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Simpson's Rule

Simpson's Rule

A numerical integration method that approximates definite integrals by fitting parabolic segments to function values at equally-spaced intervals.

Simpson's Rule

Simpson's Rule is a sophisticated numerical integration technique that significantly improves upon simpler methods like the trapezoidal rule by approximating curves with quadratic functions rather than straight lines.

Mathematical Foundation

The rule is derived from fitting parabola to sets of three points. The formula for Simpson's Rule over an interval [a,b] is:

∫[a,b] f(x)dx ≈ (h/3)[f(a) + 4f((a+b)/2) + f(b)]

where h = (b-a)/2 represents the sub-interval width.

Composite Simpson's Rule

For better accuracy over wider intervals, the Composite Simpson's Rule divides the integration region into an even number of equal subintervals:

∫[a,b] f(x)dx ≈ (h/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + 2f(x₄) + ... + 4f(xₙ₋₁) + f(xₙ)]

Error Analysis

The error term for Simpson's Rule is proportional to h⁴f⁽⁴⁾(ξ), where:

  • h is the step size
  • f⁽⁴⁾ represents the fourth derivative
  • ξ is some point in the interval

This makes it significantly more accurate than the trapezoidal rule (error ∝ h²) for smooth functions.

Applications

Simpson's Rule finds extensive use in:

Implementation Considerations

When implementing Simpson's Rule:

  1. The number of subintervals must be even
  2. Function evaluations should be stored to avoid redundant calculations
  3. adaptive quadrature versions can automatically adjust step size

Historical Context

Named after Thomas Simpson, an 18th-century English mathematician, though similar methods were known earlier to Johannes Kepler. The rule represents a key development in the history of numerical methods.

Limitations

Despite its advantages, Simpson's Rule has some constraints:

  • Requires evenly spaced points
  • May struggle with discontinuous functions
  • Not optimal for highly oscillatory functions
  • Needs smooth functions for best accuracy

Related Methods