Calculating the inverse of $k$-diagonal circulant matrices and cyclic banded matrices is a more challenging problem than calculating their determinants. Algorithms that directly involve or specify linear or quadratic complexity for the inverses of these two types of matrices are rare. This paper presents two fast algorithms that can compute the complexity of a $k$-diagonal circulant matrix within complexity $O(k^3 \log n+k^4)+kn$, and for $k$-diagonal cyclic banded matrices it is $O(k^3 n+k^5)+kn^2$. Since $k$ is generally much smaller than $n$, the cost of these two algorithms can be approximated as $kn$ and $kn^2$.

翻译：计算$k$-对角循环矩阵和循环带状矩阵的逆矩阵比计算其行列式更具挑战性。直接涉及或指定线性或二次复杂度的这两类矩阵求逆算法较为罕见。本文提出两种快速算法：对于$k$-对角循环矩阵，可在复杂度$O(k^3 \log n+k^4)+kn$内完成计算；对于$k$-对角循环带状矩阵，复杂度为$O(k^3 n+k^5)+kn^2$。由于$k$通常远小于$n$，这两种算法的成本可近似为$kn$和$kn^2$。