Numerical Analysis 2.2. Stopping Criteria for Numerical Iterations скачать в хорошем качестве

Numerical Analysis 2.2. Stopping Criteria for Numerical Iterations

The video discusses the stopping criteria for iterations in numerical methods. It addresses the fundamental question of how to determine when a generated sequence of approximations is "close enough" to the actual solution (the root of an equation). Key Stopping Conditions Discussed: The instructor explains three common conditions used to decide when to stop generating elements (P k ​ ) of a sequence: Absolute Difference [01:41]: * Condition: ∣P k ​ −P k−1 ​ ∣ lt ϵ 1 ​ This is a numerical analog for the absolute error. If the difference between consecutive terms is small enough, it suggests the sequence is settling near a limit. Relative Difference [02:44]: * Condition: ∣P k ​ ∣ ∣P k ​ −P k−1 ​ ∣ ​ lt ϵ 2 ​ This acts as a numerical analog for relative error, which is often more useful when the values of P are very large or very small. Function Value [03:44]: * Condition: ∣f(P k ​ )∣ lt ϵ 3 ​ Since the goal is to find the root where f(p)=0, the iteration stops if the function's value at the current approximation is sufficiently close to zero. Limitations and Risks: The video emphasizes that none of these conditions are perfect [04:11]: Conditions 1 & 2 can be satisfied by divergent sequences (like the harmonic series) where terms get closer together even though the sum goes to infinity [04:23]. Condition 3 can fail if the function is very flat; the function value might be near zero even if the current point is far from the actual root [05:42]. Practical Recommendations: Combination: It is recommended to use a combination of these conditions (e.g., 1 and 3, or 2 and 3) in practice [06:25]. Max Iterations: Always include a maximum iteration limit (e.g., 1000 steps) in your code to prevent infinite loops in cases where the sequence does not converge [06:49].