%
% @file modes.m
%
% @brief
%
% @copyright
% Copyright (©) 2021 EPFL (Ecole Polytechnique Fédérale de Lausanne)
% SPC (Swiss Plasma Center)
%
% SPClibs is free software: you can redistribute it and/or modify it under
% the terms of the GNU Lesser General Public License as published by the Free
% Software Foundation, either version 3 of the License, or (at your option)
% any later version.
%
% SPClibs is distributed in the hope that it will be useful, but WITHOUT ANY
% WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
% FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.
%
% You should have received a copy of the GNU Lesser General Public License
% along with this program. If not, see <https://www.gnu.org/licenses/>.
%
% @author
% (in alphabetical order)
% @author Trach-Minh Tran <trach-minh.tran@epfl.ch>
%
N=8;
kmodes=N-1;

%%  modes on fine grid %%
xh=(1/N).*(0:1:N);
wh=zeros(N+1,kmodes);
for k=1:kmodes
    wh(:,k) = sin((k*pi).*xh);
end

%%  Restriction %%
R=[
1 0 0 0 0 0 0 0 0
0 0.5 1 0.5 0 0 0 0 0
0 0 0 0.5 1 0.5 0 0 0
0 0 0 0 0 0.5 1 0.5 0
0 0 0 0 0 0 0 0 1
];

%% Null space of Restriction %%
ns = [
     0     0     0     0
     2     0     0     0
    -1    -1     0     0
     0     2     0     0
     0    -1    -1     0
     0     0     2     0
     0     0    -1    -1
     0     0     0     2
     0     0     0     0];

figure
subplot(211)
plot(ns,'o-');
title('Basis of Null space of Restriction')
subplot(212)
plot(R','o-');
title('Basis of Range of Prolongation')


%%  modes on coarse grid %%
N2h = N/2;
x2h=(1/N2h).*(0:1:N2h);
w2h = R*wh;

x=0:0.01:1.;
figure
for k=1:kmodes
    subplot(3,3,k)
    plot(xh,wh(:,k),'o', x, sin((k*pi).*x),'b-', x2h, w2h(:,k), 'ro-');
    grid on
end

figure
for k=1:N/2
    subplot(2,2,k)
    plot(xh,wh(:,k),'o-', xh,wh(:,N-k),'r*-')
    grid on
end