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The characteristic exponent ? 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.