Numerical Analysis 6.5. Cubic Spline Interpolation скачать в хорошем качестве

Numerical Analysis 6.5. Cubic Spline Interpolation

This video is a detailed lecture on Spline Interpolation, specifically focusing on Cubic Splines as part of a Numerical Analysis course. Introduction to Spline Functions The lecture begins by defining a spline function of degree k over an interval [a,b] divided by a mesh. A spline function s(x) must satisfy two main properties: It is k−1 times continuously differentiable across the entire interval [00:51]. On each sub-interval [x i ​ ,x i+1 ​ ], the function is a polynomial of degree at most k [01:02]. Types of Splines: The lecturer mentions linear splines (degree 1), quadratic splines (degree 2), and cubic splines (degree 3), which are preferred in applications like computer graphics because they produce smooth curves [01:35]. Cubic Spline Interpolation The core of the lecture explores how to construct a cubic spline that passes through a set of data points (x i ​ ,y i ​ ). Conditions: To solve for the 4n parameters of the piecewise cubic polynomials, several conditions must be met: Interpolation: The segments must pass through the given data points [04:37]. Continuity: The first and second derivatives must match at the internal mesh points to ensure smoothness [06:17]. Natural Spline: Since the number of conditions is two fewer than the number of parameters, two additional conditions are needed. For a natural cubic spline, the second derivatives at the endpoints (x 0 ​ and x n ​ ) are set to zero [09:11]. Solving the System: The lecturer demonstrates a mathematical "trick" by writing the polynomial in the form s i ​ (x)=a i ​ +b i ​ (x−x i ​ )+c i ​ (x−x i ​ ) 2 +d i ​ (x−x i ​ ) 3 . This leads to a tridiagonal, diagonally dominant matrix system that can be solved quickly and accurately using Gaussian elimination [18:45], [22:44]. Key Theorems and Properties Minimum Curvature: A theorem states that the natural cubic spline is the "smoothest" possible interpolating function, meaning it minimizes the integral of the square of the second derivative (a measure of curvature) [27:16], [29:36]. Clamped Splines: The lecturer briefly mentions clamped splines, where the first derivatives at the endpoints are specified instead of the second derivatives [26:38]. Error Bounds: The lecture concludes with error estimates. For a mesh with maximum spacing h, the approximation error for the function is of order O(h 4 ), while the first and second derivatives have errors of order O(h 3 ) and O(h 2 ), respectively [30:32].