    \begin{algorithmic}[1]
        \State Input:\begin{itemize}
            \item Algebraic system of difference equations named $\Sigma'$
            \item Time measured data allowing prolongation of the system.
            \item For $\bar{\mu}=\mu_1,\ldots,\mu_n$ the finite set of parameters, the data $R_{\mu_i}$ of permissible intervals for each parameter value.
        \end{itemize}
        \State Output: Parameter values of $\Sigma'$.
    \Procedure{Detect solvability}{} 
        \State Define $\Sigma prolong$ as an indefinite time prolongation of $\Sigma'$.
        \State Redefine $\Sigma':=[]$, an `empty system' of no equations.
        \State Denote $e_1,\ldots$ the equations of $\Sigma prolong$.
        \State Define $J(t)=\left[\dfrac{\partial e_i}{\partial\mu_j}\right]$ where $e_i\in \Sigma',i=1,\ldots,r$, or 0 if $\Sigma'$ is empty.
        \State $i:=1$
        \For{$rank(J(t)):=s<n$ i.e. is not of full rank, }
            \Procedure{Move equation}{}
            \State $\Sigma':=\Sigma'\cup\{e_i\}$
            \State $\Sigma prolong:=\Sigma prolong\backslash\{e_i\}$
            \State Compute $rank(J(t))$ (note $\Sigma'$ has updated)
            \If{$rank(J(t))<s+1$} $\Sigma':=\Sigma'\backslash\{e_i\}$
            \EndIf
            \State $i=i+1$
            \If{$rank(J(t)<n$} repeat procedure Move Equation.
            \EndIf
            \EndProcedure
        \EndFor
    \Procedure{Blackbox solver and filter}{} Note Detect Solvability runs until $J(t)$  \phantom{---------}has full rank and outputs a polynomial system $\Sigma'$ that has solutions. Run any \phantom{---------}algebraic solver.
        \State Filter solutions by intersecting solution set with $R_{\mu_i}$.
        \EndProcedure
        \EndProcedure
    \end{algorithmic}

\begin{algorithmic}
[1]
\State Input:\begin{itemize}
\item Algebraic system of difference equations named $\Sigma'$
\item Time measured data allowing prolongation of the system.
\item For $\bar{\mu}=\mu_1,\ldots,\mu_n$ the finite set of parameters, the data $R_{\mu_i}$ of permissible intervals for each parameter value.
\end{itemize}
\State Output: Parameter values of $\Sigma'$.
\Procedure{Detect solvability}{}
\State Define $\Sigma prolong$ as an indefinite time prolongation of $\Sigma'$.
\State Redefine $\Sigma':=[]$, an `empty system' of no equations.
\State Denote $e_1,\ldots$ the equations of $\Sigma prolong$.
\State Define $J(t)=\left[\dfrac{\partial e_i}{\partial\mu_j}\right]$ where $e_i\in \Sigma',i=1,\ldots,r$, or 0 if $\Sigma'$ is empty.
\State $i:=1$
\For{$rank(J(t)):=s<n$ i.e. is not of full rank, }
\Procedure{Move equation}{}
\State $\Sigma':=\Sigma'\cup\{e_i\}$
\State $\Sigma prolong:=\Sigma prolong\backslash\{e_i\}$
\State Compute $rank(J(t))$ (note $\Sigma'$ has updated)
\If{$rank(J(t))<s+1$} $\Sigma':=\Sigma'\backslash\{e_i\}$
\EndIf
\State $i=i+1$
\If{$rank(J(t)<n$} repeat procedure Move Equation.
\EndIf
\EndProcedure
\EndFor
\Procedure{Blackbox solver and filter}{} Note Detect Solvability runs until $J(t)$ \phantom{---------}has full rank and outputs a polynomial system $\Sigma'$ that has solutions. Run any \phantom{---------}algebraic solver.
\State Filter solutions by intersecting solution set with $R_{\mu_i}$.
\EndProcedure
\EndProcedure
\end{algorithmic}
{
    "title": "symbolic-numeric algorithm for parameter estimation in discrete-time   models with $\\exp$",
    "id": "2401.16220",
    "abstract": "determining unknown parameter values in dynamic models is crucial for accurate analysis of the dynamics across the different scientific disciplines. discrete-time dynamic models are widely used to model biological processes, but it is often difficult to determine these parameters. in this paper, we propose a robust symbolic-numeric approach for parameter estimation in discrete-time models that involve non-algebraic functions such as exp. we illustrate the performance (precision) of our approach by applying our approach to the flour beetle (lpa) model, an archetypal discrete-time model in biology. unlike optimization-based methods, our algorithm guarantees to find all solutions of the parameter values given time-series data for the measured variables.",
    "categories": "q-bio.qm cs.sc cs.sy eess.sy math.ac math.ds",
    "doi": "",
    "created": "2024-01-29",
    "updated": "",
    "authors": [
        "yosef berman",
        "joshua forrest",
        "matthew grote",
        "alexey ovchinnikov",
        "sonia rueda"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.16220"
}
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