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  • 1
    Electronic Resource
    Electronic Resource
    An elementary noniterative quadrature-type method for the numerical solution of a nonlinear equation (1986)
    Springer
    Computing 37 (1986), S. 269-275 
    Details
    ISSN: 1436-5057
    Keywords: Primary 65H05 ; secondary: 65D32 ; Cauchy-type principal value integrals ; convergence ; Gauss- and Lobatto-Chebyshev quadrature rules ; nonlinear equations ; numerical integration ; roots ; transcendental equations ; zeros
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wurde eine einfache nicht-iterative Methode für die numerische Berechnung einer einfachen Nullstelle einer nichtlinearen differenzierbaren algebraischen oder transzendenten Funktion längs eines endlichen reellen Intervalles vorgestellt. Die Methode gründet sich auf die Berechnung eines Integrales, das die Funktion enthält, mittels der Gauß- und der Lobatto-Tschebyscheff-Quadraturformeln und die anschließende gleichsetzung der erhaltenen Resultate. Die Konvergenz der Methode wird unter schwachen Annahmen bewiesen; numerische Resultate sind für zwei klassiche transzendente Gleichungen angegeben.
    Notes: Abstract A simple noniterative method for the numerical determination of one simple root of a nonlinear differentiable algebraic or transcendental function along a finite real interval is proposed. This method is based on the computation of an integral involving the above function both by the Gauss- and the Lobatto-Chebyshev quadrature rules for regular integrals and equating the obtained results. The convergence of the method is proved under mild assumptions and numerical results for two classical transcendental equations are presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02252519
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    Paper (German National Licenses)
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  • 2
    Electronic Resource
    Electronic Resource
    Numerical evaluation of analytic functions by Cauchy's theorem (1991)
    Springer
    BIT 31 (1991), S. 276-285 
    Details
    ISSN: 1572-9125
    Keywords: primary: 65E05 ; secondary: 30E20, 65D32 ; analytic functions ; asymptotic estimates ; Cauchy formula ; Cauchy theorem ; circle ; contour ; complex contour integrals ; error bounds ; error term ; numerical integration ; Taylor series ; trapezoidal quadrature rule
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The use of the Cauchy theorem (instead of the Cauchy formula) in complex analysis together with numerical integration rules is proposed for the computation of analytic functions and their derivatives inside a closed contour from boundary data for the analytic function only. This approach permits a dramatical increase of the accuracy of the numerical results for points near the contour. Several theoretical results about this method are proved. Related numerical results are also displayed. The present method together with the trapezoidal quadrature rule on a circular contour is investigated from a theoretical point of view (including error bounds and corresponding asymptotic estimates), compared with the numerically competitive Lyness-Delves method and rederived by using the Theotokoglou results on the error term. Generalizations for the present method are suggested in brief.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01931287
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On the simultaneous determination of zeros of analytic or sectionally analytic functions (1986)
    Springer
    Computing 36 (1986), S. 239-247 
    Details
    ISSN: 1436-5057
    Keywords: Primary ; 65H05 ; secondary ; 30C15 ; 30E20 ; Analytic functions ; polynomials ; Riemann-Hilbert boundary value problem ; sectionally analytic functions ; simultaneous iterative methods ; single-step method ; zeros
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Es wird gezeigt, wie die Totalschritt- und Einschritt-Iterations-Verfahren für die gleichzeitige Bestimmung von einfachen Nullstellen von Polynomen sowie ihre Verbesserungen (mit einer kleinen Modifikation) für die Bestimmung von einfachen Nullstellen analytischer Funktionen (im inneren oder äußeren einer einfachen glatten abgeschlossenen Kontur in der komplexen Ebene) oder stückweise analytischer Funktionen (im äußeren ihrer Unstetigkeitsbögen) benutzt werden können. Numerische Ergebnisse, die mit der Einschrittmethode erhalten wurden, werden auch präsentiert.
    Notes: Abstract It is shown how the total-step and single-step iterative methods, as well as their improvements, for the simultaneous determination of simple zeros of polynomials can be used (with one slight modification) for the determination of simple zeros of analytic functions (inside or outside a simple smooth closed contour in the complex plane) or sectionally analytic functions (outside their arcs of discontinuity). Numerical results, obtained by the single-step method, are also presented.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02240070
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    Paper (German National Licenses)
    Fulltext
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