Topic
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 novel method to interpolate a set of data points as well as unit tangent vectors or unit normal vectors at the data points by means of a B-spline curve interpolation technique using geometric algorithms is introduced.
Abstract: We introduce a novel method to interpolate a set of data points as well as unit tangent vectors or unit normal vectors at the data points by means of a B-spline curve interpolation technique using geometric algorithms The advantages of our algorithm are that it has a compact representation, it does not require the magnitudes of the tangent vectors or normal vectors, and it has C^2 continuity We compare our method with the conventional curve interpolation methods, namely, the standard point interpolation method, the method introduced by Piegl and Tiller, which interpolates points as well as the first derivatives at every point, and the piecewise cubic Hermite interpolation method Examples are provided to demonstrate the effectiveness of the proposed algorithms
50 citations
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TL;DR: In this article, a new class of collocation methods using cubic splines for solving elliptic partial differential equations (PDEs) is presented and a convergence analysis is carried out for a broad class of elliptic PDEs.
Abstract: This paper presents a new class of collocation methods using cubic splines for solving elliptic partial differential equations (PDEs). The error bounds obtained for these methods are optimal. The methods are formulated and a convergence analysis is carried out for a broad class of elliptic PDEs. Experimental results confirm the optimal convergence and indicate that these methods are computationally more efficient than methods based on either collocation with Hermite cubics or on the Galerkin method with cubic splines. Various direct and iterative methods have been applied for the solution of cubic spline collocation methods.
50 citations
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01 Sep 1998TL;DR: In this paper, uniform cubic spline polynomials are used to derive consistency relations and these relations are then used to develop a numerical method for computing smooth approximations to the solution and its first, second as well as third derivatives for a second order boundary value problem.
Abstract: In this paper, we use uniform cubic spline polynomials to derive some new consistency relations These relations are then used to develop a numerical method for computing smooth approximations to the solution and its first, second as well as third derivatives for a second order boundary value problem The present method outperforms other collocations, finite-difference and splines methods of the same order Numerical illustrations are provided to demonstrate the practical use of our method
50 citations
09 Jul 2012
TL;DR: In this paper, a new rational Krylov method was proposed for solving the nonlinear eigenvalue problem, where the degree of the interpolating polynomial and the interpolation points are not fixed in advance.
Abstract: This paper proposes a new rational Krylov method for solving the nonlinear eigenvalue problem: $A(\lambda)x = 0$. The method approximates $A(\lambda)$ by Hermite interpolation where the degree of the interpolating polynomial and the interpolation points are not fixed in advance. It uses a companion-type reformulation to obtain a linear generalized eigenvalue problem (GEP). To this GEP we apply a rational Krylov method that preserves the structure. The companion form grows in each iteration and the interpolation points are dynamically chosen. Each iteration requires a linear system solve with $A(\sigma)$, where $\sigma$ is the last interpolation point. The method is illustrated by small- and large-scale numerical examples. In particular, we illustrate that the method is fully dynamic and can be used as a global search method as well as a local refinement method. In the last case, we compare the method to Newton's method and illustrate that we can achieve an even faster convergence rate.
50 citations