We develop spectral methods for ODEs and operator eigenvalue problems that are based on a least-squares formulation of the problem. The key tool is a method for rectangular generalized eigenvalue problems, which we extend to quasimatrices and objects combining quasimatrices and matrices. The strength of the approach is its flexibility that lies in the quasimatrix formulation allowing the basis functions to be chosen arbitrarily (e.g. those obtained by solving nearby problems), and often giving high accuracy. We also show how our algorithm can easily be modified to solve problems with eigenvalue-dependent boundary conditions, and discuss reformulations as an integral equation, which often improves the accuracy.

中文翻译：

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用于 ODE 特征值问题的最小二乘谱方法

我们为 ODE 和算子特征值问题开发了基于问题的最小二乘公式的谱方法。关键工具是用于矩形广义特征值问题的方法，我们将其扩展到拟矩阵和组合拟矩阵和矩阵的对象。该方法的优势在于其灵活性在于准矩阵公式允许任意选择基函数（例如通过解决附近问题获得的基函数），并且通常提供高精度。我们还展示了如何轻松修改我们的算法以解决依赖于特征值的边界条件的问题，并讨论作为积分方程的重构，这通常会提高准确性。