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  • 1
    Electronic Resource
    Electronic Resource
    On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems (1981)
    Springer
    Computing 26 (1981), S. 123-139 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die Benutzung algebraischer Eigenwerte zur näherungsweisen Berechnung der Eigenwerte von Sturm-Liouville-Operatoren ist bekanntlich nur für die Grundschwingung und einige weitere Harmonische zufriedenstellend. In dieser Arbeit zeigen wir, wie man den asymptotischen Fehler, der bei verwandten aber einfachen Sturm-Liouville-Operatoren auftritt, dazu benutzen kann, um gewisse Klassen algebraischer Eigenwerte so zu korrigieren, daß die gleichmäßig gute Approximationen liefern.
    Notes: Abstract The use of algebraic eigenvalues to approximate the eigenvalues of Sturm-Liouville operators is known to be satisfactory only when approximations to the fundamental and the first few harmonics are required. In this paper, we show how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02241779
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  • 2
    Electronic Resource
    Electronic Resource
    A numerical technique for solving the scalar wave equation for Gaussian and smoothed-out profiles (1981)
    Springer
    Optical and quantum electronics 13 (1981), S. 217-224 
    Details
    ISSN: 1572-817X
    Source: Springer Online Journal Archives 1860-2000
    Topics: Electrical Engineering, Measurement and Control Technology , Physics
    Notes: Abstract A numerical technique, based on transforming the scalar wave equation to a finite interval before discretization is applied, is proposed for the solution of the scalar wave equation defining the modal fields and propagation constants of optical fibres. It is used to compute the propagation constants and modal fields of weakly guiding fibres of circular cross-section when the refractive index profile has a Gaussian or smoothed-out form. The smoothed-out profiles are studied because they vary continuously from the Gaussian to the step profile. Results are compared with and shown to be accurately approximated by simply explicit functions, dependent only on the fibre parameter.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00620324
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    On the correction of finite difference eigenvalue approximations for sturm-liouville problems with general boundary conditions (1984)
    Springer
    BIT 24 (1984), S. 401-412 
    Details
    ISSN: 1572-9125
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract When finite difference and finite element methods are used to approximate continuous (differential) eigenvalue problems, the resulting algebraic eigenvalues only yield accurate estimates for the fundamental and first few harmonics. One way around this difficulty would be to estimate the error between the differential and algebraic eigenvalues by some independent procedure and then use it to correct the algebraic eigenvalues. Such an estimate has been derived by Paine, de Hoog and Anderssen for the Liouville normal form with Dirichlet boundary conditions. In this paper, we extend their result to the Liouville normal form with general boundary conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934900
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  • 4
    Electronic Resource
    Electronic Resource
    Finite difference methods for the numerical differentiation of non-exact data (1984)
    Springer
    Computing 33 (1984), S. 259-267 
    Details
    ISSN: 1436-5057
    Keywords: 65D25 ; 62M15 ; Finite difference methods ; numerical differentiation ; regularization ; Wiener filtering ; minimum variance ; amplification factor
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung In dieser Arbeit werden Ergebnisse über die numerische Güte von Mehrpunktdifferenzenformeln für die Differentation empirischer Funktionen hergeleitet. Insbesondere wird gezeigt, daß Mehrpunktdifferenzenoperatoren konstruiert werden können, die asymptotisch verzerrungsfrei sind und einen für abnehmende Schrittweite und zunehmende Punkteanzahl beschränkten Amplifikationsfaktor besitzen.
