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\hypertarget{group__numpp__differentiation__finite__backward}{}\section{Backward Finite Differentiation}
\label{group__numpp__differentiation__finite__backward}\index{Backward Finite Differentiation@{Backward Finite Differentiation}}
\subsection*{Functions}
\begin{DoxyCompactItemize}
\item 
{\footnotesize template$<$typename Func , typename T , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+floating\+\_\+point$<$\+T$>$\+::value$>$$>$ }\\C\+O\+N\+S\+T\+E\+X\+PR auto \hyperlink{group__numpp__differentiation__finite__backward_gaf772b22eb473fffdb4c4eb2333d21937}{numpp\+::differentiation\+::finite\+::backward} (Func \&\&f, T x)
\item 
{\footnotesize template$<$typename Func , typename T , typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$      std\+::is\+\_\+floating\+\_\+point$<$\+T$>$\+::value and      std\+::is\+\_\+arithmetic$<$\+U$>$\+::value    $>$$>$ }\\C\+O\+N\+S\+T\+E\+X\+PR auto \hyperlink{group__numpp__differentiation__finite__backward_ga33f8373a02912cf6b680d0d8955bea3a}{numpp\+::differentiation\+::finite\+::backward} (Func \&\&f, T x, U y)
\item 
{\footnotesize template$<$typename Func , typename T , typename U $>$ }\\constexpr auto \hyperlink{group__numpp__differentiation__finite__backward_ga3015b47eec3f468c07e5d72e88ea22dc}{numpp\+::differentiation\+::finite\+::backward} (Func \&\&f, T \&\&x, U \&\&y, T \&\&h)
\end{DoxyCompactItemize}


\subsection{Detailed Description}
Calculates backward finite derivative $\frac{f(x-h)-f(x)}{h}$


\begin{DoxyTemplParams}{Template Parameters}
{\em Func} & Function whose finite difference derivative should be calcuated \\
\hline
{\em T} & Argument type (e.\+g. double)\\
\hline
\end{DoxyTemplParams}

\begin{DoxyParams}{Parameters}
{\em f} & Functor or function whose derivative we want to obtain \\
\hline
{\em x} & Point where the derivative is calculated \\
\hline
{\em h} & Disturbtion to x argument, should be around e-\/7 usually\\
\hline
\end{DoxyParams}
\begin{DoxyWarning}{Warning}
If you are unsure about parameter h use function with automatically calculated overload!
\end{DoxyWarning}
Eventual error performed by this calculation is not provided.

For better accuracy and numerical stability see\+:

\hyperlink{group__numpp__differentiation__symbolic}{Symbolic Differentiation} \hyperlink{group__numpp__differentiation__automatic}{Automatic Differentiation}

\begin{DoxyReturn}{Returns}
Derivative at point x
\end{DoxyReturn}


\subsection{Function Documentation}
\mbox{\Hypertarget{group__numpp__differentiation__finite__backward_gaf772b22eb473fffdb4c4eb2333d21937}\label{group__numpp__differentiation__finite__backward_gaf772b22eb473fffdb4c4eb2333d21937}} 
\index{Backward Finite Differentiation@{Backward Finite Differentiation}!backward@{backward}}
\index{backward@{backward}!Backward Finite Differentiation@{Backward Finite Differentiation}}
\subsubsection{\texorpdfstring{backward()}{backward()}\hspace{0.1cm}{\footnotesize\ttfamily [1/3]}}
{\footnotesize\ttfamily template$<$typename Func , typename T , typename  = std\+::enable\+\_\+if\+\_\+t$<$std\+::is\+\_\+floating\+\_\+point$<$\+T$>$\+::value$>$$>$ \\
C\+O\+N\+S\+T\+E\+X\+PR auto numpp\+::differentiation\+::finite\+::backward (\begin{DoxyParamCaption}\item[{Func \&\&}]{f,  }\item[{T}]{x }\end{DoxyParamCaption})}

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Automatically calculated disturbtion h to argument x, taken from Numerical Recipees (add source)\mbox{\Hypertarget{group__numpp__differentiation__finite__backward_ga33f8373a02912cf6b680d0d8955bea3a}\label{group__numpp__differentiation__finite__backward_ga33f8373a02912cf6b680d0d8955bea3a}} 
\index{Backward Finite Differentiation@{Backward Finite Differentiation}!backward@{backward}}
\index{backward@{backward}!Backward Finite Differentiation@{Backward Finite Differentiation}}
\subsubsection{\texorpdfstring{backward()}{backward()}\hspace{0.1cm}{\footnotesize\ttfamily [2/3]}}
{\footnotesize\ttfamily template$<$typename Func , typename T , typename U , typename  = std\+::enable\+\_\+if\+\_\+t$<$      std\+::is\+\_\+floating\+\_\+point$<$\+T$>$\+::value and      std\+::is\+\_\+arithmetic$<$\+U$>$\+::value    $>$$>$ \\
C\+O\+N\+S\+T\+E\+X\+PR auto numpp\+::differentiation\+::finite\+::backward (\begin{DoxyParamCaption}\item[{Func \&\&}]{f,  }\item[{T}]{x,  }\item[{U}]{y }\end{DoxyParamCaption})}

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Automatically calculated disturbtion h to argument x, taken from Numerical Recipees (add source) 
\begin{DoxyParams}{Parameters}
{\em y} & Calculated value of f(x) for improved efficiency\\
\hline
\end{DoxyParams}
\mbox{\Hypertarget{group__numpp__differentiation__finite__backward_ga3015b47eec3f468c07e5d72e88ea22dc}\label{group__numpp__differentiation__finite__backward_ga3015b47eec3f468c07e5d72e88ea22dc}} 
\index{Backward Finite Differentiation@{Backward Finite Differentiation}!backward@{backward}}
\index{backward@{backward}!Backward Finite Differentiation@{Backward Finite Differentiation}}
\subsubsection{\texorpdfstring{backward()}{backward()}\hspace{0.1cm}{\footnotesize\ttfamily [3/3]}}
{\footnotesize\ttfamily template$<$typename Func , typename T , typename U $>$ \\
constexpr auto numpp\+::differentiation\+::finite\+::backward (\begin{DoxyParamCaption}\item[{Func \&\&}]{f,  }\item[{T \&\&}]{x,  }\item[{U \&\&}]{y,  }\item[{T \&\&}]{h }\end{DoxyParamCaption})}

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.


\begin{DoxyParams}{Parameters}
{\em y} & Calculated value of f(x) for improved efficiency \\
\hline
{\em h} & Specified disturbtion to argument x\\
\hline
\end{DoxyParams}
