Commit 9bdab8c8 authored by Conor McCoid's avatar Conor McCoid
Browse files

Extrap: started equivalence of qN methods

parent f8ca51ac
......@@ -456,4 +456,19 @@ x_n & \dots & x_{n+k} \\ r_n & \dots & r_{n+k} \\ \vdots & & \vdots \\ r_{n+k-1}
1 & \dots & 1 \\ r_n & \dots & r_{n+k} \\ \vdots & & \vdots \\ r_{n+k-1} & \dots & r_{n+2k-1} }}.
\end{equation}
\section{Equivalence of many quasi-Newton methods}
Suppose we have a method such that
\begin{equation}
\begin{bmatrix} 1 & \dots & 1 \\ \fxi{n} & \dots & \fxi{n+k} \end{bmatrix} \vec{u} = F_{n,k} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \\ \hat{\vec{x}} = \begin{bmatrix} \vec{x}_n & \dots & \vec{x}_{n+k} \end{bmatrix} \vec{u} = X_{n,k} \vec{u}
\end{equation}
then one can write an equivalent method:
\begin{equation*}
\hat{\vec{x}} = X_{n,k} \vec{u} = \vec{x}_{n+i} - X_{n,k} (\vec{e}_i - \vec{u}) = \vec{x}_{n+i} - X_{n,k} \hat{\vec{u}}
\end{equation*}
where $\hat{\vec{u}}$ satisfies
\begin{equation*}
F_{n,k} \hat{\vec{u}} = \begin{bmatrix} 0 \\ \fxi{n+i} \end{bmatrix}.
\end{equation*}
\end{document}
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