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Numerical optimization of eigenvalues of Hermitian matrix functions. (English) Zbl 1307.65043
This paper presents an efficient algorithm based on the analytical properties of eigenvalues of an analytic and Hermitian matrix-valued function, which can be used to optimize any eigenvalue problem where lower bounds on the second derivatives of the eigenvalue function can be calculated analytically or numerically. Among these problems the authors include computation of quantities related to dynamical systems, minimization of the largest and maximization of the smallest eigenvalue, minimization of the sum of the \(j\) largest eigenvalues, computation of numerical radius and distance to uncontrollability. Numerical examples related to these applications are also discussed in detail and comparisons with other algorithms are provided, which show the effectiveness of the presented algorithm. Global convergence of the algorithm is proved. Remarkably, a MATLAB implementation of the algorithm and a user guide are available on the web page of the first author.

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
90C26 Nonconvex programming, global optimization
65F60 Numerical computation of matrix exponential and similar matrix functions
Software:
Matlab; CTDSX
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