Convergence rate analysis and improved iterations for numerical radius computation
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We analyze existing methods for computing the numerical radius and introduce improved algorithms. Current methods include level-set and cutting-plane based approaches, but until now, no formal convergence rate results have been established for these techniques. We first introduce an improved level-set method that is often significantly faster than the existing approach of Mengi and Overton. We then establish the first rate of convergence results for any numerical radius method, showing how the convergence of Uhlig's cutting-plane method varies from superlinear to linear based on the normalized curvature at outermost points in the field of values. Moreover, we introduce a more efficient cutting-plane method whose convergence rate we also derive. Finally, as cutting-plane methods become very expensive when the field of values closely resembles a circular disk centered at the origin, we introduce a third algorithm combining both approaches to remain efficient in all cases.
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