Commit a400f4a1 by conmccoid

### Extrap: initial commit on map of methods

parent 0bf5ca77
 ... ... @@ -6,6 +6,7 @@ \usepackage{graphicx} \usepackage{hyperref} \usepackage{subcaption} \usepackage{tikz} \newcommand{\dxdy}[2]{\frac{d #1}{d #2}} \newcommand{\dxdyk}[3]{\frac{d^{#3} #1}{d {#2}^{#3}}} ... ... @@ -604,4 +605,45 @@ This is trivial to show. If $Q_k$ is derived from another method but shares the column space of $F_{n,k-1}$ then there is still mathematical equivalence between methods 2 and 3. If one minimizes with respect to a different norm then the methods correspond to other methods discussed here. \section{Map of equivalences} \begin{figure} \centering \begin{tikzpicture} \matrix (m) [column sep=2cm, row sep=1em]{ \node[align=left] (multisecant) {$\begin{bmatrix} \vec{1}^\top \\ F_{n,k} \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$,\\ $X_{n,k} \vec{u} = \hat{\vec{x}}$}; & \node (overdetermined) {$\begin{bmatrix} \vec{1}^\top \\ B^\top F_{n,k} \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$}; & \node (preMPE) {}; & \node (Anderson) {Anderson mixing}; \\ & & & \node (MPE) {MPE}; \\ \node[align=left] (preBroyden) {$\begin{bmatrix} \vec{1}^\top & \vec{1}^\top \\ F_{n,k} & B \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$,\\ $\begin{bmatrix} X_{n,k} & C \end{bmatrix} \vec{u} = \hat{\vec{x}}$}; & & & \node (GMRES) {GMRES}; \\ & \node (prepreRRE) {}; & \node (preRRE) {}; & \node (RRE) {RRE}; \\ & & & \node (GCR) {GCR}; \\ & \node (prepreMMPE) {}; & \node (preMMPE) {}; & \node (MMPE) {MMPE}; \\ \node (Broyden) {Generalized Broyden}; & \node (preTEA) {}; & \node (preBiCG) {}; & \node (BiCG) {BiCG}; \\ & & & \node (TEA) {TEA}; \\ }; \path[->,very thick] (multisecant) edge node[above] {$k \neq d$} (overdetermined) edge node[right] {$k < d$} (preBroyden) (preBroyden) edge node[right,align=left] {$(C \Delta)^\top (X_{n,k} \Delta) = 0$, \\ $B = \hat{J}_{n-1,k} C$} (Broyden) (overdetermined) edge node[above] {$B=F_{n,k-1}$} (preMPE) (preMPE) edge node[above] {+ relaxation} (Anderson) (prepreRRE) edge node[above] {$B=F_{n,k} \Delta$} (preRRE) (preRRE) edge[red] (RRE) (prepreMMPE) edge node[above] {$B=\begin{bmatrix} \vec{q}_1 & \dots & \vec{q}_k \end{bmatrix}$} (preMMPE) (preMMPE) edge[red] (MMPE) (preTEA) edge node[above] {$B=\begin{bmatrix} \vec{q} & A^\top \vec{q} & \dots \end{bmatrix}$} (preBiCG) (preBiCG) edge[blue] (BiCG) (BiCG) edge[red] (TEA); \draw[-, very thick] (overdetermined) edge (preTEA); \draw[->,very thick,red] (preMPE) |- (MPE); \draw[->,very thick,blue] (preMPE) |- (GMRES); \draw[->,very thick,blue] (preRRE) |- (GCR); \end{tikzpicture} \caption{Interconnectivity of extrapolation, acceleration and quasi-Newton methods. Red arrows indicated $\fxi{n} = \vec{x}_{n+1}-\vec{x}_n$ while blue arrows indicate $\fxi{n} = (A-I) \vec{x}_n + \vec{b}$ and $\fxi{n+1} = A \fxi{n}$.} \end{figure} \end{document} \ No newline at end of file
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