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Monotone cubic interpolation
About: Monotone cubic interpolation is a research topic. Over the lifetime, 1740 publications have been published within this topic receiving 38111 citations.
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TL;DR: A bivariate C^1 cubic super spline is constructed on Powell-Sabin type-1 split with the additional smoothness at vertices in the original triangulation being C^2, which permits the Hermite interpolation up to the second order partial derivatives exactly on all the vertices of the original Triangulation.
Abstract: A bivariate C^1 cubic super spline is constructed on Powell-Sabin type-1 split with the additional smoothness at vertices in the original triangulation being C^2, which permits the Hermite interpolation up to the second order partial derivatives exactly on all the vertices in the original triangulation. The locally supported dual basis and computational details using derivatives around each vertex are given.
12 citations
TL;DR: In this article, the authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties.
Abstract: Recently, in [Electronic Transaction on Numerical Analysis, 41 (2014), pp. 420-442] authors introduced a new class of rational cubic fractal interpolation functions with linear denominators via fractal perturbation of traditional nonrecursive rational cubic splines and investigated their basic shape preserving properties. The main goal of the current article is to embark on univariate constrained fractal interpolation that is more general than what was considered so far. To this end, we propose some strategies for selecting the parameters of the rational fractal spline so that the interpolating curves lie strictly above or below a prescribed linear or a quadratic spline function. Approximation property of the proposed rational cubic fractal spine is broached by using the Peano kernel theorem as an interlude. The paper also provides an illustration of background theory, veined by examples.
11 citations
TL;DR: It is shown that if the chosen partitioning of the state space is sufficiently small, the resulting suboptimal controller leads to a stable closed loop system.
Abstract: We consider a class of nonlinear quadratic
regulator problems where the system dynamics are affine in the control.
It has been shown recently that an optimal feedback control law
for this class of problems can be given in terms of the solution of a
state dependent algebraic Ricatti equation (ARE) at each instance of
time. However, in most practical
problems it is not possible to find an analytic solution to the ARE and hence
numerical schemes to calculate suboptimal controls are required. In this paper,
we consider one such scheme based on cubic basis spline interpolation. It is
shown that if the chosen partitioning of the state space is sufficiently small,
the resulting suboptimal controller leads to a stable closed loop system.
11 citations
TL;DR: In this paper, a local Hermite scheme incorporating a tension parameter for each spline segment is employed, the imposed shape constraints being concerned with preservation of convexity at the knots and the sign of the discrete torsion over spline segments.
Abstract: The interpolation of discrete spatial data-a sequence of points and unit tangents-by G(1) Pythagorean-hodograph (PH) quintic spline curves, under shape constraints, is addressed. To achieve this, a local Hermite scheme incorporating a tension parameter for each spline segment is employed, the imposed shape constraints being concerned with preservation of convexity at the knots and the sign of the discrete torsion over each spline segment. An asymptotic analysis in terms of the tension parameters is developed, and it is shown that satisfaction of the prescribed shape constraints can always be achieved for each spline segment by a suitable choice of the free angular parameters that characterize each PH quintic Hermite segment. In particular, it is proved that the cubic-cubic criterion (Farouki, R. T., Giannelli, C., Manni, C. & Sestini, A. (2008) Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures. Comput. Aided Geom. Design, 25, 274-297; Sestini, A., Landolfi, L. & Manni, C. (2013) On the approximation order of a space data dependent PH quintic Hermite interpolation scheme. Comput. Aided Geom. Design, 30, 148-158) for specifying these free parameters ensures satisfaction of the desired shape-preserving properties, requiring only mild application of the tension parameters that does not compromise the overall fairness of the interpolant. The performance of the method is illustrated through some computed examples.
11 citations
TL;DR: This paper considers the optimality and the evaluation of the constants that appear in the expressions of error bounds for interpotating spline functions over a uniform mesh of the real line when the nodes are uniformly shifted.
11 citations