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 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 direct solves with a wide variety of symmetric Toeplitz matrices. 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$\mathcal{O}(n \log n)$时间内将极广泛类别的对称Toeplitz矩阵近似至机器精度，并应用于时间序列模型拟合。具体而言，对于满足$\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$（其中$S(\omega)$分段光滑）的对称Toeplitz矩阵$\mathbf{\Sigma}$，我们给出近似表达式$\mathbf{\mathcal{F}} \mathbf{\Sigma} \mathbf{\mathcal{F}}^H \approx \mathbf{D} + \mathbf{U} \mathbf{V}^H$，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$r \ll n$。我们在时间序列背景下研究这些矩阵，对此结构提供理论解释，并将其与现有谱域近似框架建立关联。随后，我们完整讨论该近似的数值组装方法，并证明其在改进Whittle型似然近似中的效率，包括显著算例：对于$\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$的矩阵，仅需秩$r = 2$的修正即可将标准Whittle近似的精度从3位有效数字提升至14位。本文方法与分析的应用远超时间序列分析范围，为多种对称Toeplitz矩阵的高精度直接求解提供算法。本文分析主要依赖于振荡积分的渐近展开，同时也为高斯对数似然的现有谱域近似方法何时可能产生严重偏差提供了新视角。