Embedded Simpson refinement indicator

Let $a<b$, let $n\ge 2$ be even, and let $f$ be real-valued at the nodes of the uniform grids with $n$ and $2n$ subintervals. The coarse Simpson value is $S_0=Q_S(f;a,b,n)$ and the refined value is $S_1=Q_S(f;a,b,2n)$, using Composite Simpson rule on an even uniform grid. The signed refinement indicator is $(S_1-S_0)\mathrm{/}15$ and its magnitude is ${\eta }_n(f;a,b):=\frac{\mathrm{\mid }{S}_1-S_0\mathrm{\mid }}{15}\mathrm{.}$

For the local two-panel case $n=2$, $S_1$ is also the sum of the two one-block Simpson values on the halves of $[a,b]$.

This definition alone is not a bound on the true error.

For data supplied only at the grid nodes, the calls to the Simpson procedure mean calls on the extension $\tilde{f}:[a,b]\to \mathbb{R}$ that agrees with those data at the nodes and is zero elsewhere. This extension is evaluable everywhere as required by the procedure. Its finite sums use only the prescribed nodes, so any other evaluable extension gives the same $S_0$, $S_1$ and indicator; no off-grid values or regularity are required.

The factor $1\mathrm{/}15$ has asymptotic Richardson justification under the hypotheses of Conditional asymptotic validity of the Simpson /15 indicator; it remains a heuristic indicator at any finite grid.