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  • 1
    Electronic Resource
    Electronic Resource
    Smoothing noisy data with spline functions (1985)
    Springer
    Numerische Mathematik 47 (1985), S. 99-106 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D ; 65K ; CR: G.1.2 ; G.1.1
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A procedure for calculating the trace of the influence matrix associated with a polynomial smoothing spline of degree2m−1 fitted ton distinct, not necessarily equally spaced or uniformly weighted, data points is presented. The procedure requires orderm 2 n operations and therefore permits efficient orderm 2 n calculation of statistics associated with a polynomial smoothing spline, including the generalized cross validation. The method is a significant improvement over an existing method which requires ordern 3 operations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389878
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  • 2
    Electronic Resource
    Electronic Resource
    QR factorization of Toeplitz matrices (1986)
    Springer
    Numerische Mathematik 49 (1986), S. 81-94 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65F05 ; CR: G1.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary This paper presents a new algorithm for computing theQR factorization of anm×n Toeplitz matrix inO(mn) operations. The algorithm exploits the procedure for the rank-1 modification and the fact that both principal (m−1)×(n−1) submatrices of the Toeplitz matrix are identical. An efficient parallel implementation of the algorithm is possible.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389431
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  • 3
    Electronic Resource
    Electronic Resource
    An efficient method for calculating smoothing splines using orthogonal transformations (1986)
    Springer
    Numerische Mathematik 50 (1986), S. 311-319 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65D, 65K ; CR: G.1.2, G.1.1, G.3
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A procedure for calculating the mean squared residual and the trace of the influence matrix associated with a polynomial smoothing spline of degree 2m−1 using an orthogonal factorization is presented. The procedure substantially overcomes the problem of ill-conditioning encountered by a recently developed method which employs a Cholesky factorization, but still requires only orderm 2 n operations and ordermn storage.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01390708
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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
    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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  • 6
    Electronic Resource
    Electronic Resource
    An approximation theory for boundary value problems on infinite intervals (1980)
    Springer
    Computing 24 (1980), S. 227-239 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Ein mögliches Verfahren zur numerischen Lösung von Randwertproblemen auf unendlichen Intervallen besteht darin, das unendliche Intervall durch ein endliches zu ersetzen und zusätzliche Randbedingungen am entfernten Intervallende aufzuerlegen, in denen das asymptotische Verhalten der Lösung zum Ausdruck kommt. In dieser Arbeit wird für eine recht allgemeine Klasse von Differentialgleichungen die Menge aller zusätzlichen Randbedingungen charakterisiert, für welche die Lösung des „endlichen” Problems bei wachsender Intervallänge gegen die Lösung des „unendlichen” Problems konvergiert. Weiters wird die Konvergenzgeschwindigkeit abgeschätzt, und es werden die „optimalen” Randbedingungen beschrieben, die zu einer möglichst schnellen Konvergenz führen.
    Notes: Abstract Boundary value problems for ordinary differential equations on infinite intervals are often solved by restricting the problem to a large but finite interval and imposing certain supplementary boundary conditions at the far end. The success of this procedure depends on the proper choice of these conditions. For a rather general class of problems we give a characterization of all possible supplementary boundary conditions which work, examine the rate of convergence of the solution of the “finite” problem to that of the original “infinite” problem as the interval length of the finite problem tends to infinity, and describe the supplementary boundary conditions for which this rate is optimal.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02281727
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  • 7
    Electronic Resource
    Electronic Resource
    An algorithm for solving multi-point boundary value problems (1987)
    Springer
    Computing 38 (1987), S. 219-234 
    Details
    ISSN: 1436-5057
    Keywords: 65 L 10 ; Boundary value problems ; multipoint BC ; polychotomy
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir beschreiben ein Verfahren zur Berechnung von Lösungen von Mehrpunkt-Randwertproblemen, wobei die Unterteilungsstruktur des Lösungsraumes speziell ausgenützt wird. Nach der Integration des Systems mit Hilfe der Mehrziel-Methode wird das entstandene lineare Gleichungssystem mit einem speziellen Entkopplungs-Algorithmus gelöst, der berücksichtigt, daß die Randbedingungen für jedes Teilintervall verschiedenartig sein können (im Gegensatz zum Zweipunkt-Fall). Die Theorie wird durch einige Beispiele bestätigt.
    Notes: Abstract We describe a method for computing solutions of multi-point boundary value problems, where the polychotomic structure of the underlying solution space is specifically exploited. After the system is integrated in a multiple shooting way, the resulting linear system is solved by a special decoupling algorithm that takes into account the fact that the dichotomy on each interval is potentially different (in contrast to the two point case). A number of examples sustains the theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02240097
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  • 8
    Electronic Resource
    Electronic Resource
    A note on polychotomy (1990)
    Springer
    Computing 45 (1990), S. 1-15 
    Details
    ISSN: 1436-5057
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Bisher haben sich Untersuchungen über Wachstumsschranken für Lösungen von Randwertproblemen entweder mit 2-Punkt-oder mit Mehrpunkt-Bedingungen beschäftigt. Wenn jedoch einige der 2-Punkt-Bedingungen unabhängig von den übrigen Bedingungen sind, dann konnte man nur ein gröberes Polychotomie-Resultat angeben. In dieser Note wird gezeigt, daß eine solche (entkoppelte) 2-Punkt-Bedingung tatsächlich einen dichotomen Unterraum induziert. Dazu werden die Begriffe Dichotomie und Polychotomie genauer untersucht und es werden geeignete Projektionen zur Beschreibung der Struktur des Lösungsraumes hergeleitet.
    Notes: Abstract Previous analyses on bounds for the growth of solutions of BVP were dealing either with two point or multipoint conditions. For BVP where some of the two point boundary conditions are separated from the rest one could only give a cruder polychotomy result for a multipoint case as such. In this note we show that such a (decoupled) two point condition actually induces a dichotomic subspace. This is done by reconsidering the notions of dichotomy and polychotomy and deriving appropriate projection mappings which describe the solution space structure.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02250581
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  • 9
    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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  • 10
    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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