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Mathematics of Computation

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Finite-difference methods and the eigenvalue problem for nonselfadjoint Sturm-Liouville operators


Author: Alfred Carasso
Journal: Math. Comp. 23 (1969), 717-729
MSC: Primary 65.62
MathSciNet review: 0258291
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Abstract: In this paper we analyze the convergence of a centered finite-difference approximation to the nonselfadjoint Sturm-Liouville eigenvalue problem where $ [{\text{unk}}]$ has smooth coefficients and $ a(x) \geqq {a_0} > 0$ on [0, 1]. We show that the rate of convergence is $ O(\Delta {x^2})$ as in the selfadjoint case for a scheme of the same accuracy. We also establish discrete analogs of the Sturm oscillation and comparison theorems. As a corollary we obtain the result

$\displaystyle \mathop {\lim \sup }\limits_{M \to \infty ;{\Delta _x} \to 0;(M +... ...frac{{\vert\vert{V^p}\vert\vert\infty }} {{{\Lambda _p}}}} } \right\} < \infty $

) where $ \Delta x = 1/(M + 1)$ is the mesh size and $ {\Lambda _p},{V^p}$ are the characteristic pairs of $ L$, the $ M \times M$ matrix which approximates $ [{\text{unk}}]$, and $ {V^p}$ is normalized so that $ \vert\vert{V^p}\vert{\vert _2} = 1$.

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DOI: https://doi.org/10.1090/S0025-5718-1969-0258291-7
Article copyright: © Copyright 1969 American Mathematical Society