@@ -496,4 +496,33 @@ To solve the system(s) one can multiply by a matrix $A^\top$:

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 $\bbr^d$.

\section{Multisecant equations}

The multisecant equations presented previously may be summarized as

\begin{equation}

F_{n,k}\Delta = \hat{J} X_{n,k}\Delta

\end{equation}

where all components have been previously defined.

The square matrix $\hat{J}$ is an approximation of the Jacobian for the vector-valued function $\fxi{}$ for somewhere in the simplex of the columns of $X_{n,k}$.

Generally, it can be thought of as an approximation of $J(\vec{x}_n)$ or $J(\vec{x}_{n+k})$ but could also represent $J(\vec{x}_{n+i})$ for any $i$ between 0 and $k$, due to the equivalence of these equations explained previously.

These equations appear in many, if not all, quasi-Newton methods.

The update for Broyden's method is chosen such that