Commit 8d9f4f6b authored by Conor McCoid's avatar Conor McCoid
Browse files

Extrap: minor adjustments to map and FPI example

parent 27113bb0
% Example - Extrapolation: fixed point iteration on linear function
% solve the equation x = Ax + b for random A and b
d = 5; n=2*d;
d = 10; n=2*d;
A = rand(d); b = rand(d,1); x_true = -(A - eye(d)) \ b;
X = b;
for i=1:n
......@@ -28,7 +28,7 @@ end
[x_GMRES,~,~,~,res_GMRES]=gmres(A-eye(d),-b,d,0,n,eye(d),eye(d),X(:,1));
figure(1)
semilogy(1:n,Error_v1,'b*--',1:n,Error_v2,'k.--',1:length(res_GMRES),res_GMRES,'ro')
semilogy(1:n,Error_v1,'b*--',1:n,Error_v2,'k.--')
xlabel('Iteration')
ylabel('Error in solution')
title('Error vs iteration')
......
......@@ -7,6 +7,7 @@
\usepackage{hyperref}
\usepackage{subcaption}
\usepackage{tikz}
\usetikzlibrary{calc,positioning}
\newcommand{\dxdy}[2]{\frac{d #1}{d #2}}
\newcommand{\dxdyk}[3]{\frac{d^{#3} #1}{d {#2}^{#3}}}
......@@ -605,18 +606,16 @@ 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=1cm, row sep=1em]{
& & & & \node[align=left] (Anderson) {Anderson\\mixing}; \\
\node[align=left] (multisecant) {$\begin{bmatrix} \vec{1}^\top \\ F_{n,k} \end{bmatrix} \vec{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}$,\\ $X_{n,k} \vec{u} = \hat{\vec{x}}$}; &
\matrix (m) [column sep=0.8cm, row sep=1em]{
& & & & \node[align=center] (Anderson) {Anderson\\mixing}; \\
\node[align=center] (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 (MPE) {MPE}; \\
& & & & \node (GMRES) {GMRES}; \\
\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[align=center] (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 (prepreRRE) {}; & & \node (preRRE) {}; & \node (RRE) {RRE}; \\
& & & & \node (GCR) {GCR}; \\
& \node (prepreMMPE) {}; & & \node (preMMPE) {}; & \node (MMPE) {MMPE}; \\
......
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