Theoretical Principles & Computational Physics Guide
While Loop: Indefinite Iteration & Convergence Thresholds
Dr. Alorika Chatterjee & Dr. Anirban Shaw
Science & Physics Learners
Python 3 / NumPy / SciPy
1. Core Principles & Theoretical Framework
Convergence Control: Repeats calculations until an error tolerance $|x_{k+1} - x_k| < \epsilon_{\text{tol}}$ is satisfied.
Stopping Conditions: Ideal for root-finding, iterative relaxation, and trajectory termination (e.g. projectile ground impact).
Safety Guards: Includes iteration caps (
max_iter) to prevent infinite loops when convergence fails.2. Mathematical Formulation & Algorithmic Pipeline
Convergence criterion for iterative scientific algorithms:
$$ |x_{k+1} - x_k| < \epsilon_{\text{tol}} \quad \text{or} \quad \frac{|x_{k+1} - x_k|}{|x_{k+1}|} < \epsilon_{\text{rel}} $$
3. Numerical Mechanics & Implementation Logic
- State Initialization: Define initial guess $x_0$, tolerance $\epsilon$, and loop counter $k = 0$.
- Iterative Advancement: Compute $x_{k+1} = g(x_k)$ and calculate absolute/relative discrepancy.
- Break Condition: Exit when tolerance is reached or iteration limit $k \ge k_{\max}$ is exceeded.
4. Relevant Computational Practice Exercises
Exercise 1
Implement the Babylonian method to find $\sqrt{S}$ using $x_{k+1} = \frac{1}{2}(x_k + S/x_k)$ until $|x_{k+1} - x_k| < 10^{-10}$.
Exercise 2
Simulate radioactive decay $N(t + \Delta t) = N(t) - \lambda N(t) \Delta t$, continuing until $N(t) \le 0.01 N_0$.
Exercise 3
Compute projectile flight time under quadratic air drag, stopping when height $y(t) \le 0$.
- William H. Press et al., "Numerical Recipes", Cambridge University Press (2007).