--ZusammenfassungConvex Spline Interpolants with Minimal Curvature. 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. . The problem to construct a cubic spline interpolant which is convex if the given data set is of this type may be not solvable. If a solution exists the convex spline interpolant is in general not uniquely determined. In this case it is appropriate to select that spline which has minimal curvature. These requirements lead to a quadratic programming problem of special structure. In the present note a corresponding dual problem without constraints is stated having the advantage that it can be solved effectively. The main result representing smooth duality statements is theorem 7.
A MS Subject
2.In order to formulate the situation in more detail denote by A a grid on the interval
Das betrachtete Verfahren gehört zu der Klasse von Minimterungsverfahren, die das Minimum einer quadratischen Funktion von n Argumenten in höchstens n Schritten liefern. Für nichtquadratische konvexe Zielfunktionen wird gezeigt, daß das Verfahren mindestens die Konvergenzordnung n√2 besitzt.
Abstract.In this paper a new technique for avoiding exact Jacobians in ROW methods is proposed. The Jacobians f~ are substituted by matrices A. satisfying a directional consistency condition Anf, = f'f. + O(h). In contrast to general W-methods this enables us to reduce the number of order conditions and we construct a 2-stage method of order 3 and families of imbedded 4-stage methods of order 4(3). The directional approximation of the Jacobians has been realized via rank-1 updating as known from quasi-Newton methods.
Abstract-ZusammenfassungOn the R-Order of Coupled Sequences II. As in the preceding paper [3], the question whether some of the sequences {x., i} coupled by a system (3) of inequalities converge at least with a certain R-order z is reduced to the nonnegative solvability of the system (4) of linear inequalities. Further, the optimal R-order z implied by (3) is characterized as the spectral radius of a certain matrix composed of exponents appearing in (3).l~ber die R-Ordnung gekoppelter Folgen II. Die Bestimmung der R-Ordnung z yon Folgen {x,. ~}, die fiber ein System (3) yon Ungleichungen miteinander verkoppelt sind, wird wie in der vorangegangenen Arbeit [3] auf die nichtnegative L6sbarkeit des Systems (4) von linearen Ungleichnngen zurfickgeffihrt. Weiterhin wird gezeigt, dab die beste R-Ordnung z, welche sich aus (3) herteiten 1513t, gleich dem Spektralradius einer bestimmten Matrix ist, welche aus in (3) auftretenden Exponenten gebildet wird.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.