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Apply some math formatting
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henrikt-ma committed Jun 2, 2020
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Expand Up @@ -939,11 +939,11 @@ \subsection{The Number of Equations Needed for Initialization}\doublelabel{the-n

\begin{nonnormative}
In general, for the case of a pure (first order) ordinary
differential equation (ODE) system with n state variables and m output
variables, we will have n+m unknowns in the simulation problem. The ODE
initialization problem has n additional unknowns corresponding to the
differential equation (ODE) system with $n$ state variables and $m$ output
variables, we will have $n+m$ unknowns in the simulation problem. The ODE
initialization problem has $n$ additional unknowns corresponding to the
derivative variables. At initialization of an ODE we will need to find
the values of 2n+m variables, in contrast to just n+m variables to be
the values of $2n+m$ variables, in contrast to just $n+m$ variables to be
solved for during simulation.
\end{nonnormative}

Expand All @@ -957,17 +957,17 @@ \subsection{The Number of Equations Needed for Initialization}\doublelabel{the-n

Here we have three variables with unknown values: two dynamic
variables that also are state variables, \lstinline!x1! and \lstinline!x2!, i.e.,
n=2, one output variable \lstinline!y!, i.e., m=1, and one input variable \lstinline!u! with
$n=2$, one output variable \lstinline!y!, i.e., $m=1$, and one input variable \lstinline!u! with
known value. A consistent solution of the initial value problem
providing initial values for \lstinline!x1!, \lstinline!x2!, \lstinline!der(x1)!,
\lstinline!der(x2)!, and \lstinline!y! needs to be found. Two additional initial
equations thus need to be provided to solve the initialization problem.

Regarding DAEs, only that at most n additional equations are
needed to arrive at 2n+m equations in the initialization system. The
Regarding DAEs, only that at most $n$ additional equations are
needed to arrive at $2n+m$ equations in the initialization system. The
reason is that in a higher index DAE problem the number of dynamic
continuous-time state variables might be less than the number of state
variables n. As noted in \autoref{initialization-initial-equation-and-initial-algorithm} a tool may add/remove
variables $n$. As noted in \autoref{initialization-initial-equation-and-initial-algorithm} a tool may add/remove
initial equations to fulfill this requirement, if appropriate
diagnostics are given.
\end{example}
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