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A class of stochastic algorithms for the Wigner equation
A class of stochastic algorithms for the numerical treatment of the Wigner equation is introduced. The algorithms are derived using the theory of pure jump processes with a general state space. The class contains several new algorithms as well as some of the algorithms previously considered in the literature. The approximation error and the efficiency of the algorithms are analyzed. Numerical experiments are performed in a benchmark test case, where certain advantages of the new class of algorithms are demonstrated
Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems. II
We describe a list of open problems in random matrix theory and the theory of
integrable systems that was presented at the conference Asymptotics in
Integrable Systems, Random Matrices and Random Processes and Universality,
Centre de Recherches Mathematiques, Montreal, June 7-11, 2015. We also describe
progress that has been made on problems in an earlier list presented by the
author on the occasion of his 60th birthday in 2005 (see [Deift P., Contemp.
Math., Vol. 458, Amer. Math. Soc., Providence, RI, 2008, 419-430,
arXiv:0712.0849]).Comment: for Part I see arXiv:0712.084
How long does it take to compute the eigenvalues of a random symmetric matrix?
We present the results of an empirical study of the performance of the QR
algorithm (with and without shifts) and the Toda algorithm on random symmetric
matrices. The random matrices are chosen from six ensembles, four of which lie
in the Wigner class. For all three algorithms, we observe a form of
universality for the deflation time statistics for random matrices within the
Wigner class. For these ensembles, the empirical distribution of a normalized
deflation time is found to collapse onto a curve that depends only on the
algorithm, but not on the matrix size or deflation tolerance provided the
matrix size is large enough (see Figure 4, Figure 7 and Figure 10). For the QR
algorithm with the Wilkinson shift, the observed universality is even stronger
and includes certain non-Wigner ensembles. Our experiments also provide a
quantitative statistical picture of the accelerated convergence with shifts.Comment: 20 Figures; Revision includes a treatment of the QR algorithm with
shift
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