Simpson's 1/3 and 3/8 Rules: Composite and Adaptive Integration

All flashcards

Single-panel Simpson's 1/3 rule

Approximates an integral using endpoint and midpoint values: $\int_a^b f(x)\,dx\approx\frac{b-a}{6}\left[f(a)+4f\left(\frac{a+b}{2}\right)+f(b)\right]$.

Composite Simpson's 1/3 error

For a sufficiently smooth integrand and even $n$, the error is $E=-\frac{(b-a)h^4}{180}f^{(4)}(\xi)$ for some $\xi\in[a,b]$, where $h=(b-a)/n$.

Simpson's 3/8 rule

Uses four equally spaced samples to integrate their cubic interpolant: $\int_a^b f(x)\,dx\approx\frac{3h}{8}[f(a)+3f(a+h)+3f(a+2h)+f(b)]$, where $h=(b-a)/3$.

Adaptive Simpson integration

Compares a coarse Simpson estimate with estimates on smaller panels, refining regions where the difference exceeds an error tolerance.

Simpson integration on uneven grids

Unequal sample spacing generally requires adjusted weights from an irregular-grid formula; standard composite Simpson weights cannot be used unchanged.

Quiz questions

1. For a single-panel Simpson's 1/3 rule on $[a,b]$, what factor multiplies the weighted sum of the three function values?Answer: $(b-a)/6$
2. Why does Simpson's 1/3 rule integrate cubic polynomials exactly?Answer: The cubic contribution about the midpoint is odd and integrates to zero over the symmetric interval.
3. For a sufficiently smooth function, how does the composite Simpson's 1/3 truncation error scale with step size $h$?Answer: It scales as $h^4$.
4. Which polynomial does a single-panel Simpson's 3/8 rule integrate to approximate the integral?Answer: The cubic interpolant through four equally spaced samples.
5. If each sampled function value in composite Simpson's 1/3 rule has absolute round-off error at most $\varepsilon_r$, what bound does the notes give for the total round-off contribution?Answer: $(b-a)\varepsilon_r$