Analysis and calculus · Maths EE idea · Accessible
Trapezium rule vs Simpson's rule
A research question to start from
How does the error of the trapezium rule and Simpson's rule depend on the number of strips, and why does Simpson's rule converge so much faster?
A starting point, not your question: change the case, the comparison or the limit until it is yours. The research-question builder helps you check it.
Why it works as a maths EE
Familiar methods, with error orders you can predict, prove for simple cases and check numerically.
Mathematics you would need
- Numerical integration
- Taylor expansions
- Orders of error
- Log–log graphs
Much of this goes beyond the DP course. That is expected in a maths EE, but you must understand and explain everything you use.
One possible line of attack
- Derive both rules.
- Measure error against strip number for functions with known integrals.
- Explain the error orders with Taylor series.
- Find functions where Simpson's rule loses its advantage and explain why.
Scope and difficulty
Accessible. Accessible; the counterexamples give depth.
Pitfalls
- Only numerical tables.
- Using functions whose integral you cannot check.
Where to start reading
Search a library catalogue or a university's open lecture notes for: trapezium rule error term derivation; Simpson's rule error fourth order. Prefer textbooks, lecture notes and journal articles to a single website, and cite everything you use (how to reference a maths EE).