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. In particular, for a 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 $\mathcal{O}(n \log n)$ 时间内将非常广泛的一类对称Toeplitz矩阵近似至机器精度。具体地，对于元素满足 $\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$（其中 $S(\omega)$ 为分段光滑函数）的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，且矩阵 $\mathbf{U}$ 与 $\mathbf{V}$ 属于 $\mathbb{C}^{n \times r}$（$r \ll n$）。通过将此类矩阵置于时间序列背景下研究，我们为这种结构提供了理论解释，并将其与现有频谱域近似框架建立联系。随后，我们完整讨论了该近似的数值组装方法，并证明了其在改进Whittle型似然近似中的效率——包括一个显著案例：对矩阵 $\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$，仅秩 $r=2$ 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。本文的方法与分析不仅适用于时间序列分析，更为广泛类型的对称Toeplitz矩阵提供了超高精度直接求解的算法。文中分析主要依赖于振荡积分的渐近展开，同时为理解现有高斯对数似然频谱域近似方法的潜在失效场景提供了新视角。