@@ -357,6 +357,46 @@ Let $\vec{r}(\vec{x}_i) = \vec{x}_{i+1} - \vec{x}_i$ be the residual function.

Then MPE may be expressed as equation (\ref{eq:uqn}) with $\fxi{i}=\vec{r}(\vec{x}_i)$ and $\vec{v}_i =\vec{r}(\vec{x}_{n+i-1})$.

RRE and MMPE may also be expressed as such, using $\vec{v}_i =\vec{r}(\vec{x}_{n+i})-\vec{r}(\vec{x}_{n+i-1})$ and $\vec{v}_i$ some fixed vector independent of $\vec{r}(\vec{x})$, respectively.

TEA also has a connection to quasi-Newton methods, though in a more roundabout way.

Recall the equation that forms the foundation of the quasi-Newton methods expressed in this section:

\begin{equation}

F_k = \hat{J} X_k.

\end{equation}

This equation assumes that our starting position is $\vec{x}_n$.

If our starting position is instead $\vec{x}_{n+j}$ then the equation becomes

\begin{equation}

F_{n+j,k} = \hat{J}_{n+j} X_{n+j,k}.

\end{equation}

We can create any number of these equations as long as we have enough values of $\fxi{n+j}$ and $\vec{x}_{n+j}$.

Using all of these equations to determine a direction would create an overdetermined system.

One way to reduce the size of such a system is to take the inner product with some vector $\vec{q}$: