CompanionMatrix_DMD.m

function [KEv, KModes,Norms] = CompanionMatrix_DMD( Data,eps_norm )

% Code is provided by Hassan Arbabi
% and modified by Keisuke Fujii

% Dynamic Mode Decomposition as presented by 
% "Spectral analysis of nonlinear flows" by Rowley et al., Journal of FLuid
% Mechanics, 2009


% inputs : 
% Data - the data matrix: each column is a a set of measurements done at
% each instant - the sampling frequency is assumed to be constant

% outputs: 
% 1 - KModes- Koopman or Dynamic Modes: structures that scale exponentially
% with time - the exponent is given by Koopman or Dynamic Eigenvalues and
% could be imaginary as well

% 2- KEv - Koopman or Dynamic Eigenvalues: the exponents and frequencies
% that make up the time-varaiation of data in time

% 3- Norms - Euclidean (vector) norm of each mode, used to sort the data




X=Data(:,1:end-1);

% pseudo inverse approach
c = pinv(X)*Data(:,end);

m = size(Data,2)-1;
C = spdiags(ones(m,1),-1,m,m);
C(:,end)=c;                  %% companion matrix

[P,D,Z]=eig(full(C));                           %% Ritz evalue and evectors

KEunsrtd = diag(D);                     %% Koopman Evalues
KMunsrtd = X*P;                         %% Koopman Modes

if nargout>2 % norm
    Cnorm = sum(abs(KMunsrtd).^2,1);  %% Euclidean norm of Koopman Modes
    [Norms, ind]=sort(Cnorm,'descend');      %% sorting based on the norm
    % ind     = ind(abs(Norms)>eps_norm); % absolute tolerance
    ind     = ind(abs(Norms)/abs(Norms(1))>eps_norm); % relative tolerance
    
    KEv= KEunsrtd(ind);           %% Koopman Eigenvalues
    
    KModes = KMunsrtd(:,ind);                %% Koopman Modes
    Norms = Norms(1:length(ind))' ;
end
end


%=========================================================================%
% Hassan Arbabi - 08-17-2015
% Mezic research group
% UC Santa Barbara
% arbabiha@gmail.com
%=========================================================================%