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.

翻译：我们开发了基于问题最小方格的光谱方法,用于解决脱氧核糖核酸和操作器电子元值问题。关键工具是处理长方形普遍乙基值问题的方法,我们将这种方法推广到准矩阵和矩阵相结合的准矩阵和物体。这种方法的优点在于其灵活性在于准矩阵的提法,允许任意选择基础功能(例如通过解决附近问题获得的功能),而且常常提供很高的准确性。我们还表明如何可以轻而易举地修改我们的算法,以便解决以离子值为依存的边界条件的问题,并讨论重订作为整体方程式的问题,这往往能提高准确性。