Commit 3e40b830 authored by Philipp Arras's avatar Philipp Arras
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......@@ -192,18 +192,18 @@ function:
\begin{align*}
R: \mathbb R^{s.shape} \to \mathbb R^{d.shape},
\end{align*}
its gradient at the position \texttt{s=position} is a linear map of the following shape:
\footnote{There are various ways to think about derivatives and gradients of
multi-dimensional functions. A different view on gradients would be that at a
given point $s=s_0$ the gradient is a matrix with \texttt{s.size} columns and
\texttt{d.size} rows. Obviously, it is not feasible to store such a matrix on a
computer due to its size. There we think of this matrix in terms of a linear
map which maps an array of shape \texttt{s.shape} to an array of shape
\texttt{d.shape}. This linear map shall be implemented on the computer in terms
of a function. Think of this map as a linear approximation to \texttt{R} based
at \texttt{s\_0}.}
its gradient at the position \texttt{s=position} is a linear map of the
following shape: \footnote{There are various ways to think about derivatives and
gradients of multi-dimensional functions. A different view on gradients would
be that at a given point $s=\text{position}$ the gradient is a matrix with
\texttt{s.size} columns and \texttt{d.size} rows. Obviously, it is not
feasible to store such a matrix on a computer due to its size. There we think
of this matrix in terms of a linear map which maps an array of shape
\texttt{s.shape} to an array of shape \texttt{d.shape}. This linear map shall
be implemented on the computer in terms of a function. Think of this map as a
linear approximation to \texttt{R} based at \texttt{position}.}
\begin{align*}
\left. \frac{dR}{ds}\right|_{s=position} = R': \mathbb R^{s.shape} \to \mathbb R^{d.shape}
\left. \frac{dR}{ds}\right|_{s=\text{position}} = R': \mathbb R^{s.shape} \to \mathbb R^{d.shape}
\end{align*}
What needs to be implemented is a function \texttt{R\_prime(position, s0)} which
takes the arguments \texttt{position} (which is an array of shape \texttt{s.shape}
......@@ -315,8 +315,4 @@ Why is this already sufficient?
R^\dagger$. All in all, we need an implementation for $R$ and $R^\dagger$.
\end{itemize}
% \section*{Bibliography test}
% RESOLVE was first presented in \cite{Resolve2016}.
% \printbibliography
\end{document}
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