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Algorithm for the computation of the pseudospectral radius and the numerical radius of a matrix. (English) Zbl 1082.65043

This paper concerns the convergence of a discrete-time dynamical system \[ x_k= Ax_{k-1},\quad A\in \mathbb{C}^{n\times n}.\tag{1} \] For \(k\to\infty\), the eigenvalues of the matrix \(A\) provide all information for analyzing (1), as is known. For finite time (and non-normal \(A\)), additional insight is obtained from the \(\varepsilon\)-pseudospectral radius and the numerical radius of \(A\), both quantities being measures of the robust stability of (1), for which the paper gives globally convergent algorithms, which depend on computing the eigenvalues of symplectic pencils and Hamiltonian matrices. An example of a complex \(50\times 50\) random matrix \(A= [a_{jk}]\) with \(\text{Re}(a_{jk})\) and \(\text{Im}(a_{jk})\) standardized normal, is given.

MSC:

65F15 Numerical computation of eigenvalues and eigenvectors of matrices
65F30 Other matrix algorithms (MSC2010)
15A60 Norms of matrices, numerical range, applications of functional analysis to matrix theory
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