Efficient Calculations for Inverse of $k$-diagonal Circulant Matrices and Cyclic Banded Matrices
Chen Wang, Hailong Yu, Chao Wang
TL;DR
The paper tackles efficient computation of inverses for two circular-structured sparse matrix classes: $k$-diagonal circulant matrices ($k$-CM) and $k$-diagonal cyclic banded matrices ($k$-CBM). It develops determinant-based formulas and inverse representations built on recursive relations, avoiding FFTs and enabling finite-field applicability. For $k$-CM it achieves a determinant complexity of $O(k^3 \log n)$ and an inverse representation cost of $O(k^3 \log n + k^4) + kn$, while for $k$-CBM it achieves $O(k^3 n)$ for the determinant and $O(k^3 n + k^5) + kn^2$ for the full inverse. Collectively, these results yield highly efficient, compact inverse representations suitable for large-scale problems where traditional methods are impractical or inapplicable.
Abstract
$k$-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.