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  • 1
    Electronic Resource
    Electronic Resource
    Die Determinantenmethode zur Berechnung des charakteristischen Exponenten der endlichen Hillschen Differentialgleichung (1980)
    Springer
    Numerische Mathematik 35 (1980), S. 405-420 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.16
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The characteristic exponent ν of the finite Hill differential equation $$y''(x) + \left( {\lambda + \sum\limits_{\kappa = 1}^k {(2t_\kappa ) \cos (2\kappa x)} } \right) y(x) = 0$$ can be evaluated from the relations $$\sin ^2 \left( {\frac{\pi }{2}v} \right) = \frac{{\pi ^2 }}{4} \det C^{(0)} \det S^{(0)}$$ or $$\cos ^2 \left( {\frac{\pi }{2}v} \right) = \det C^{(1)} \det S^{(1)} ,$$ whereS (μ) andC (μ) are certain infinite band matrices. According to Mennicken [3] the convergence of the infinite determinants can be accelerated by splitting up suitable infinite products. In the present paper this method is discussed under numerical aspects, moreover the formulas for the infinite products are simplified in such way that the complex Gamma-function is no longer needed. Finally, the presented determinental method is compared with other methods by means of some numerical examples.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399008
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  • 2
    Electronic Resource
    Electronic Resource
    Ein Verfahren höherer Konvergenzordnung zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung (1976)
    Springer
    Numerische Mathematik 27 (1976), S. 53-65 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The characteristic exponent ν of Mathieu's differential equation $$y''(x) + 4(\lambda + 2t\cos 2x)y(x) = 0$$ satisfies the relation $$\sin ^2 \left( {\frac{\pi }{2}v} \right) = \sin ^2 (\pi \sqrt \lambda )\det S^{(0)} \det C^{(0)} ,$$ if λ≠n 2 (n∈ℕ), and an analogous equation for λ≠(n+1/2)2, whereS (0) andC (0) are certain infinite tridiagonal matrices. We calculate the determinants ofS (0)=(σ n,m 0 ∞ andC (0) using $$\det S^{(0)} = \prod\limits_{n = 0}^\infty {(1 - \beta _n ){\text{ }}\det B,{\text{ }}B = \left( {\frac{{\sigma _{n,m} }}{{1 - \beta _n }}} \right)_0^\infty ,}$$ where the constants (1−β n ) are chosen in such way that the infinite product may be evaluated by trigonometric functions and the finite determinants detB N converge like a series with termsO(N −12).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399084
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  • 3
    Electronic Resource
    Electronic Resource
    Berechnung des charakteristischen Exponenten der endlichen Hillschen Differentialgleichung durch Numerische Integration (1979)
    Springer
    Numerische Mathematik 32 (1979), S. 31-50 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; CR: 5.16
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01397648
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