archimedes.quadrature.simpson

archimedes.quadrature.simpson(

n: int,

a: float,

b: float,

) → QuadratureRule

Composite Simpson’s rule quadrature.

Simpson’s rule fits a parabola through the two ends and the midpoint of an interval: \(\int_a^b f(x) \, dx \approx \tfrac{h}{3}(f_0 + 4 f_1 + f_2)\)

The quadrature rule tiles n equal-width segments across the domain \([a, b]\), with each segment using the Simpson quadrature rule.

Parameters:

• n (int) – Number of integration segments

• a (float) – Bounds of the target interval

• b (float) – Bounds of the target interval

Returns:

rule – Composite Simpson’s rule with 3 * n nodes on \([a, b]\), exact for a cubic on each segment.

Return type:

QuadratureRule

Raises:

ValueError – If n < 1.

See also

gauss_lobatto

Equivalent single-segment quadrature rule.

composite_quad

General tiling for piecewise quadrature rules.

Notes

Simpson’s rule is equivalent to the 3-point Gauss-Lobatto quadrature rule, which is exact for cubics. The composite Simpson’s rule is exact for piecewise cubics.