Efficient Calculations for k-diagonal Circulant Matrices and Cyclic Banded Matrices

Abstract

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 $kn^2$.