Results and discussion

This page presents and interprets the comparison plots for the three integration methods: Trapezoidal rule, Simpson’s rule, and Romberg integration. Figures show:

1. Absolute differences between C++ and Python results.
2. Relative errors in the C++ implementation versus analytic solution.
3. Relative errors in the Python implementation versus analytic solution.

All plots use a log–log scale and vary the number of sampling points \(N\) from \(2^3\) to \(2^{23}\).

Relative errors: C++ vs Analytic Solution

The trapezoidal rule decays as \(O(h^2)\), where \(h\) is the step size. In the log-log plot, this appears as a straight line with slope \(-2\).

Simpson’s rule decays as \(O(h^4)\), appearing as a straight line with slope \(-4\) until it reaches machine precision around \(N \approx 2^{12}\).

Romberg integration instead achieves machine precision for essentially all \(N\) values, showing rapid convergence to the analytic solution.

Relative errors: Python vs Analytic Solution

The Python implementations mirror the C++ convergence rates. Slightly larger absolute errors at small \(N\) for Romberg's Python implementation reflect differences in floating‑point routines and array handling in NumPy.

Absolute differences: C++ vs Python

Romberg differences plateau at around \(10^{-15}\), indicating both implementations reach machine precision and agree within rounding error. Simpson and Trapz differences reach \(10^{-13}\) for large \(N\). Small deviations are due to differences in loop ordering and floating‑point summation between C++ and NumPy.

The two languages produce numerically consistent results at the level of machine precision.