Certified adaptive Simpson from derivative enclosures

Input finite dyadic endpoints $a<b$, a positive dyadic tolerance $\tau$, integers $D\ge 0$ and $P\ge 1$, and a function $f\in C^4([a,b])$. Also supply two validated enclosure routines:

• $V(x)$ returns a finite real interval containing the exact value $f(x)$;
• $Y([u,v])$ returns a finite real interval containing every $f^{(4)}(x)$ for $x\in [u,v]$.

If either routine does not meet its enclosure contract, this algorithm makes no certification claim; soundness is a mathematical caller precondition and cannot be detected from samples. An explicit oracle failure or malformed or non-finite interval result returns failure. The intervals and operations are those of Real intervals, complex rectangles and ball enclosures and Outward-rounded interval, rectangle and ball operations. The function $\mathrm{ru}$ is upward directed rounding from Dyadic floating-point format and directed rounding. A midpoint that is not exactly representable or a non-finite interval operation returns failure with no enclosure; malformed or non-finite oracle results are covered by the explicit failure rule above.

Define $A([u,v],d,p,t)$ recursively. Form the exact midpoint $m=(u+v)\mathrm{/}2$; if it is not representable in the format, return failure. Query $V(u),V(m),V(v)$ and let $Y([u,v])=[y_{-},y_{+}]$. Put $M=\max (\mathrm{\mid }{y}_{-}\mathrm{\mid },\mathrm{\mid }{y}_{+}\mathrm{\mid })$ and let $L=v-u$ be the exact difference of the dyadic endpoints. Form an outward-rounded interval $W=[v,v]-[u,u]$ by interval subtraction, and compute $S_P:=(W\mathrm{/}[6,6])(V(u)+4V(m)+V(v))$ using outward interval operations. Compute $B_P=\mathrm{ru}(M{L}^5\mathrm{/}2880)$ by upward rounding the exact real expression; if it overflows, return failure. Set $C_P=S_P+[-B_P,B_P]$ by outward interval addition. If any interval operation has a non-finite status, return failure. Treat the endpoints of $C_P$ as exact dyadic rationals and report their exact rational midpoint $q_P$ and half-width $r_P$; compare $r_P$ with $t$ exactly.

1. If $r_P\le t$, return (certified, enclosure $C_P$, estimate $q_P$, error bound $r_P$).
2. If $r_P>t$ and either $d=0$ or $p<3$, return (inconclusive, enclosure $C_P$, estimate $q_P$, error bound $r_P$).
3. Otherwise split at $m$, allocate $p_L=\lfloor (p-1)\mathrm{/}2\rfloor$ and $p_R=\lceil (p-1)\mathrm{/}2\rceil$, and recurse with depth $d-1$ and tolerance $t\mathrm{/}2$ on each half. A failed child or non-finite interval sum returns failure. Otherwise outward-add the child enclosures to obtain $C$, with exact midpoint $q$ and half-width $r$; return certified if $r\le t$, and inconclusive otherwise.

The public call is $A([a,b],D,P,\tau )$. certified asserts an error at most $r\le \tau$; inconclusive still returns a valid enclosure and bound but does not assert the requested tolerance was met. The derivative enclosure routine is an explicit mathematical precondition, not something inferred from sampled values or the Simpson indicator. Outward rounding of all interval operations, and the separate Peano remainder bound, are essential.