Theoretical Principles & Computational Physics Guide
Series Expansion: Taylor & Maclaurin Polynomials with Truncation Error
Dr. Alorika Chatterjee & Dr. Anirban Shaw
Science & Physics Learners
Python 3 / NumPy / SciPy
1. Core Principles & Theoretical Framework
Polynomial Approximation: Expands transcendental functions into computable polynomial series around a point of interest.
Convergence Radius: Computes successive terms iteratively until the incremental term falls below numerical precision.
Truncation Error: Balances order of expansion $N$ against floating-point round-off accumulation.
2. Mathematical Formulation & Algorithmic Pipeline
Taylor series expansion and error remainder:
$$ f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(x_0)}{n!}(x - x_0)^n = \sum_{n=0}^N t_n(x) + R_N(x), \quad t_n = t_{n-1} \cdot \frac{x}{n} $$
3. Numerical Mechanics & Implementation Logic
- Recurrence Relation: Compute each new term from the preceding term ($t_n = t_{n-1} \cdot \frac{x}{n}$) to avoid expensive factorials.
- Convergence Loop: Terminate when $|t_n| < \epsilon_{\text{tol}}$ or when maximum terms $N_{\max}$ are reached.
- Domain Reduction: Map arguments into primary fundamental intervals (e.g. $[0, 2\pi]$) for trigonometric stability.
4. Relevant Computational Practice Exercises
Exercise 1
Implement the Taylor series for $e^x$ using term-to-term recurrence and compare against <code>math.exp(x)</code> for $x = -15, 0.5, 10$.
Exercise 2
Compute $\sin(x)$ expansion and verify that truncation error scales as $\mathcal{O}(x^{2N+1}/(2N+1)!)$.
Exercise 3
Demonstrate numerical catastrophic cancellation when evaluating $e^{-x}$ for large positive $x$ via direct series versus $1 / e^x$.
- William H. Press et al., "Numerical Recipes: The Art of Scientific Computing", Cambridge University Press (2007).