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An alternating maximization method for approximating the hump of the matrix exponential. (English) Zbl 1378.65097
If $$A\in\mathbb{C}^{n\times n}$$ is stable (eigenvalues in the left-half plane), then the curve $$\Gamma(t)=\|e^{tA}\|_2$$, $$t\geq0$$ has humps where local maxima $$\Gamma_h=\Gamma(t_h)$$ are attained. Define $$\gamma(t,v)=\|e^{tA}v\|_2$$, with $$t\geq0$$ and $$v\in\mathbb{C}^n$$, $$\|v\|_2=1$$, then $$\max_{\|v\|_2=1}\gamma(t_h,v)=\gamma(t_h,v_h)=\Gamma_h=\max_{t\in I}\gamma(t,v_h)$$ where $$I$$ is an interval containing $$t_h$$. Note that $$v_h$$ is a right singular vector of $$e^{t_hA}$$ with singular value $$\Gamma_h$$. This is used to compute a (local) hump iteratively by alternatingly computing the maximum of $$\gamma(t,v)$$ over $$v$$ for $$t$$ fixed, and use the result to fix $$v$$ and compute a maximum over $$t$$. A convergence analysis is given and it is proved that the method converges to the local maximum or alternates between accumulation points where $$\gamma(t,v)$$ is not maximal. Implementation aspects (e.g. finding a starting vector for $$v$$ and a stopping criterion) are discussed and the method is extensively tested numerically.

MSC:
 65F60 Numerical computation of matrix exponential and similar matrix functions 65K10 Numerical optimization and variational techniques 49J05 Existence theories for free problems in one independent variable 15A18 Eigenvalues, singular values, and eigenvectors
Eigtool; Expokit
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References:
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