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  • 1
    Electronic Resource
    Electronic Resource
    Convexity preserving interpolation with exponential splines (1986)
    Springer
    Computing 36 (1986), S. 335-342 
    Details
    ISSN: 1436-5057
    Keywords: 65D07 ; 41A15 ; 90C20 ; Convexity conditions ; weakly coupled system of inequalities ; numerical algorithms ; quadratic programming ; unconstrained dual program with tridiagonal Hessian
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Während die Aufgabe der konvexen Interpolation im Falle kubischer Splines im allgemeineen nicht lösbar ist, kann jetzt gezeigt werden, daß sie bei Verwendung von Exponentialsplines Lösungen besitzt sofern die vorkommenden Parameter hinreichend roß sind, und es werden konkrete hinreichende und notwendige Lösbarkeitsbedingungen hergeleitet. Da eindeutige Lösbarkeit in der Regel nicht vorliegt, wird eine Auswahlfunktion eingeführt und das entstehende quadratische Optimierungsproblem nach Dualisierung numerisch behandelt.
    Notes: Abstract Sufficient and necessary conditions are derived under which interpolating splines are convex if the data set is in convex position. In order to select one of the interpolants, by means of a well-known objective function a quadratic optimization problem is stated which can be solved effectively by passing to a dual program.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02240208
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  • 2
    Electronic Resource
    Electronic Resource
    Convex spline interpolants with minimal curvature (1985)
    Springer
    Computing 35 (1985), S. 219-229 
    Details
    ISSN: 1436-5057
    Keywords: 65K05 ; 65D07 ; 90C20 ; 41A15 ; Existence conditions ; quadratic programming ; dual optimization problem ; tridiagonal Hessian ; Newton's method
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Das Problem, konvexe Interpolationssplines mit minimaler mittlerer Krümmung zu ermitteln, führt auf eine speziell strukturierte quadratische Optimierungsaufgabe. In der vorliegenden Note wird eine zugehörige duale Aufgabe aufgestellt, die ohne Nebenbedingungen auskommt, deren Zielfunktion stückweise quadratisch ist und die daher eine effektive numerische Behandlung erlaubt.
    Notes: Abstract The problem of finding convex spline interpolants with minimal mean curvature leads to a quadratic optimization problem of special structure. In the present note a corresponding dual problem without constraints is derived. Its objective function is piecewise quadratic and therefore admits an effective numerical treatment.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02260507
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  • 3
    Electronic Resource
    Electronic Resource
    Positive interpolation with rational quadratic splines (1987)
    Springer
    Computing 38 (1987), S. 261-267 
    Details
    ISSN: 1436-5057
    Keywords: 65D07 ; 41A15 ; Existence conditions ; construction os positive splines ; minimization of the curvature
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Unter positiver Interpolation wird die Aufgabenstellung verstanden, zur einer nichtnegativen Datenmenge nichtnegative Interpolierende zu konstruieren. Im Fallen rational-quadratischer Splines wird eine notwendige und hinreichende Bedingung für die Durchführbarkeit positiver Interpolation hergeleitet, und es wird gezeigt, daß diese sich bei passender Wahl der vorkommenden Parameter stets erfüllen läßt.
    Notes: Abstract A necessary and sufficient criterion is presented under which the property of positivity carry over from the data set to rational quadratic spline interpolants. The criterion can always be satisfied if the occuring parameters are properly chosen.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02240100
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  • 4
    Electronic Resource
    Electronic Resource
    Positivity of cubic polynomials on intervals and positive spline interpolation (1988)
    Springer
    BIT 28 (1988), S. 340-352 
    Details
    ISSN: 1572-9125
    Keywords: 65D17 ; 41A15 ; 90C20 ; Positivity of cubic polynomials ; nonnegative spline interpolants minimal curvature ; dual program ; return-formula
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A criterion for the positivity of a cubic polynomial on a given interval is derived. By means of this result a necessary and sufficient condition is given under which cubicC 1-spline interpolants are nonnegative. Further, since such interpolants are not uniquely determined, for selecting one of them the geometric curvature is minimized. The arising optimization problem is solved numerically via dualization.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01934097
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