numerical-integration

# Numerical Integration ## When to Use Use this skill when working on numerical-integration problems in numerical methods. ## Decision Tree 1. **Identify Integral Type** - Definite integral over finite interval? - Improper integral (infinite bounds or singularities)? - Multiple dimensions? 2. **Select Quadrature Method** - Smooth function, finite interval: Gaussian quadrature - Oscillatory integrand: specialized methods (Filon, Levin) - Singularity at endpoint: adaptive methods - `scipy.integrate.quad(f, a, b)` for general 1D 3. **Adaptive Integration** - Let algorithm subdivide where needed - Specify error tolerances (rtol, atol) - `scipy.integrate.quad(f, a, b, epsabs=1e-8, epsrel=1e-8)` 4. **Multiple Dimensions** - `scipy.integrate.dblquad` for 2D - `scipy.integrate.tplquad` for 3D - Monte Carlo for higher dimensions 5. **Verify Accuracy** - Compare with known analytic solutions - Check convergence by refining tolerance - `sympy_compute.py integrate "f(x)" --var x --from a --to b` ## Tool Commands ### Scipy_Quad ```bash uv run python -c "from scipy.integrate import quad; import numpy as np; result, err = quad(lambda x: np.sin(x), 0, np.pi); print('Integral:', result, 'Error:', err)" ``` ### Scipy_Dblquad ```bash uv run python -c "from scipy.integrate import dblquad; result, err = dblquad(lambda y, x: x*y, 0, 1, 0, 1); print('Integral:', result)" ``` ### Sympy_Integrate ```bash uv run python -m runtime.harness sc...