Boole’s Rule

← Numerical Methods

Source inspiration: (Mathew 2000-2019).

Animations

Each animation below shows the Boole’s rule diagram building up panel by panel over \([0, 2]\) for \(y = e^{-x}\sin(8x^{2/3})+1\). Each frame adds one new Boole panel (spanning 4 subintervals = 5 nodes); the running sum is annotated. Boole’s rule fits a 4th-degree polynomial through 5 nodes — achieving \(O(h^6)\) accuracy.

Julia source scripts that generated these animations are linked under each case.

Case 1 — Boole’s Rule, \(y = e^{-x}\sin(8x^{2/3})+1\), \(n=8\) subintervals

Behavior: Each panel uses 5 nodes and the formula \(\frac{2h}{45}(7f_0 + 32f_1 + 12f_2 + 32f_3 + 7f_4)\). Boole’s rule is exact for polynomials up to degree 5, giving \(O(h^6)\) accuracy — the highest of the Newton–Cotes family shown here.

Julia source

Boole’s Rule animation: 2 quartic panels build over [0,2]; each panel spans 4 sub-intervals fitted by a degree-4 polynomial; running sum annotated each frame

Derivation Notes (Planned)

Short derivations will be added to explain the core equations and assumptions.

Worked Example (Planned)

A compact numerical example with intermediate steps will be included.

Implementation Notes (Planned)

Implementation details, numerical stability notes, and practical pitfalls will be added.

Legacy Animation Inventory (Stub)

  • Legacy module page: Boole’s Rule (ok)
  • Animation links found in module Animations paragraph: 4
  • Unique animation portals: 1

Animation Portals

  1. Boole’s Rule (ok)
  • Animation item links found: 1

Animation Items

  1. Boole’s Rule (ok)
  • Main animated GIF count: 1
  • http://localhost:8000/a2001/Animations/Quadrature/Boole/Booleaa.gif

References

Mathew, John H. 2000-2019. Numerical Analysis - Numerical Methods Modules. https://web.archive.org/web/20190808102217/http://mathfaculty.fullerton.edu/mathews/n2003/NumericalUndergradMod.html.