循环积和式的精确递归计算：一致矩阵内的一组不同对角线
Exact recursive calculation of circulant permanents: A band of different diagonals inside a uniform matrix
摘要：
本文给出了一个有限阶循环矩阵积算式的递推关系系统，该积算式在一个一致矩阵（k = 1，2，3）上包含一个k个任意值对角线，并给出了基于带缺陷矩阵积算式的递推关系的推导方法.所提出的变系数线性递归方程组为分析循环积和式、快速线性时间计算和在大矩阵极限下求其渐近解提供了有力的工具。后一个问题是一个开放的基本问题。它的解决方案对于统一分析大范围的自然界的P-困难问题非常重要，包括多体系统物理学、临界现象、量子计算、量子场论、混沌理论、分形、图论、数论、组合学、密码学等问题。
英文摘要：
We present a finite-order system of recurrence relations for a permanent of circulant matrices containing a band of k any-value diagonals on top of a uniform matrix (for k = 1, 2, and 3) as well as 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 #P-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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