[Math] ny matlab code to find out maximum eigenvalue of the randomly generated matrices

MATLAB

I am running a matlab code for computing the Drazin inverse of the matrix $A$.

Initial value of the iteration method
is $X_0 = \beta A^{k}$, where $k = index (A)$(For $A\in \mathbb{C}^{n\times
n}$, the smallest nonnegative integer $k$ such that $rank(A^{k+1}) =
rank(A^k)$ is called the index of $A$). .

Parameter $\beta$ satisfies: $0<\beta < \frac{2}{\lambda_{max}(A^{k+1})}$.

I want to test the method for the randomly generated matrices so I need a matlab code to determine the maximum eigen value of the matrix $A^k$ so that I may easily choose the value of $\beta$.

Could anybody help me with this. Thanks

Best Answer

Are you looking for the largest eigenvalue or the eigenvalue with the largest magnitude? For magnitude,

a=rand(1000);
max(abs(eig(a)))

is much slower especially if you want to repeat it multiple times because it will compute all of the eigenvalues and then pick the max. You might want to use

a=rand(1000);
eigs(a,1)

which will compute and return only the largest magnitude eigenvalue.

Related Solutions

[Math] How to determine stopping criteria in iterative methods

For iterative methods the difference from one iteration to the other gets lower and lower as it is supposed to converge. Therefore absolute error or absolute squared error from one iteration to the other is usually defined as the stopping criterion. Of course there is one other way, that is to fix the number of iterations which is trivial. Important of choosing a threshold for the absolute error from one iteration to the next provides you a certain level of quality. If you set it to $10^{-8}$ it will take more time for the convergence but the quality of the approximation will be of course better than the case where you set it to $10^{-4}$. It is all related to the trade-off between speed of convergence and the quality of convergence which is determined by the user.

In your example, the thresholds are determined in a similar way. It is a subjective process but the quality can be objectively evaluated.

[Math] Need matlab help related with the iteration method

The prep before your loop should stay the same. The appropriate script is

A = ...; % as you have given
beta = ...; % whatever you want
X0 = beta*A'; % calculate initial estimate

% (these initial values may need to be changed, I don't have a copy of
%    matlab in front of me)    
dklast = NaN; dk = NaN; % initialise to begin loop

iter = 0;
maxiter = 100;

while (abs(dk/dklast - beta - 1) > 1e-4) && (iter < maxiter) % loop until tolerance met
    iter = iter + 1; % keep count of iteration

    X1 = (1+beta)*X0 -beta*X0*A*X0; % calculate new iterate

    dklast = dk; % move old difference "new estimate to previous iterate"
    dk = norm(X1-X0,'fro'); % determine new difference 

    X0 = X1; % copy current iterate to "old" iterate for next iteration

end

I am wondering why you are using this convergence test at all. I would recommend using

dk = norm(X1*X0-I,'fro');

which measures how close X1 is to the left inverse of $A$. Your termination criteria would then be

while dk > (some_tolerance) && iter < maxiter
    ....
end

As you currently have, you are measuring how much X1 changes from X0, which may be small, but still not an approximate inverse (or pseudoinverse) for $A$.

Best Answer

Related Solutions

[Math] How to determine stopping criteria in iterative methods

[Math] Need matlab help related with the iteration method

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