Extrap: finished equivalence of qN for pure multisecant equation methods

Research/Extrapolation methods/notes.tex

@@ -470,5 +470,19 @@ where $\hat{\vec{u}}$ satisfies
 \begin{equation*}
 F_{n,k} \hat{\vec{u}} = \begin{bmatrix} 0 \\ \fxi{n+i} \end{bmatrix}.
 \end{equation*}
+Replace $\hat{\vec{u}}$ with $\Delta \tilde{\vec{u}}$ where $\Delta$ is some matrix representing a differentiation operator.
+The conditions on $\Delta$ are that it has $k-1$ rows and $k$ columns, and its columns sum to zero.
+Then an equivalent method is
+\begin{equation*}
+\hat{\vec{x}} = \vec{x}_{n+i} - X_{n,k} \Delta \tilde{\vec{u}}
+\end{equation*}
+where
+\begin{equation*}
+F_{n,k} \Delta \tilde{\vec{u}} = \begin{bmatrix} 0 \\ \fxi{n+i} \end{bmatrix}.
+\end{equation*}
+Some standard choices of $\Delta$:
+\begin{equation*}
+\begin{bmatrix} -1 & \dots & -1 \\ 1 \\ & \ddots \\ & & 1 \end{bmatrix}, \quad \begin{bmatrix} -1 \\ 1 & \ddots \\ & \ddots & -1 \\ & & 1 \end{bmatrix}.
+\end{equation*}
 
 \end{document}
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