Commit 7adfa993 authored by conmccoid's avatar conmccoid
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Extrap: additional notes on equivalence class

parent a86928b2
......@@ -485,4 +485,16 @@ Some standard choices of $\Delta$:
\begin{bmatrix} -1 & \dots & -1 \\ 1 \\ & \ddots \\ & & 1 \end{bmatrix}, \quad \begin{bmatrix} -1 \\ 1 & \ddots \\ & \ddots & -1 \\ & & 1 \end{bmatrix}.
\end{equation*}
The equations to solve for $\vec{u}$, $\hat{\vec{u}}$ and $\tilde{\vec{u}}$ are solved in some sense.
For example, if $\vec{f} : \bbr^d \to \bbr^d$ and $k< d$ then each of the presented systems is underdetermined.
If $k>d$ then they are overdetermined.
To solve the system(s) one can multiply by a matrix $A^\top$:
\begin{equation*}
A^\top F_{n,k} \vec{u} = A^\top \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad A^\top F_{n,k} \Delta \tilde{\vec{u}} = A^\top \begin{bmatrix} 0 \\ \fxi{n+i} \end{bmatrix}.
\end{equation*}
The first column of $A^\top$ should be $\begin{bmatrix} 1 & 0 \end{bmatrix}^\top$ to preserve the first equation of the system.
Note that $\hat{\vec{x}}$ is always in the span of $\set{\vec{x}_{n+i}}_{i=0}^k$.
Therefore, if $k<d$ then $\hat{\vec{x}}$ is in a restricted subspace of $\bbr^d$.
\end{document}
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