We show the existence and nature of convergence to a limiting set of roots for polynomials in a three-term recurrence of the form $p_{n+1}\left(z\right)=Q_k\left(z\right)p_n\left(z\right)+\gamma p_{n-1}\left(z\right)$ as $n$ $\to$ $\infty$, where the coefficient $Q_k\left(z\right)$ is a $k^{th}$ degree polynomial, and $z,\gamma \in \mathbb{C}$. These results are then extended for approximating roots of such polynomials for any given $n$. Such relations for the evaluation are motivated by the large computational efforts of the iterative numerical methods in solving for eigenvalues, and their errors. We first apply this relation to the eigenvalue problems represented by tridiagonal matrices with a periodicity $k$ in its entries, providing a more accurate method for the evaluation of the spectra of a $k$-periodic chain and a reduction in computing effort from $\mathcal{O}\left(n^2\right)$ to $\mathcal{O}\left(n\right)$. These results are combined with the spectral rules of Kronecker products, for efficient and accurate evaluation of spectra of spatial lattices and other periodic graphs that need not be represented by a banded or a Toeplitz type matrix.

我们在形式的三项递归中展示了收敛到多项式的有限根集的存在和性质 ${磷}_{n+1}\left(z\right)={问}_{克}\left(z\right){磷}_n\left(z\right)+\gamma {磷}_{n-1}\left(z\right)$ 作为 $n$ $\to$ $\infty$，其中系数 ${问}_{克}\left(z\right)$ 是一个 ${克}^{吨H}$ 度多项式，和 $z,\gamma \in \mathbb{C}$. 然后将这些结果扩展为近似任何给定的多项式的根$n$. 这种评估关系的动机是迭代数值方法在求解特征值及其误差方面的大量计算工作。我们首先将此关系应用于由具有周期性的三对角矩阵表示的特征值问题$克$ 在其条目中，提供了一种更准确的方法来评估光谱的 $克$- 周期链和减少计算工作量 $\mathcal{哦}\left(n^2\right)$ 到 $\mathcal{哦}\left(n\right)$. 这些结果与 Kronecker 乘积的光谱规则相结合，可以高效准确地评估空间晶格和其他不需要由带状或托普利茨型矩阵表示的周期图的光谱。