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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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  • 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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  • 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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  • 4
    Electronic Resource
    Electronic Resource
    Numerical determination of a class of generalized stress intensity factors (1979)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 14 (1979), S. 949-959 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A modification of the collocation method for the numerical solution of Cauchy-type singular integral equations appearing in plane elasticity and, especially, crack problems is proposed. This modification, based on a variable transformation, applies to the case when the unknown function of the singular integral equation behaves like A(x - c)α + B(x - c)β, where α 〈 0, 0 〈 β - α 〈 1, near an endpoint c of the integration interval. In plane elasticity such a point is either a crack tip or a corner point of the boundary of the elastic medium. Thus the method seems to be quite efficient for the numerical evaluation of generalized stress intensity factors near such points. A successful application of the method to the classical plane elasticity problem of an antiplane shear crack terminating at a bimaterial interface was also made.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620140702
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  • 5
    Electronic Resource
    Electronic Resource
    A remark on the numerical evaluation of stress intensity factors by the method of singular integral equations (1979)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 14 (1979), S. 1710-1714 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A modification of the numerical techniques of solution of Cauchy-type singular integral equations and determination of stress intensity factors at crack tips in plane elastic media is proposed. This technique presents some advantages under appropriate geometry conditions.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620141112
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  • 6
    Electronic Resource
    Electronic Resource
    On the numerical evaluation of two-dimensional principal value integrals (1980)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 15 (1980), S. 629-634 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620150414
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  • 7
    Electronic Resource
    Electronic Resource
    A superconvergence result for the natural extrapolation formula for the numerical determination of stress intensity factors (1985)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 21 (1985), S. 1391-1401 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The best approach for the numerical determination of stress intensity factors at crack tips in plane and antiplane elasticity problems is frequently the numerical solution of the corresponding Cauchy-type singular integral equation by the Gauss-Chebyshev method, followed by the application of the natural extrapolation formula for the numerical determination of the stress intensity factors. It is shown here that this approach converges for Hölder-continuous and discontinuous (with jump discontinuities) loading distributions along the crack (or cracks) and that in all cases the rate of convergence is greater than that believed up to now. This superconvergence result is based on a theorem on the numerical equivalence of the Gauss-Chebyshev direct method to a relevant indirect method for the numerical solution of Cauchy-type singular integral equations, also proved here. Numerical results in various crack problems corroborate the theoretical ones.
    Additional Material: 4 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620210804
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  • 8
    Electronic Resource
    Electronic Resource
    A remark on the direct numerical determination of stress intensity factors at crack tips (1982)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 18 (1982), S. 1416-1419 
    Details
    ISSN: 0029-5981
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: A numerical method for the direct determination of stress intensity factors at crack tips from the numerical solution of the corresponding singular integral equations is proposed. This method is based on the Gauss-Chebyshev method for the numerical solution of singular integral equations and is shown to be equivalent to the Lobatto-Chebyshev method for the numerical solution of the same class of equations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/nme.1620180913
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  • 9
    Electronic Resource
    Electronic Resource
    The Gauss-Laguerre quadrature rule for finite-part integrals (1993)
    Chichester : Wiley-Blackwell
    Communications in Numerical Methods in Engineering 9 (1993), S. 439-450 
    Details
    ISSN: 1069-8299
    Keywords: Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The classical Gauss-Laguerre quadrature rule for the semi-infinite integration interval [0,∞] is modified and applied to the case of the weight function exp(-x)/x corresponding to finite-part (or, equivalently, hypersingular) integrals. The new set of orthogonal polynomials is constructed and it is seen to consist of linear combinations of the classical Laguerre polynomials with appropriately determined coefficients. The zeros of these modified Laguerre polynomials are seen to be distinct, but one of these lies outside the integration interval. Formulae for the corresponding weights are also given and numerical values and results are presented. The present results generalize the corresponding results for the finite interval [0,1] to semi-infinite intervals and they are applicable to a variety of applied mechanics and related problems, where finite-part integrals appear in a natural way.
    Additional Material: 2 Tab.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/cnm.1640090509
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  • 10
    Electronic Resource
    Electronic Resource
    BEAMS ON TENSIONLESS ELASTIC FOUNDATION: APPROXIMATE QUANTIFIER ELIMINATION WITH CHEBYSHEV SERIES (1996)
    Chichester [u.a.] : Wiley-Blackwell
    International Journal for Numerical Methods in Engineering 39 (1996), S. 663-686 
    Details
    ISSN: 0029-5981
    Keywords: beams ; bending ; Chebyshev approximations ; quantifier elimination ; Sturm sequences ; tensionless elastic foundation ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics , Technology
    Notes: The well-known Sturm's theorem (based on Sturm's sequences) for the determination of the number of distinct real zeros of polynomials in a finite or infinite real interval has been already used in elementary quantifier elimination problems including applied mechanics and elasticity problems. Here it is further suggested that this theorem can also be used for quantifier elimination, but in more complicated problems where the functions involved are not simply polynomials, but they may contain arbitrary transcendental functions. In this case, it is suggested that the related transcendental equations/inequalities can be numerically approximated by polynomial equations/inequalities with the help of Chebyshev series expansions in numerical analysis. The classical problem of a straight isotropic elastic beam on a tensionless elastic foundation, where the deflection function (incorporating both the exponential function and trigonometric functions) should be continuously positive (this giving rise to a quantifier elimination problem along the length of the beam) is used as an appropriate vehicle for the illustration of the present mixed (symbolic-numerical) approach. Two such elementary beam problems are considered in some detail (with the help of the Maple V computer algebra system) and the related simple quantifier-free formulae are established and seen to coincide with those already available in the literature for the same beam problems. More complicated problems, probably necessitating the use of more advanced computer algebra techniques (together with Sturm's theorem), such as the Collins well-known and powerful cylindrical algebraic decomposition method for quantifier elimination, can also easily be employed in the present approximate (because of the use of Chebyshev series expansions) symbolic-numerical computational environment.
    Additional Material: 1 Ill.
    Type of Medium: Electronic Resource
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