Berechnung des charakteristischen Exponenten der endlichen Hillschen Differentialgleichung durch Numerische Integration

Summary

The characteristic exponent ν of the finite Hill equation

$$(*) y''(x) + \left( {\lambda + 2\sum\limits_{k = 1}^l {t_k \cos (2kx)} } \right) y(x) = 0$$

satisfies the equations

$$\cos (\pi v) = 2y_1 \left( {\frac{\pi }{2}} \right) y'_2 \left( {\frac{\pi }{2}} \right) - 1 = 2y_2 \left( {\frac{\pi }{2}} \right) y'_1 \left( {\frac{\pi }{2}} \right) + 1,$$

wherey ₁,y ₂ are the canonical fundamental solutions of (*). For calculatingy ₁,y ₂ the Taylor expansion method of a high orderp (10≦p≦40) turns out to be the best of all known methods of numerical integration. In this paper the Taylor method for solving (*) is formulated, an extensive error analysis-including the rounding errors—is performed. If the parameters in (*) are not too large, the computed error bounds will be rather realistic.