$k$-对角循环矩阵和循环带状矩阵逆的高效计算
Efficient Calculations for Inverse of $k$-diagonal Circulant Matrices and Cyclic Banded Matrices
AI总结：
针对$k$-对角循环矩阵和循环带状矩阵的逆，提出基于递归公式的高效计算方法，具有线性或二次复杂度，且不依赖快速傅里叶变换，适用于有限域计算。
摘要：
$k$-对角循环矩阵和循环带状矩阵广泛应用于圆形线性系统的数值模拟和信号处理中。直接涉及或明确这两种矩阵逆的线性或二次复杂度的算法十分罕见。我们发现，$k$-对角循环矩阵的逆可由一个递归公式唯一确定，该公式可在 $O(k^3 \log n+k^4)$ 内导出。类似地，循环带状矩阵的逆也可由一系列递归公式唯一确定，这些递归的初始项可在 $O(k^3 n+k^5)$ 内计算。求解这两类矩阵完整逆的额外开销分别为 $kn$ 和 $kn^2$。我们的计算实现了快速表示，大部分过程由显式公式定义。此外，求逆 $k$-对角循环矩阵的大多数算法依赖快速傅里叶变换，不适用于有限域，而我们的算法可应用于有限域中的计算。
英文摘要：
$k$-diagonal circulant matrices and cyclic banded matrices are widely used in numerical simulations and signal processing of circular linear systems. Algorithms that directly involve or specify linear or quadratic complexity for the inverses of these two types of matrices are rare. We find that the inverse of a $k$-diagonal circulant matrix can be uniquely determined by a recursive formula, which can be derived within $O(k^3 \log n+k^4)$. Similarly for the inverse of a cyclic banded matrix, its inverse can be uniquely determined by a series of recursive formulas, with the initial terms of these recursions computable within $O(k^3 n+k^5)$. The additional costs for solving the complete inverses of these two types of matrices are $kn$ and $kn^2$. Our calculations enable rapid representation with most processes defined by explicit formulas. Additionally, most algorithms for inverting $k$-diagonal circulant matrices rely on the Fast Fourier Transform, which is not applicable to finite fields, while our algorithms can be applied to computations in finite fields.
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