Perron eigenvector
d
Is there's a matlab function that calculates the Perron vector of a matrix ?
(the eigenvector that corresponds to the "Perron root" or the "Perron–Frobenius eigenvalue") ?
thank you,
D
d
(of course, for a strictly positive matrix A, [e,v] = eigs(A,1) would do)
thanks again,
D
Matt J
> hi,
> Is there's a matlab function that calculates the Perron vector of a matrix ?
> (the eigenvector that corresponds to the "Perron root" or the "Perron–Frobenius eigenvalue") ?
Something like this perhaps:
[V,D]=eig(myMatrix);
[trash,idx]=max(abs(diag(D)));
PerronVector=V(:,idx);
Bruno Luong
Not quite, first we have to note that using max(diag(D)) is correct. But even with that change, I don't believe there is any warranty the above would return Perron vector.
For example let
Q=[0 1 0 0;
1 0 0 0;
0 0 0 1;
0 0 1 0];
We have
V1 = [1; 1; 0; 0]
V2 = [0; 0; 1; 1]
are eigen vectors corresponding to rho(Q)=1, and both are Perron vectors. However there is no thing to prevent Matlab to return
V1 = [1; 1; -1; -1]
V2 = [2; 2; -1; -1]
for example. Both of course are not Perron's vector. That's where the difficulty arises.
Bruno
Bruno Luong
[V D]=eig(M);
lambda = diag(D);
rho = max(lambda);
tol = 1e-10; % use some small tolerance
ind = find(lambda>=rho*(1-tol));
V = V(:,ind);
% Use linear programming to find a combination of V that
% has non-negative elements
% Vperron := V*x
% Vperron >= 0 <=> -V*x w= 0
% sum(V*x) = 1 <=> sum(V,1)*x = 1
A = -V;
b = zeros(size(V,1),1);
Aeq = sum(V,1);
beq = 1;
dummy = ones(size(V,2),1);
x = linprog(dummy, A, b, Aeq, beq);
Vperron = V*x
% Bruno
d
...
> > Something like this perhaps:
> >
> > [V,D]=eig(myMatrix);
> > [trash,idx]=max(abs(diag(D)));
> > PerronVector=V(:,idx);
>
> Not quite, first we have to note that using max(diag(D)) is correct. But even with that change, I don't believe there is any warranty the above would return Perron vector.
> for example. Both of course are not Perron's vector. That's where the difficulty arises.
>
> Bruno
thank you for responding.
I would like to add that I only need the vector corresponding to the largest (absolute) eigenvalue, therefore I would rather not compute all eigenvalues+eigenvectors. I'm dealing with quite large, somewhat sparse, matrices (~700x700).
D
Bruno Luong
>
>
> I would like to add that I only need the vector corresponding to the largest (absolute) eigenvalue, therefore I would rather not compute all eigenvalues+eigenvectors. I'm dealing with quite large, somewhat sparse, matrices (~700x700).
>
Do you have an example where
[V D]=eigs(A,1,'lm')
fails?
Bruno
d
> I suggest to use LINPROG (opt toolbox) to eventually "fix" the eigen vector.
>
> [V D]=eig(M);
> lambda = diag(D);
.
.
> x = linprog(dummy, A, b, Aeq, beq);
> Vperron = V*x
>
> % Bruno
hi Bruno, just saw your suggestion now.
it actually had no problem with the size of the matrix, and was even faster then another code I found:
http://www.mathworks.de/matlabcentral/fileexchange/22763-perron-root-computation
the two methods gave similar results, which I think is a good sign.
thank you very much
D
Bruno Luong
>
> hi Bruno, just saw your suggestion now.
> it actually had no problem with the size of the matrix, and was even faster then another code I found:
> http://www.mathworks.de/matlabcentral/fileexchange/22763-perron-root-computation
>
> the two methods gave similar results, which I think is a good sign.
> thank you very much
It looks like this code is from people who know about this kind of problem, so you should perhaps use their.
Bruno
chintu gurbani
clc
clear all
n=4;
I=eye(n,n);
gama(:,:,1)=I;
P(:,:,1)=D_Matrix( n)
for i=1:2
P(:,:,i+1)=D_Matrix( n)*P(:,:,i);
for j=1:20
gama(:,:,j+1)=P(:,:,i+1)*gama(:,:,j);
end
end
[RV1, EV1, LV1]=eig(P(:,:,1));
[RV, EV, LV]=eig(P(:,:,i+1));
function [ D ] = D_Matrix( n)
for i=1:n
for j=1:n
D(i,j)=floor(n*rand);
end
D(i,:)=D(i,:)/sum(D(i,:));
end
end
Reference: Cooperative control with distributed gain adaptation and
connectivity estimation for directed networks‡