Least-Squares Spectral Methods for ODE Eigenvalue Problems

Abstract

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, a good choice (e.g., those obtained by solving nearby problems) leading to rapid convergence, 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.

Rectangular eigenvalue problems

“…Variant 2 Finite set of boundary conditions. Suppose next that (1) is coupled with a finite set of μ > 0 homogeneous linear boundary conditions, as is considered (along with other possibilities) in [17]. An example would be −(4/π 2 )u = λu on [−1, 1] with boundary conditions u(±1) = 0, with eigenvalues 1, 4, 9, .…”

“…where 0 denotes the zero matrix of dimensions μ × n. Various methods can be employed to make this equation square, as discussed in [17] in the context of the Ito-Murota formulation. The simplest, analogous to what is called the "tau method" of imposing boundary conditions in spectral methods [8], is to let Q − denote the m × (n − μ) matrix or quasimatrix consisting of Q with its final μ columns removed and then consider…”

“…The columns of Q are orthonormal with respect to an inner product that combines sums (or integrals, when m or μ is infinite) associated with both and ∂ . For discussions of QR factorization and singular value decomposition of such mixed objects, see [17]. One could analyze what relative weighting is most appropriate in balancing the two halves, but the expectation is that in applications it will not make much difference since we are aiming for residuals close to zero.…”

“…The closest previous contributions we know of on this topic are by Manzhos and coauthors, who have developed what they call "rectangular collocation" methods for eigenvalue problems in quantum chemistry [21,22], and by the first two authors [17]. The emphasis in [17] is on spectral methods for ODEs, and the linear algebra is carried out by the method of Ito and Murota [19], involving the singular value decomposition (SVD) of a matrix with twice as many columns as there are basis functions. (Important earlier related papers are [7] and [35].)…”

“…We will mainly deal with fully discrete m × n rectangular matrices, always with m > n. As pointed out in [17], however, it makes good sense conceptually to consider the limit in which the columns are functions of one or more continuous variables, so that instead of matrices, we have quasimatrices; see [4,32] and [12, chap. 6].…”