\) norms, the algorithm depends on computing the eigenvalues of associated Hamiltoniam matrices. The criss-cross method is implemented in MATLAB and extensively tested.

MSC:

65L15 Numerical solution of eigenvalue problems involving ordinary differential equations
65L20 Stability and convergence of numerical methods for ordinary differential equations
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