I heard that subspace iteration plus Ritz acceleration could improve the performance a lot for solving clustered eigenvalues, for the eigenvalues and eigenvectors could converge linearly with ratio $\lambda_{p+1}/\lambda_j$, $j=1,\ldots,p$.  For Hermitian Matrix this is even faster.This is really fantastic.  Now for any $n\times n$ Complex Matrix $C$ with sufficient eigenvectors that is not deficit, we can construct an algorithm to make all the eigenvalues be solved in an extremely efficient way:Add a big number to the diagonal elements to make the eigenvalue large in magnitude.Add a zero column together with a zero row to make it an $n+1\times n+1$ matrix $C'$.Modify $C'_{n+1\times n+1}$ with a very small nonzero number.Use Subspace Iteration Algorithm with $n$ orthogonal $n+1$ dimensional vectors.  According to the proof of the Algorithm it should converge very fast, for the first $n$ eigenvalues are respectively huge to the last one.Subtract the big number to the solved $n$ eigenvalues to get the original ones.Does anyone think it possible?
About Subspace Iteration for Eigenvalues
linear algebra;eigensystem;matrix