Balanced simultaneous schur decomposition for joint eigenvalue estimation

We address the problem of joint eigenvalue estimation for the non-defective commuting set of matrices A. We propose a procedure revealing the joint eigenstructure by simultaneous diagonalization of A with simultaneous Schur decomposition (SSD) and balance procedure alternately for performance considerations and also to overcome the convergence difficulties of previous methods based only on simultaneous Schur form and unitary transformations. We show that the SSD procedure can be well incorporated with the balancing algorithm in a pingpong manner, i.e., each optimizes a cost function and at the same time serves as an acceleration procedure for the other. Numerical experiments conducted in a multi-dimensional harmonic retrieval application suggest that the method presented here converges considerably faster with an analyzable performance than the methods based on only unitary transformation for matrices which are not near to normality.