Let A be a large sparse matrix (not invertible). A has repeated eigenvalues (say $\lambda_i$) which has degenerate eigenvectors (say more than 2). what is the best way (or algorithm) to identify generalized eigenvectors corresponding to $\lambda_i$.
I know the method is to solve for n generalized eigenvectors by $$(A-\lambda I)x_k=x_{k-1}$$ for $k=2,3, \dots n$, or $$(A-\lambda I)^n x=0$$ I tried to identify the eigenvectors of the latter method using MATLAB function $eigs(A,n,'sm')$.
Q1. which one is the best out of these two? Is there any other way to identify generalized eigenvectors accurately?. Accuracy is the main concern.
Q2. I tried algorithms in http://www.mathworks.com/help/matlab/math/systems-of-linear-equations.html#brzoiix. But not sure about the best choice.
Q3. How to confirm there are exactly n eigenvectors of $\lambda_i$ are degenerate so that I need to identify n generalized eigenvectors.