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Polynomial Interpolation
Published in Jeffery J. Leader, Numerical Analysis and Scientific Computation, 2022
(−1≤x≤1). In the Problems you are asked to create and plot several interpolants to it. Large oscillations develop near the endpoints. This type of behavior is typical; the polynomial interpolant must often make a large excursion in order to be able to turn smoothly and ultimately interpolate the function at the next node. For the Runge function, the amplitude of the turns actually increases as n increases–that is, the approximation gets worse as n increases, in the sense that max|f(x)−p(x)| grows and in fact diverges. This possible failure of polynomial interpolants to converge as n→∞ is known as the Runge phenomenon.
Robotics
Published in Jian Chen, Bingxi Jia, Kaixiang Zhang, Multi-View Geometry Based Visual Perception and Control of Robotic Systems, 2018
Jian Chen, Bingxi Jia, Kaixiang Zhang
For the case of path following or trajectory tracking, a set of points are generally given to represent the desired path or trajectory. For control development, a sufficient smooth curve representation is required, i.e., the first and second derivatives exist and are bounded. High order polynomial functions can be used for curve fitting by solving the least square solution of conditions formed by the points. However, the numerical stability decreases with higher order and the fitting accuracy decreases with lower order. Besides, the Runge’s phenomenon exists which is a problem of oscillation at the edges of an interval that occurs when using polynomial interpolation with polynomials of high degree over a set of equispaced interpolation points. It shows that going to higher order does not always improve accuracy of curve fitting. Considering the flexibility and accuracy of curve fitting, the spline curves can be used for representation. A spline is a special function defined piecewise by polynomials. It uses low order polynomials in each interval and ensures the continuity between the intervals. Considering the fundamental case of one-dimensional fitting, a set of points {xi}i∈ [1,N] at time instants {ti}i∈ [1,N] are given. There are various forms of spline functions that can be used for curve fitting. Without generality, the following two cubic spline functions are used:
Interpolation and Extrapolation
Published in James P. Howard, Computational Methods for Numerical Analysis with R, 2017
Further, while the higher-degreed polynomial is guaranteed to pass through all of the points given, it may fluctuate wildly between two given points, a pattern known as Runge’s phenomenon. In many applied problems, such as modelling an income curve, we might know a priori or have very strong reason to believe that such fluctuation does accurately reflect the underlying data. In these applied problems, this fluctuation should be eliminated to support analysis.
Multi-objective trajectory optimization of the 2-redundancy planar feeding manipulator based on pseudo-attractor and radial basis function neural network
Published in Mechanics Based Design of Structures and Machines, 2023
Shenquan Huang, Shunqing Zhou, Luchuan Yu, Jiajia Wang
The path point with time information is defined as the trajectory point. The allocation of time information is a key problem in trajectory optimization. It can directly affect the trajectory performance. Based on the I-RBFNN algorithm in Section 2.2, a time-adaptive allocation strategy is used to endow path points with reasonable time information. If a high-order polynomial interpolation function is used to fit discrete points, the Runge phenomenon will easily occur. To avoid the Runge phenomenon during the trajectory planning, the constrained function in Eq. (17) is considered where k is the number of sub-trajectory segments. M is the number of selected time points in the ith sub-trajectory. tj is the jth time point in the ith sub-trajectory.
A real 3D scene rendering optimization method based on region of interest and viewing frustum prediction in virtual reality
Published in International Journal of Digital Earth, 2022
Pei Dang, Jun Zhu, Jianlin Wu, Weilian Li, Jigang You, Lin Fu, Yiqun Shi, Yuhang Gong
The problem of predicting the VR viewing frustum can be simplified to predict the state of the next time according to the past state through extrapolation. Extrapolation methods mainly include Lagrange interpolation, the Hermite interpolation method, and a Kalman filter. The Lagrange method is a polynomial-based method, which has less computation and is suitable for smooth motion trajectory prediction. However, when the interpolation times are high, the Runge phenomenon will appear, resulting in a large deviation of the interpolation results. Hermite interpolation is an optimization of the Lagrange interpolation method, but it requires the same derivative value at the viewpoint and has great restrictions on use, so it is not suitable for viewing frustum prediction. A Kalman filter is a linear optimal filtering algorithm (Meinhold and Singpurwalla 1983) that uses the mean square error and the criterion of the minimum mean square error to predict the rotation of the VR helmet. It has high accuracy and no requirements for the movement of the viewing frustum (Gómez and Maravall 1994). In VR, to prevent the occurrence of dizziness, teleportation movement mode is mostly used. This movement mode is discontinuous and cannot realize position prediction through extrapolation. Therefore, it is more reasonable to extrapolate and predict the rotation of the VR helmet using a Kalman filter (Zhang and Zhang 2010).
Characterizing Tradeoffs in Memory, Accuracy, and Speed for Chemistry Tabulation Techniques
Published in Combustion Science and Technology, 2023
Elizabeth Armstrong, John C. Hewson, James C. Sutherland
We will explore which minimizes overshooting errors that can be experienced with high-degree polynomial interpolants (Runge’s phenomenon). This is also a typical range of degrees for interpolants and splines used in combustion.