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On a shooting algorithm for Sturm-Liouville eigenvalue problems with periodic and semi-periodic boundary conditions. (English) Zbl 0804.65085
The eigenvalues of a regular Sturm-Liouville system on $(0,a)$ with periodic and semi-periodic boundary conditions satisfy an equation (i) $\varphi\sb 1 (a, \lambda) + \varphi\sb 2' (a, \lambda) = D (\lambda) = \pm 2$, where $\varphi\sb 1$, $\varphi\sb 2$ are basic solutions of the differential equation satisfying initial conditions $\varphi\sb 1 (0, \lambda) = 1$, $\varphi\sb 1' (0, \lambda) = 0$, $\varphi\sb 2 (0, \lambda) = 0$, $\varphi\sb 2' (0, \lambda) = 1$. The author applies the shooting method to generate a sequence $\lambda\sp{(k)}$, $\mu\sp{(k)}$ of numbers and functions $\varphi\sb i (x, \lambda\sp{(k)})$, $\varphi\sb i (x, \mu\sp{(k)})$, $i=1,2$, such that $\lambda\sp{(k)}$, $\mu\sp{(k)}$ converge to eigenvalues $\lambda, \mu$ of the periodic and semi-periodic boundary value problems respectively. Starting values for $\lambda\sp{(k)}$, $\mu\sp{(k)}$ are based on the known behavior of $D(\lambda)$. The new algorithm is applied to three test problems, one of which is the Mathieu equation. Calculations made on a CYBER 860 computer for the first ten pairs of eigenvalues are accurate to $10\sp{-4}$.

65L15Eigenvalue problems for ODE (numerical methods)
34B24Sturm-Liouville theory
34L15Eigenvalues, estimation of eigenvalues, upper and lower bounds for OD operators
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