Certified action bounds control one exponential-integrator step

For a nonzero action request, [F11] makes the scaled-Taylor output certificate a valid ${ϵ}_{\mathrm{r}\mathrm{e}\mathrm{d}}$; put $c(s)=e^{sH}\beta e_1$ in [F2]. By [F8] and ${\rho }_H\ge \mathrm{\parallel }H{\mathrm{\parallel }}_2$, $\mathrm{\parallel }c(s){\mathrm{\parallel }}_2\le \beta e^{s{\rho }_H}$; [F10] gives $\mathrm{\mid }{e}_j^{\ast }c(s)\mathrm{\mid }\le \mathrm{\parallel }c(s){\mathrm{\parallel }}_2$. Since ${\rho }_B\ge {\mu }_2(B)$, [F9] bounds the initial term in [F2] by $e^{{\rho }_B\tau }{\eta }_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$ and its integral by $\beta \tau e^{\tau \max ({\rho }_B,{\rho }_H)}(\gamma +\sqrt{{\sum }_i{\eta }_i^2}+{\eta }_{\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}})$. The reduced and final projection terms are at most ${\nu }_V{ϵ}_{\mathrm{r}\mathrm{e}\mathrm{d}}+{\eta }_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{j}}$ by [F7]. Thus [F3]'s ${ϵ}_{\mathrm{a}\mathrm{c}\mathrm{t}}$ bounds the full exponential-action error. For a zero input, the algorithm returns the exact zero action with bound zero.

For $0\le \theta \le 1$, put $s=(1-\theta )h$. Then $0\le s\le h$ and $\omega s\le \max (\omega ,0)h$: for $\omega \ge 0$ this follows from $s\le h$, and for $\omega <0$ it follows from $s\ge 0$. The semigroup bound in [A1] and scalar monotonicity [F9] therefore give $\mathrm{\parallel }{e}^{(1-\theta )hA}{\mathrm{\parallel }}_2\le M{e}^{\omega s}\le B_h$. For each unit vector $v\in {\mathbb{C}}^d$, [F5] and componentwise linearity in [F12] yield ${\phi }_1(hA)v={\int }_0^1{e}^{(1-\theta )hA}vd\theta$ and the analogous identity with weight $\theta$ for ${\phi }_2$. These curves are continuous by [F12]. Apply the norm inequality in [F12] and the matrix-action inequality in [F7]: the two norms are at most $B_h{\int }_0^{1}d\theta =B_h$ and $B_h{\int }_0^1\theta d\theta =B_h\mathrm{/}2$. Taking the operator-norm maximum over unit vectors in [F7], valid since $d\ge 1$ is an input condition, proves both asserted phi bounds.

For exponential Euler, the exact formula [F4] applies $e^{hA}$ to $y$ and $h{\phi }_1(hA)$ to $g(t,y)$. [F3] supplies action-error bounds for $e^{hA}y$ and the full augmented action whose top block is ${\phi }_1(hA){\tilde{q}}_0$ by [F1]; coordinate projection cannot increase its error. Replacing $g(t,y)$ by ${\tilde{q}}_0$ changes the phi action by at most $B_h{\eta }_0$ by step 1.2 and [A1]. Multiplication by $h$ and the certified combination error ${\alpha }_E$ give the exponential-Euler ${\delta }_n$.

For ETD2, the predictor's difference from the exact predictor $a_h(t,y)$ is at most ${ϵ}_0+h({ϵ}_1+B_h{\eta }_0)+{\alpha }_a={\eta }_a$ by the triangle inequality, using the exponential-Euler estimate from step 2.1. Therefore the exact stage forcing differs from the evaluated ${\tilde{q}}_a$ by at most $L{\eta }_a+{\eta }_a^g$, using [A1].

Using the phi bounds proved in step 1.2, in the final ETD2 update, the ${\phi }_1$ action at $g(t,y)$ contributes at most ${ϵ}_1+B_h{\eta }_0$; the ${\phi }_2$ action at $g(t,y)$ contributes at most ${ϵ}_{20}+(B_h\mathrm{/}2){\eta }_0$; and the stage ${\phi }_2$ action contributes at most ${ϵ}_{2a}+(B_h\mathrm{/}2)(L{\eta }_a+{\eta }_a^g)$. Each phi action certificate follows from [F1] and step 1.1. The exponential action contributes ${ϵ}_0$. Multiplying the three phi terms by $h$, adding the certified final arithmetic error ${\alpha }_2$, and applying the triangle inequality gives exactly the ETD2 ${\delta }_n$ in [F3].

Under the stated service contracts, each started backend/provider call terminates, including every call nested inside an Arnoldi or reduced-action request. Outside these services the procedure has a fixed finite sequence of branches: at most two action requests for method $E$ and four for method $2$, together with finitely many field, membership, enclosure, and vector-arithmetic requests. A missing certificate or insufficient pre-call allowance returns immediately, and every started service returns in finite time by its contract, so the entire algorithm terminates. For the counters, begin with $C=0$ and $W=0$. Before a service starts, its certified upper count $b$ satisfies $b\le C_{\max }-C$; its valid returned actual count $c$ obeys $0\le c\le b$, so the update gives $0\le C+c\le C_{\max }$. Induction over the finitely many completed top-level services proves the count bound, with their nested work already included in their returned records. At each such update, the algorithm also adds the actual service work record componentwise to $W$. The same finite induction makes each component exactly the sum of its accrued actual counters; an early return retains these accrued totals. Zero-input action shortcuts contribute zero action work. This proves termination and the advertised accounting without adding calls or presuming that a missing mathematical certificate succeeds.