Table of contents
2. How to avoid the Runge phenomenon?
foreword
Data fitting refers to knowing a finite number of data points, and seeking to approximate these data point approximation functions, so the function does not need to pass through the known data points, and only requires the minimum total deviation of the function on these points. In path planning based on polynomial curves, by constructing boundary conditions and using polynomial functions to plan a smooth path, it can be regarded as an interpolation because it passes through the first and last points. However, when a polynomial function is used for interpolation, as the order of the polynomial increases, the interpolation result will deviate from the original function, which we call the Runge Phenomenon.
1. Runge phenomenon
The Runge phenomenon was discovered in 1901 by the German mathematician Carl Runge, who gave a simple function in his research results on the risk of high-degree polynomial interpolation
The above formula is called the Runge function, and this function has a rather abnormal property: as the order of polynomial function interpolation increases, the interpolation error becomes larger instead.
In order to facilitate intuitive understanding, we draw the scatter points of the Runge function in the [-1, 1] interval according to the interval of 0.1, and use the polyfit function of MATLAB to use the polynomial of degree 5, polynomial of degree 9, polynomial of degree 14 and polynomial of degree 16 The degree polynomial is used for fitting (MATLAB's polyfit function specifically refers to the use of polynomial curves to fit a limited number of scattered points. In this paper, fitting is used instead of interpolation, and the conclusions obtained below remain unchanged). The results are shown in Figure 1 below.
It can be seen from Figure 1 that when the polynomial degree is low, the global fitting effect with the scatter point is poor; as the polynomial degree increases, the local fitting effect of the scatter point is very good, but when it is close to -1 and 1 There is a large oscillation at the end point. The above phenomenon shows that the use of high-degree polynomial fitting does not always improve the accuracy. Polynomials produced in this way may actually diverge as the degree of the polynomial increases, usually at the endpoints near the interpolation points.
2. How to avoid the Runge phenomenon?
To avoid the Runge phenomenon that occurs when high-order polynomials are fitted, the scattered points are usually interpolated using multi-segment splines. In fact, in the B-spline curve in the previous section, we have been exposed to the idea of multi-segment spline curve interpolation. The B-spline curve uses the mechanism of basis functions and node vectors. A complete B-spline curve is formed by It is formed by connecting several spline curves with lower degrees. The spline function belongs to piecewise smooth interpolation. Its basic idea is to use a low-degree polynomial to approximate each small interval formed by two adjacent value points, and to ensure that the connection of each value point is smooth (that is, the derivative is continuous).