Uniform interval linear residual certificate with inconclusive status

Input $n\ge 1$, nonempty families $\mathcal{A}\subseteq {\mathbb{C}}^{n\times n}$ and $\mathcal{b}\subseteq {\mathbb{C}}^n$, each enclosed entrywise by finite complex rectangles, together with stored point proposals $R$ and $\widehat{x}$. The represented family need only be a subset of the supplied product of rectangles; correlations between entries may be discarded. All stored point data are exact dyadics.

1. Use outward rectangle operations to evaluate the expression DAGs

$$E(A)=I-RA,c(A,b)=R(b-A\widehat{x})$$

over the full input rectangles. If an operation has no finite enclosure or returns a domain or range status, return inconclusive.

2. For each complex rectangle entry $X+iY$, outward-enclose its modulus by computing $\sqrt{X^2+Y^2}$ and taking the upper endpoint. Combine those bounds by outward sums of squares and square roots to obtain $q\ge {\sup }_{A\in \mathcal{A}}\mathrm{\parallel }E(A){\mathrm{\parallel }}_F$ and $r\ge {\sup }_{A\in \mathcal{A},b\in \mathcal{b}}\mathrm{\parallel }c(A,b){\mathrm{\parallel }}_2$. If any modulus, sum, square root or final division has no finite outward result, return inconclusive. The Frobenius upper bound also bounds the operator 2-norm. If $q\ge 1$, return inconclusive with $q,r$. Otherwise return certified with the uniform residual data $(R,\widehat{x},q,r)$ and radius $r\mathrm{/}(1-q)$, rounded outward if represented as a ball.

The interval DAG encloses every value for every admitted pair, including dependencies between repeated inputs because natural interval evaluation may enlarge the range. inconclusive makes no assertion that a member is singular.