The non-linear equations necessitate the starting values to initiate the iteration process. The boundary conditions to each of the algebraic equations in the D-PT are unknown. Guessing the initial values is subjective, to say the least.
The initial (not "boundary") conditions are not "unknown": given a partition of \(M\) as \(D+\lambda\ \Delta\), D-PT uses a non-ambiguous set of initial conditions for fixed-point iteration, namely \(A^{(0)}=I\). The only "subjective" element in this procedure is the partitioning itself—but this is a feature of any perturbation theory, by definition.
Some surmised/estimated, insufficiently good starters to nonlinear equations can lead to the wrong results, irrespective of the status of the convergence issue (convergence to the wrong result is not infrequent for implicit non-linear algebraic equations).
Once again, we prove convergence to the right results under the condition \(\Vert \theta\Vert \Vert \Delta\Vert < (3-2\sqrt{2})\) with the explicit (not "surmised/estimated") initial condition \(A^{(0)}=I\). We do not understand the referee's concern here.
Comment 4
In practice, the elements of M are not ideal entries, i.e. they enter the analysis with their intrinsic inaccuracies. Matrix elements can come from some elaborate computations with finite arithmetcs (computational round-off errors) or they can be some empirical data stemming from experimental measurements (inevitably contaminated with noise–systematic, random, etc).
This is true, but since we never require or mention "ideal entries", we fail to see the relevance of this comment.
For such matrices routinely encountered in practice, the D-PT would give some non-unique eigensolutions.
This comment is unsubstantiated and false. The uniqueness of the solution of D-PT has nothing to do with "round-off errors" or the origin of the data in \(M\). Once again, we prove convergence of D-PT to a unique fixed point containing a complete set of eigenvectors of \(M\) provided that \(\Vert \theta\Vert \Vert \Delta\Vert < (3-2\sqrt{2})\).
Stated equivalently, there is no guarantee that the eigensolutions would not vary from one to another set of the eigensolutions.