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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. (English) Zbl 1063.33028
The Rayleigh-Ritz method yields a decreasing sequence ${\mu }_{r}$ that approximates the desired eigenvalue $\lambda$ of Mathieu function or of spheroidal wave function. The method produces also a sequence of elementary functions $g$ that approximates the desired eigenfunction $f$ of the same functions. This paper provides answers to some questions about the rate of convergence ${\mu }_{r}\to \lambda$ and ${g}_{r}\to f$ as $r\to \infty$, to the error bounds for the maximum norm $\parallel {g}_{r}{-f\parallel }_{\infty }$ and to other convergence properties of the method. Errors are investigated by interesting numerical experimentations.

##### MSC:
 33E10 Lamé, Mathieu, and spheroidal wave functions 65L60 Finite elements, Rayleigh-Ritz, Galerkin and collocation methods for ODE 65L70 Error bounds (numerical methods for ODE)
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