Finite-budget adaptive Runge–Kutta controller

Inputs are an IVP in the sense of First-order systems, initial value problems, and solutions on intervals, $y^{\mathrm{\prime }}=f(t,y)$ on an open domain $D\subseteq \mathbb{R}\times {\mathbb{R}}^d$ with initial data $(t_0,y_0)\in D$ and requested interval $[t_0,T]$, where $t_0\le T$ and $d\ge 1$; an initial proposal $0<h_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}\le h_{\max }$; an embedded pair with low order $p\ge 1$ as in Embedded Runge–Kutta pair and difference estimator; tolerances $\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{l}>0$ and $\mathrm{r}\mathrm{t}\mathrm{o}\mathrm{l}\ge 0$; safety $0<\sigma <1$; scale bounds $0<{\gamma }_{\min }\le 1\le {\gamma }_{\max }$; step bounds $0<h_{\min }\le h_{\max }$; and a positive integer attempt budget $N$.

Output. Return a record consisting of one of success, failure, minimum-step-rejected, or attempt-budget-exhausted, together with the accepted mesh $\mathcal{A}=((t_0,y_0),\dots ,(t_m,y_m))$. The mesh always contains the initial pair and every accepted endpoint; on success, $t_m=T$ and $y_m$ is the returned endpoint approximation. On any other status, $\mathcal{A}$ is the accepted prefix and $y_m$ is the last accepted approximation.

Every maximum and minimum below is over a displayed nonempty finite list of real numbers; existence follows from Every nonempty finite set of reals has a maximum and a minimum.

Procedure. Initialize $t:=t_0$, $y:=y_0$, $\mathcal{A}:=((t_0,y_0))$, and proposal $h_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}:=h_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$. If $t_0=T$, return success with $\mathcal{A}$. For each attempt index $k=1,\dots ,N$, perform exactly the following steps:

1. Set $h:=\min (h_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}},h_{\max },T-t)$. The last term permits a final clipped step shorter than $h_{\min }$. If $h\le 0$ before reaching $T$, return failure with $\mathcal{A}$.
2. Evaluate the shared stages in tableau order, checking before each evaluation that its time-state input lies in $D$. If a stage input is outside $D$, its evaluation is undefined, or its value is nonfinite, return failure with $\mathcal{A}$. Compute both pair outputs; if either output or any arithmetic needed to compute them is undefined, fails, or is nonfinite in the rounded realization, return failure with $\mathcal{A}$.
3. Define $s_i:=\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{l}+\mathrm{r}\mathrm{t}\mathrm{o}\mathrm{l}\max (\mathrm{\mid }(y{)}_i\mathrm{\mid },\mathrm{\mid }(Y{)}_i\mathrm{\mid })$ and $Q:={\max }_i\mathrm{\mid }(\widehat{Y}-Y{)}_i\mathrm{\mid }\mathrm{/}{s}_i$, where $Y$ is the high-order value and $\widehat{Y}$ the low-order value. If any scale or $Q$ operation fails or is nonfinite in the rounded realization, return failure with $\mathcal{A}$. If $Q\le 1$, first check $(t+h,Y)\in D$; if not, return failure with $\mathcal{A}$. Otherwise accept: set $t:=t+h$, $y:=Y$, and append $(t,y)$ to $\mathcal{A}$. If $Q>1$, reject and retain $t,y,\mathcal{A}$.
4. If the attempt was accepted and $t=T$, return success with $\mathcal{A}$. Otherwise, if it was rejected and $h\le h_{\min }$, return minimum-step-rejected with $\mathcal{A}$. These terminal checks precede the budget check, so a final accepted step that reaches $T$ succeeds and a rejection at the minimum step reports that safeguard.
5. If $k=N$, return attempt-budget-exhausted with $\mathcal{A}$. Otherwise compute $g=\{\begin{array}{ll}{\gamma }_{\max },& Q=0,\\ \min ({\gamma }_{\max },\max ({\gamma }_{\min },\sigma \mathrm{/}\sqrt[p+1]{Q})),& Q>0.\end{array}$ Set $h_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}:=\min (h_{\max },\max (h_{\min },gh))$ and continue to attempt $k+1$. A failed or nonfinite rounded computation of $g$ or $h_{\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{p}}$ returns failure with $\mathcal{A}$.

For $Q>0$, $\sqrt[p+1]{Q}$ is the unique positive $(p+1)$st root supplied by Existence and uniqueness of $n$-th roots: a unique $a^{1\mathrm{/}n}\ge 0$ with $(a^{1\mathrm{/}n}{)}^n=a$. The exponent $1\mathrm{/}(p+1)$ is a leading-error model for step selection; neither acceptance nor this controller is a global error certificate or a stiffness test.

The mathematical specification evaluates each displayed operation, including maximum, minimum, absolute value, and positive-root evaluation, exactly. A rounded implementation instead applies its fixed finite-format rounding rule to every stage, output, scale, and controller-factor arithmetic operation and uses the resulting stored high and low values in the displayed difference and acceptance test; nonfinite or failed operations return failure. The finite termination theorem applies to either realization when each attempt has finite work. The rounded realization has no a posteriori guarantee: a tolerance below its resolution can be accepted because both stored values coincide.