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
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$\begingroup$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.
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