Exact Recursive Calculation of Circulant Permanents: A Band of Different Diagonals inside a Uniform Matrix |
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Authors: | Vitaly Kocharovsky Vladimir Kocharovsky Vladimir Martyanov Sergey Tarasov |
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Affiliation: | 1.Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843, USA;2.Institute of Applied Physics, Russian Academy of Sciences, 603950 Nizhny Novgorod, Russia; (V.K.); (S.T.);3.Intel Corporation, 5000 W Chandler Blvd, Chandler, AZ 85226, USA; |
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Abstract: | We present a finite-order system of recurrence relations for the permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for and 3) and the method for deriving such recurrence relations, which is based on the permanents of the matrices with defects. The proposed system of linear recurrence equations with variable coefficients provides a powerful tool for the analysis of the circulant permanents, their fast, linear-time computing; and finding their asymptotics in a large-matrix-size limit. The latter problem is an open fundamental problem. Its solution would be tremendously important for a unified analysis of a wide range of the nature’s -hard problems, including problems in the physics of many-body systems, critical phenomena, quantum computing, quantum field theory, theory of chaos, fractals, theory of graphs, number theory, combinatorics, cryptography, etc. |
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Keywords: | permanent, circulant matrix, mé nage problem, NP-hard problem, critical phenomena, quantum computing |
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