Finite-precision one-stage Rosenbrock step with residual status

Input: a dimension $d\ge 1$, an ODE field $f:D\to {\mathbb{R}}^d$ on the open domain $D\subseteq \mathbb{R}\times {\mathbb{R}}^d$, $(t_n,y_n)\in D$, $h>0$, $\gamma >0$, a finite real Jacobian matrix ${\tilde{J}}_n\in {\mathbb{R}}^{d\times d}$, a residual tolerance $\epsilon \ge 0$, a unit roundoff $u>0$ with $du<1$, the selected stored finite set $\mathbb{F}$ of Operationwise real and complex floating point arithmetic with rounded square roots and fused multiply-add, a terminating field-evaluation routine that returns either the exact value $f(t,y)$ or evaluation-failure for each requested $(t,y)\in D$, a terminating partial exact-storage check store, and a terminating certificate evaluator evaluator meeting the input contract of Reusable Householder QR solve for a shifted matrix with residual status. For an exact real scalar, vector, or matrix $v$, store(v) returns $v$ if every entry belongs to $\mathbb{F}$ and returns unavailable otherwise; it does not round. Apply it before each factorization call.

1. Form $M=I_d-\gamma h{\tilde{J}}_n$ by exact outer arithmetic and set $M_s:=\mathrm{store}(M)$. If storage returns unavailable, return $(y_n,\mathtt{\text{representation-unavailable}})$. Call factor(M_s,u,evaluator) of Reusable Householder QR solve for a shifted matrix with residual status. If it returns a factorization failure status, return $(y_n,\text{that status})$; otherwise retain the returned factor object.
2. Request the field evaluation at $(t_n,y_n)$. If it returns evaluation-failure, return $(y_n,\mathtt{\text{field-evaluation-failure}})$. Otherwise set $b=f(t_n,y_n)$ to the returned exact value and set $b_s:=\mathrm{store}(b)$. If storage returns unavailable, return $(y_n,\mathtt{\text{representation-unavailable}})$. Call the retained factor object's solve(b_s,\varepsilon). If the solve status is not success, return $(y_n,\text{that status})$ and retain its residual bound for diagnosis.
3. If the solve returns $k_n$ and a certified residual upper bound ${\rho }_n\le \epsilon$, return $(y_n+h{k}_n,\mathtt{\text{success}},{\rho }_n)$.

The name Rosenbrock–Euler applies when $\gamma =1$. The exact-arithmetic stage equation is $M{k}_n=f(t_n,y_n)$, exactly the map of The Rosenbrock–Euler step map with an exact or approximate Jacobian. Its residual $M{k}_n-f(t_n,y_n)$ is the update residual of the linear stage equation in the sense of Implicit residuals and their update and derivative units. For this mixed-arithmetic specification, evaluations and outer affine combinations are exact real operations and every matrix/RHS passed to factorization or solve must pass store; only the factorization, triangular solve, and residual dot products use rounded operations, while residual-bound formation is delegated to the terminating certificate evaluator. A full hardware-rounding theorem for outer operations needs additional certified evaluation and update-error terms. This implementation has one shifted-matrix factorization and one stage solve. It does not turn a small residual into a forward-error claim without an inverse bound.