    Notes: Abstract In this paper, we derive results about the numerical performance of multi-point (moving average) finite difference formulas for the differentiation of non-exact data. In particular, we show that multi-point differentiators can be constructed which are asymptotically unbiased and have a bounded amplification factor as the steplength decreases and the number of points increases.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02242272
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  • 5
    Electronic Resource
    Electronic Resource
    The numerical solution of parabolic differential equations using variational methods (1969)
    Springer
    Numerische Mathematik 13 (1969), S. 129-145 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The question of constructing stable numerical representations for the solutions of initial-boundary value problems for parabolic differential equations is examined.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02163231
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  • 6
    Electronic Resource
    Electronic Resource
    A stability analysis for the extended Kantorovich method applied to the torsion problem (1971)
    Springer
    Numerische Mathematik 17 (1971), S. 239-246 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary It is shown that the extended Kantorovich method introduced by Kerr [4] cannot be unconditionally stable. On the basis of this stability result, it is concluded that Kerr's claims for the extended Kantorovich method, though justified, are optimistic.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01436380
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  • 7
    Electronic Resource
    Electronic Resource
    Numerical differentiation procedures for non-exact data (1974)
    Springer
    Numerische Mathematik 22 (1974), S. 157-182 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The numerical differentiation of data divides naturally into two distinct problems: (i) the differentiation of exact data, and (ii) the differentiation of non-exact (experimental) data. In this paper, we examine the latter. Because methods developed for exact data are based on abstract formalisms which are independent of the structure within the data, they prove, except for the regularization procedure of Cullum, to be unsatisfactory for non-exact data. We therefore adopt the point of view that satisfactory methods for non-exact data must take the structure within the data into account in some natural way, and use the concepts of regression and spectrum analysis as a basis for the development of such methods. The regression procedure is used when either the structure within the non-exact data is known on independent grounds, or the assumptions which underlie the spectrum analysis procedure [viz., stationarity of the (detrended) data] do not apply. In this latter case, the data could be modelled using splines. The spectrum analysis procedure is used when the structure within the nonexact data (or a suitable transformation of it, where the transformation can be differentiated exactly) behaves as if it were generated by a stationary stochastic process. By proving that the regularization procedure of Cullum is equivalent to a certain spectrum analysis procedure, we derive a fast Fourier transform implementation for regularization (based on this equivalence) in which an acceptable value of the regularization parameter is estimated directly from a time series formulation based on this equivalence. Compared with the regularization procedure, which involvesO(n 3) operations (wheren is the number of data points), the fast Fourier transform implementation only involvesO(n logn).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01436965
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  • 8
    Electronic Resource
    Electronic Resource
    Improved estimates of statistical regularization parameters in fourier differentiation and smoothing (1986)
    Springer
    Numerische Mathematik 48 (1986), S. 671-697 
    Details
    ISSN: 0945-3245
    Keywords: AMS (MOS): 65D25 ; 65D10 ; CR: G1.2
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We investigate the statistical methods of cross-validation (CV) and maximum-likelihood (ML) for estimating optimal regularization parameters in the numerical differentiation and smoothing of non-exact data. Various criteria for optimality are examined, and the (asymptotic) notions of strong optimality, weak optimality and suboptimality are introduced relative to these criteria. By restricting attention to certainN-dimensional Hilbert spaces of smooth and stochastic functions, whereN is the number of data, we give regularity conditions on the data under which λCV, the regularization parameter predicted by CV, is strongly optimal with respect to the predictive mean-square signal error. We show that λML is at best weakly optimal with respect to this criterion but is strongly optimal with respect to the innovation variance of the data. For numerical differentiation, λCV and λML are both shown to be suboptimal with respect to the predictive mean-square derivative error.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399688
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  • 9
    Electronic Resource
    Electronic Resource
    Asymptotic structure in torsional free oscillation data for the Earth (1974)
    [s.l.] : Nature Publishing Group
    Nature 250 (1974), S. 400-401 
    Details
    ISSN: 1476-4687
    Source: Nature Archives 1869 - 2009
    Topics: Biology , Chemistry and Pharmacology , Medicine , Natural Sciences in General , Physics
    Notes: [Auszug] Using the Liouville transformation, s U=r-\w)-** Z, r= J μ1/2 p~1/2 dt+a, a the asymptotic behaviour of nsl, can be expressed in the form Baa,=(/i7c)a y-2+^y-2 + £y-2 /i-a+0(/i-3) A for fixed l and suitably large n, where g = ò r1/2 μ–1/2 dr, and A and B are constants which ...
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1038/250400a0
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  • 10
    Electronic Resource
    Electronic Resource
    The character of non-uniqueness in the conductivity modelling problem for the earth (1970)
    Springer
    Pure and applied geophysics 80 (1970), S. 238-259 
    Details
    ISSN: 1420-9136
    Source: Springer Online Journal Archives 1860-2000
    Topics: Geosciences , Physics
    Notes: Summary In this paper, we examine the character of non-uniqueness in the conductivity modelling problem for a spherically symmetric earth in order to show that methods, for determining upper and lower bounds on the possible range of values for electrical conductivity as a function of depth, must be developed before a better picture of conductivity within the earth can be obtained.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00880211
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