Exact Recursive Calculation of Circulant Permanents: A Band of Different Diagonals inside a Uniform Matrix
We present a finite-order system of recurrence relations for the permanent of circulant matrices containing a band of <i>k</i> any-value diagonals on top of a uniform matrix (for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"...
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| Autores principales: | , , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/f715c6e390a7461a874af18fed4a205e |
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| Sumario: | We present a finite-order system of recurrence relations for the permanent of circulant matrices containing a band of <i>k</i> any-value diagonals on top of a uniform matrix (for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>k</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></mrow></semantics></math></inline-formula> 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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>♯</mo><mi>P</mi></mrow></semantics></math></inline-formula>-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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