\begin{algorithm}[H]
\caption{Dynamic Taylor}
\label{alg:dynamictaylor}
\begin{algorithmic}
\State Input:\begin{enumerate}
    \item[1] Data input for Algorithm~\ref{alg:expansionrange}, e.g. system $\Sigma$.
    \item[2] Output of Algorithm~\ref{alg:expansionrange}.
    \item[3] Set number of terms, i.e., the highest degree of Taylor polynomial, $N$.
    \item[4] Expansion point $\alpha_j\in R_{G_j}$ for each $j$.
    \item[5] Time measured data $\bar{y}(t_0),\cdots,\bar{y}(t_s)$.
\end{enumerate}
\State Output: System of polynomial equations $\Sigma'$ with parameters approximately equal to those of $\Sigma$.
\Procedure{$\Sigma$ to power series $\Sigma^*$}{} Suppose there are $K$ transcendental functions in $\Sigma$.
    \For{$j=1\to K$}
    \State $\tau_j:=\tilde{G}_j$
    \State $G_j\to T_{G_j}:=\sum_{k=0}^\infty G_j^{(k)}(\alpha_j)(\tau_j-\alpha_j)^k$, where differentiation is with respect to $\tau_j$.
    \State For every equation $e_i=G_{i_1}+\cdots+G_{i_r}+A_i$ in $\Sigma$, $e_i^*=T_{G_{i_1}}+\cdots+T_{G_{i_r}}+A_i$ is in $\Sigma^*$, \phantom{---------}where $A_i$ is algebraic.
    \EndFor
\EndProcedure
\Procedure{$\Sigma^*$ to algebraic $\Sigma'$}{}
    \For{$j=1\to K$}
    \State $\hat{T}_{G_j}:=\sum_{k=0}^N G_j^{(k)}(\alpha_j)(\tau-\alpha_j)^k$
    \State $\tau_j\mapsto \tilde{G}_j$
    \State For every equation $e_i^*=T_{G_{i_1}}+\cdots+T_{G_{i_r}}+A_i$ in $\Sigma^*$, $e_i'=\hat{T}_{G_{i_1}}(\tilde{G}_{i_1})+\cdots+\hat{T}_{G_{i_r}}(\tilde{G}_{i_r})+A_i$ \phantom{---------}is in $\Sigma'$, where $A_i$ is algebraic.
    \EndFor 
    \State Substitute $\bar{y}(t_0),\cdots,\bar{y}(t_s)$ throughout $\Sigma'$ as they appear, if known.
\EndProcedure
\end{algorithmic}
\end{algorithm} 

\begin{algorithm}
[H]
\caption{Dynamic Taylor}
\begin{algorithmic}
\State Input:\begin{enumerate}
\item[1] Data input for Algorithm~\ref{alg:expansionrange}, e.g. system $\Sigma$.
\item[2] Output of Algorithm~\ref{alg:expansionrange}.
\item[3] Set number of terms, i.e., the highest degree of Taylor polynomial, $N$.
\item[4] Expansion point $\alpha_j\in R_{G_j}$ for each $j$.
\item[5] Time measured data $\bar{y}(t_0),\cdots,\bar{y}(t_s)$.
\end{enumerate}
\State Output: System of polynomial equations $\Sigma'$ with parameters approximately equal to those of $\Sigma$.
\Procedure{$\Sigma$ to power series $\Sigma^*$}{} Suppose there are $K$ transcendental functions in $\Sigma$.
\For{$j=1\to K$}
\State $\tau_j:=\tilde{G}_j$
\State $G_j\to T_{G_j}:=\sum_{k=0}^\infty G_j^{(k)}(\alpha_j)(\tau_j-\alpha_j)^k$, where differentiation is with respect to $\tau_j$.
\State For every equation $e_i=G_{i_1}+\cdots+G_{i_r}+A_i$ in $\Sigma$, $e_i^*=T_{G_{i_1}}+\cdots+T_{G_{i_r}}+A_i$ is in $\Sigma^*$, \phantom{---------}where $A_i$ is algebraic.
\EndFor
\EndProcedure
\Procedure{$\Sigma^*$ to algebraic $\Sigma'$}{}
\For{$j=1\to K$}
\State $\hat{T}_{G_j}:=\sum_{k=0}^N G_j^{(k)}(\alpha_j)(\tau-\alpha_j)^k$
\State $\tau_j\mapsto \tilde{G}_j$
\State For every equation $e_i^*=T_{G_{i_1}}+\cdots+T_{G_{i_r}}+A_i$ in $\Sigma^*$, $e_i'=\hat{T}_{G_{i_1}}(\tilde{G}_{i_1})+\cdots+\hat{T}_{G_{i_r}}(\tilde{G}_{i_r})+A_i$ \phantom{---------}is in $\Sigma'$, where $A_i$ is algebraic.
\EndFor
\State Substitute $\bar{y}(t_0),\cdots,\bar{y}(t_s)$ throughout $\Sigma'$ as they appear, if known.
\EndProcedure
\end{algorithmic}
\end{algorithm}
{
    "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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