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📗Numerical Integration
Numerical integration is primarily based on the relation:
∫x₀xₙ f(x) dx = ∫x₀xₙ pₙ(x) dx + ∫x₀xₙ Eₙ(x) dx
Here, pₙ(x) is an interpolation polynomial and Eₙ(x) is the associated error. By varying the degree n, one obtains the Newton-Cotes formulas. In principle, higher n yields higher precision for the integral. In practice, however, values of n greater than 4 are rarely used.
Additionally, Richardson extrapolation, applied judiciously to a Newton-Cotes formula, leads to the Romberg method, one of the most accurate numerical integration techniques. Finally, Gauss-Legendre quadrature is frequently employed in more advanced numerical methods, such as the finite element method.
