Commit a86928b2 authored by Conor McCoid's avatar Conor McCoid
Browse files

Extrap: finished equivalence of qN for pure multisecant equation methods

parent 9bdab8c8
......@@ -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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