A Note on the Solution of Circulant Real Linear Systems and its Sensitivity Analysis
Alessandro Guazzini, Enrico Caricchio
TL;DR
The paper addresses efficiently solving circulant real linear systems $A x=b$ and performing sensitivity analysis. It leverages Fourier-based diagonalization $A=F \Psi F^*$ (with $A^{-1}=F \Psi^{-1} F^*$) to derive a ready-to-use, FFT-based solution and provides an explicit componentwise expression for $x_l$ in terms of $T_k$ and $\psi_k$, including separate forms for even and odd $n$. A key contribution is the demonstration that strict diagonal dominance, $a_0>\sum_{j=1}^{n-1}|a_j|$, guarantees $\mathfrak{R}(\psi_k)>0$ and hence sign-consistency between $\partial x_l/\partial b_{rl}$ and $\partial f_l/\partial b_{rl}$, linking matrix structure to interpretability in sensitivity analyses. The results enable fast computation and reliable interpretation in economic equilibrium models, such as Salop and Riordan-type frameworks, where circulant systems arise.
Abstract
Employing the Fast Fourier Transform we propose a ready-to-use solution to circulant real linear systems of equations, particularly useful when a broader theoretical analysis is involved. We also show that strict diagonal dominance of the matrix of coefficients is a sufficient condition for sign consistency between solutions and parameters in sensitivity analysis. Keywords: Circulant matrix, Real linear system of equations, Circulant structure, FFT, Sensitivity Analysis, Strict Diagonal Dominance.