2014/10/31 by Jonathan B. Baker, Baker, Jonathan, Mark Embree +3 · 2 citations
Computer Science · Physics and Astronomy · #Matrix Theory and Algorithms #Advanced Thermodynamics and Statistical Mechanics #Model Reduction and Neural Networks
paper · pdf · doi:10.48550/arxiv.1410.8741
Lyapunov equations with low-rank right-hand sides often have solutions whose\nsingular values decay rapidly, enabling iterative methods that produce low-rank\napproximate solutions. All previously known bounds on this decay involve\nquantities that depend quadratically on the departure of the coefficient matrix\nfrom normality: these bounds suggest that the larger the departure from\nnormality, the slower the singular values will decay. We show this is only true\nup to a threshold, beyond which a larger departure from normality can actually\ncorrespond to faster decay of singular values: if the singular values decay\nslowly, the numerical range cannot extend far into the right-half plane.\n