Theoretical Principles & Computational Physics Guide
Gaussian Quadrature method: Numerical Quadrature & Definite Integrals
Dr. Alorika Chatterjee & Dr. Anirban Shaw
Science & Physics Learners
Python 3 / NumPy / SciPy
1. Core Principles & Theoretical Framework
Continuous Quadrature: Evaluates definite integrals $\int_a^b f(x)dx$ when analytical anti-derivatives do not exist in closed form.
Discretized Summation: Approximates area under curves using interpolating polynomials (Trapezoidal $\mathcal{O}(h^2)$, Simpson $\mathcal{O}(h^4)$).
Physical Observables: Essential for finding total work, probability normalization $\int |\psi|^2 dx = 1$, and center of mass coordinates.
2. Mathematical Formulation & Algorithmic Pipeline
Composite numerical integration quadrature formulas:
$$ \text{Trapezoidal: } I \approx \frac{h}{2}\left[f_0 + 2\sum_{i=1}^{N-1} f_i + f_N\right], \quad \text{Simpson: } I \approx \frac{h}{3}\left[f_0 + 4\sum_{\text{odd}} f_i + 2\sum_{\text{even}} f_i + f_N\right] $$
3. Numerical Mechanics & Implementation Logic
- Interval Subdivision: Divide $[a, b]$ into $N$ equal slices of width $h = (b - a)/N$.
- Weighted Summation: Evaluate integrand at nodal points and apply quadrature weight coefficients.
- Convergence Verification: Double $N$ and verify that error scales consistent with the method's asymptotic order.
4. Relevant Computational Practice Exercises
Exercise 1
Evaluate $\int_0^{\pi/2} \sin(x)dx$ using Gaussian Quadrature method and calculate absolute error against the exact value 1.0.
Exercise 2
Perform an error study by doubling strip count $N$ from $10$ to $1000$ and plot error vs $h$ on a log-log scale.
Exercise 3
Apply the integration method to calculate the period of a large-amplitude nonlinear pendulum $T = 4\sqrt{L/g}\int_0^{\pi/2}\frac{d\phi}{\sqrt{1 - k^2\sin^2\phi}}$.
- Mark Newman, "Computational Physics", University of Michigan (2013).
- William H. Press et al., "Numerical Recipes", Cambridge University Press (2007).