Before revision:  Numerical experiments on large matrix pencils illustrate the method , comparing with previous methods, and show that the new method is  more robust and efficient , in particular, in parallel. Further , the proposed method does not fail for the existence of many pairs of extremely close eigenvalues and breaks the previous barrier of problem size .
After revision:  Numerical experiments on large matrix pencils illustrate the method , comparing with previous methods, and show that the new method is more robust and efficient and is conjectured by experiment results to be more efficient in parallelized settings. Notably , the proposed method does not fail for the existence of many pairs of extremely close eigenvalues and breaks the previous barrier of problem size .