Addition in Homogeneous Space $\tilde +$

Computation of the canonical homogeneous norm

The function ${\color{red} { \texttt{hadd}} } $ computes the sum vectors $y=x_1\tilde + x_2$ in the homogeneous Euclidean space $\mathbb{R}^n_{\mathbf{d}}$.

$\textbf{Input}: {\color{blue}x_1}, {\color{blue}x_1}, {\color{blue}{G_{\mathbf{ d } }}}, {\color{blue}P} $

$ \textbf{Output}: {\color{magenta} y} $

Use ${\color{red} { \texttt{demo_halgebra.m}} } $ from $\texttt{HCS Toolbox}$ as a demo of computation of the sum of vectors in $\mathbb{R}^n_{\mathbf{d}}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Example of demonstrating operators in homogeneous Euclidean space %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% tol=1e-6; % compuational tolerance %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Shape matrix of the wighted Euclidean norm %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P=[1 -1; -1 2]; if norm(P-P') > tol disp('Error: shape matrix P must be symmetric'); return; end; if min(real(eig(P))) < tol disp('Error: shape matrix P must be positive definite'); return; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Generator of the linear dilation in R^2 %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Gd=[3 0; 1 1]; %monotonicy check if min(real(eig(P*Gd+Gd'*P))) < tol disp('Error: dilation must be monotone'); return; end; %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Sum of vectors in homogeneous Euclidean space %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(['--Addition in homogeneous Euclidean space R^2_d--']); x=[1;-1]; % vector x in R^2 disp(['Vector x = [', num2str(x(1)),';',num2str(x(2)),']']); y=[-0.5;1]; % vector y in R^2 disp(['Vector y = [', num2str(y(1)),';',num2str(y(2)),']']); z=x+y; disp(['x+y in R^2 =[', num2str(z(1)),';',num2str(z(2)),']']); % sum of vectors in R^2 z=hadd(x,y,Gd,P); disp(['x+y in R^2_d =[', num2str(z(1)),';',num2str(z(2)),']']); % sum of vectors in homogeneous Euclidean space R^2_d %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Multiplication of a vector by a scalar in homogeneous Euclidean space %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% disp(['--Multiplication by a scalar in homogeneous Euclidean space R^2_d--']); a=0.7; disp(['Scalar a= ', num2str(a)]); x=[1;-5]; % vector x in R^2 disp(['Vector x = [', num2str(x(1)),';',num2str(x(2)),']']); z=a*x; disp(['a*x in R^2 =[', num2str(z(1)),';',num2str(z(2)),']']); % multiplication by a scalar in R^2 z=hdot(a,x,Gd); disp(['a*x in R^2_d =[', num2str(z(1)),';',num2str(z(2)),']']); % multiplication by a scalar in homogeneous Euclidean space R^2_d %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %% Inner product in homogeneous Euclidean space %% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% P=eye(2); % for comparison with canonical inner product in R^2 disp(['--Inner product in homogeneous Euclidean space R^2_d--']); x=[2;0]; % vector x in R^2 disp(['Vector x = [', num2str(x(1)),';',num2str(x(2)),']']); y=[0;1]; % vector y in R^2 disp(['Vector y = [', num2str(y(1)),';',num2str(y(2)),']']); p=x'*P*y; disp(['< x, y > in R^2 =', num2str(p)]); % inner product of vectors in R^2 p=hinner(x,y,Gd,P); disp(['< x, y > in R^2_d =', num2str(p)]); % inner product of vectors in homogenous Euclidean space R^2_d