We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in $\mathcal{O}(n \log n)$ time with applications to fitting time series models. In particular, for a symmetric Toeplitz matrix $\mathbf{\Sigma}$ with values $\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$ where $S(\omega)$ is piecewise smooth, we give an approximation $\mathbf{\mathcal{F}} \mathbf{\Sigma} \mathbf{\mathcal{F}}^H \approx \mathbf{D} + \mathbf{U} \mathbf{V}^H$, where $\mathbf{\mathcal{F}}$ is the DFT matrix, $\mathbf{D}$ is diagonal, and the matrices $\mathbf{U}$ and $\mathbf{V}$ are in $\mathbb{C}^{n \times r}$ with $r \ll n$. Studying these matrices in the context of time series, we offer a theoretical explanation of this structure and connect it to existing spectral-domain approximation frameworks. We then give a complete discussion of the numerical method for assembling the approximation and demonstrate its efficiency for improving Whittle-type likelihood approximations, including dramatic examples where a correction of rank $r = 2$ to the standard Whittle approximation increases the accuracy of the log-likelihood approximation from $3$ to $14$ digits for a matrix $\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$. The method and analysis of this work applies well beyond time series analysis, providing an algorithm for extremely accurate solutions to linear systems with a wide variety of symmetric Toeplitz matrices whose entries are generated by a piecewise smooth $S(\omega)$. The analysis employed here largely depends on asymptotic expansions of oscillatory integrals, and also provides a new perspective on when existing spectral-domain approximation methods for Gaussian log-likelihoods can be particularly problematic.

翻译：本文提出一种算法，可在$\mathcal{O}(n \log n)$时间内以机器精度逼近一大类对称Toeplitz矩阵，并应用于时间序列模型拟合。具体而言，对于满足$\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$的对称Toeplitz矩阵$\mathbf{\Sigma}$（其中$S(\omega)$为分段光滑函数），我们给出近似表示$\mathbf{\mathcal{F}} \mathbf{\Sigma} \mathbf{\mathcal{F}}^H \approx \mathbf{D} + \mathbf{U} \mathbf{V}^H$，其中$\mathbf{\mathcal{F}}$为DFT矩阵，$\mathbf{D}$为对角矩阵，矩阵$\mathbf{U}$和$\mathbf{V}$属于$\mathbb{C}^{n \times r}$且满足$r \ll n$。在时间序列背景下研究这些矩阵，我们为此结构提供了理论解释，并将其与现有谱域逼近框架建立联系。随后完整论述了构建该近似的数值方法，并展示了其在改进Whittle型似然逼近方面的效率，包括若干显著案例：对于$\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$矩阵，仅需对标准Whittle近似进行秩$r = 2$的修正，即可将对数似然逼近精度从3位提升至14位。本文方法与分析可广泛应用于时间序列分析之外，为具有分段光滑$S(\omega)$生成项的各类对称Toeplitz矩阵线性系统提供了极高精度的求解算法。所用分析主要依赖于振荡积分渐近展开，同时为现有高斯对数似然谱域逼近方法可能失效的情形提供了新的理论视角。