Animation · Adaptive Simpson quadrature

Compare a coarse Simpson estimate with two half-interval estimates. The film follows $f(x)=e^{4x}$, splitting intervals and tolerance budgets, then adding accepted integral contributions.

Before watching¶

Where would you expect refinement to concentrate for $e^{4x}$? What information must each interval return so that its parent can assemble an integral?

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Mathematical connection¶

On the same interval, let $S$ use one Simpson pair and $S_2$ use two half-width pairs. If the leading smooth-function error term dominates, the fine estimate has about $1/16$ of the coarse estimate’s error. Therefore

The first quantity estimates the error of $S_2$; it is not a proven error bound for the corrected result. Compare it with the interval’s tolerance $\tau$. If refinement is needed, split the interval and give each child budget $\tau/2$. Accepted child contributions are added when returning to the parent.

A depth limit must report unfinished refinement as failure. Sample agreement can miss unsampled structure, even with a small estimated error.

Explain what you saw¶

Follow the quantities in each direction: tolerance budgets go down the tree, integral contributions return upward. The film’s covered-interval actual error uses a known exact integral for this example; a general adaptive integrator does not know that error.

Use the player controls to pause, replay, or view fullscreen. The animation has no narration; its labels and the explanation above carry the content.

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