Fast Algorithms for Solving FLS R -Factor Block Circulant Linear Systems and Inverse Problem of A X = b

Abstract

Block circulant and circulant matrices have already become an ideal research area for solving various differential equations. In this paper, we give the definition and the basic properties of FLS $R$-factor block circulant (retrocirculant) matrix over field $\mathbb{F}$. Fast algorithms for solving systems of linear equations involving these matrices are presented by the fast algorithm for computing matrix polynomials. The unique solution is obtained when such matrix over a field $\mathbb{F}$ is nonsingular. Fast algorithms for solving the unique solution of the inverse problem of $\mathcal{A}X=b$ in the class of the level-2 FLS $(R,r)$-circulant(retrocirculant) matrix of type $(m,n)$ over field $\mathbb{F}$ are given by the right largest common factor of the matrix polynomial. Numerical examples show the effectiveness of the algorithms.