Weddle

Weddle's Rule is a numerical integration technique used to approximate the definite integral of a function. It is a higher-order Newton-Cotes formula, derived by interpolating the integrand with a polynomial of degree six. Consequently, it requires seven equally spaced points within the interval of integration.

The general form of Weddle's Rule for approximating the integral of a function f(x) from a to b, with n subintervals (where n is a multiple of 6), is given by:

∫_a^b f(x) dx ≈ (3h/10) [ f(x₀) + 5f(x₁) + f(x₂) + 6f(x₃) + f(x₄) + 5f(x₅) + f(x₆) ] + (3h/10) [ f(x₆) + 5f(x₇) + f(x₈) + 6f(x₉) + f(x₁₀) + 5f(x₁₁) + f(x₁₂) ] + ...

where:

• h = (b - a) / n (the width of each subinterval)
• x_i = a + i*h (the points at which the function is evaluated)

Weddle's Rule is generally more accurate than Simpson's Rule or the Trapezoidal Rule for smooth functions, due to its higher order of precision. However, it also requires more function evaluations. It's particularly effective when the function is well-approximated by a sixth-degree polynomial.

The rule's derivation stems from integrating the sixth-degree Lagrange interpolating polynomial through the given points. The coefficients in the formula (3/10, 15/10, 3/10, 18/10, 3/10, 15/10, 3/10) are obtained by performing this integration. Note that the rule is applied repeatedly over consecutive intervals of width 6h.