We develop the ultraspherical rectangular collocation (URC) method, a collocation implementation of the sparse ultraspherical method of Olver \& Townsend for two-point boundary-value problems. The URC method is provably convergent, the implementation is simple and efficient, the convergence proof motivates a preconditioner for iterative methods, and the modification of collocation nodes is straightforward. The convergence theorem applies to all boundary-value problems when the coefficient functions are sufficiently smooth and when the roots of certain ultraspherical polynomials are used as collocation nodes. We also adapt a theorem of Krasnolsel'skii et al.~to our setting to prove convergence for the rectangular collocation method of Driscoll \& Hale for a restricted class of boundary conditions.

翻译：本文发展了超球矩形配置（URC）方法，该方法是一种针对两点边值问题的 Olver & Townsend 稀疏超球方法的配置实现。URC 方法具有可证明的收敛性，其实现简单且高效；收敛性证明为迭代方法提供了预条件子的理论基础，且配置节点的修改过程直接简便。当系数函数足够光滑且采用特定超球多项式的根作为配置节点时，该收敛定理适用于所有边值问题。此外，我们将 Krasnolsel'skii 等人的定理适配至本文框架，证明了 Driscoll & Hale 矩形配置方法在受限边界条件类下的收敛性。