Numerical Analysis 2.7. Order of Convergence скачать в хорошем качестве

Numerical Analysis 2.7. Order of Convergence

This video is an educational lecture covering Section 2.7 on the Order of Convergence of a sequence in numerical analysis. Key Concepts Covered Definition of Order of Convergence: A sequence p k ​ converges to p with order α≥1 if there exists a constant C such that: ∣p k+1 ​ −p∣≤C∣p k ​ −p∣ α [00:24]. If α=1, the convergence is linear (requires C lt 1). If α=2, the convergence is quadratic [02:19]. Asymptotic Error Constant (λ): Defined as the limit of the ratio ∣p k ​ −p∣ α ∣p k+1 ​ −p∣ ​ as k→∞ [03:43]. Fixed Point Iteration (g(x)): Convergence is linear if the derivative ∣g ′ (p)∣ lt 1 and non-zero [14:49]. Convergence is of order m if g ′ (p),g ′′ (p),…,g (m−1) (p)=0 but g (m) (p)  =0 [15:36]. Newton’s Method and Multiplicity: For a simple root (multiplicity m=1), Newton's method has quadratic convergence (α=2) [25:02]. For a multiple root (m gt 1), Newton's method converges only linearly (α=1) [25:46]. The lecturer demonstrates a modified Newton's method using a function μ(x) to restore quadratic convergence even for multiple roots [35:34]. Secant Method: The order of convergence is approximately 1.6 (specifically 2 1+ 5 ​ ​ ), which is faster than linear but slower than quadratic [31:08]. Practical Examples Numerical Test: The lecturer applies Newton's method to e x −2cos(x) and analyzes error ratios with different α values (1,2,3) to empirically determine the order of convergence as 2 [07:15]. Speed Comparison: A comparison between linear and quadratic convergence shows that after 6 steps, a quadratic sequence can reach an error of 10 −19 while a linear one remains at 10 −2 [13:31].