Commit 639acd67 authored by conmccoid's avatar conmccoid
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Extrap: connecting multisecant equations for qN methods

parent f4376ef4
......@@ -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
F_{n,k} \Delta = \hat{J} X_{n,k} \Delta
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
\hat{J}_{n+1} \begin{bmatrix} X_{n,1} \Delta & Q \end{bmatrix} = \begin{bmatrix} F_{n,1} \Delta & \hat{J}_n Q \end{bmatrix}
where $Q^\top X_{n,1} \Delta = 0$.
In the generalized Broyden's method $X_{n,1}$ and $F_{n,1}$ are replaced by $X_{n,k}$ and $F_{n,k}$ and $Q$ reduced in size by $k$ columns.
Broyden's family of methods may be written as
\hat{J}_{n+1} = \hat{J}_n + \fxi{n+1} \vec{v}_n^\top
where $\vec{v}_n^\top (\vec{x}_{n+1} - \vec{x}_n) = 1$.
If $\vec{v}_n$ is chosen such that
\vec{v}_n^\top X_{n-k,k} \Delta = \begin{bmatrix} 0 & \dots & 0 & 1 \end{bmatrix}
with possibly other constraints then $\hat{J}_{n+1} X_{n-k,k} \Delta = F_{n-k,k} \Delta$.
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