General remarks
We thank the referee for his/her work reviewing our paper.
The report asks us to give further details on (i) numerical stability and (ii) computational performance of our method. We feel that (i) is rather academic, because we prove the convergence of fixed-point iteration under explicit conditions on \(\theta\) and \(\lambda\) which exclude ill-conditioned eigenvalues—and, numerically, DPT is really just matrix multiplication. Nonetheless, we have added Fig. S4-S5 as well as SI G supporting the claim that DPT is both accurate and numerically stable (within its domain of convergence). Regarding (ii), we have added a timing comparison with LAPACK on large dense matrices (up to size \(N=131,072\)), showing a 30-90x speedup (Fig. 4). We also added a link to a public repository containing our code for reproducibility. Edits in the main text are highlighted in blue.
We note that the referee did not offer comments on the physical or theoretical aspects of our method. These are, however, the main focus of our Letter, in line with PRL's scope as a physics journal. We intend to give more details on the implementation of DPT on parallel architectures and other computational aspects in a more suitable venue at a later stage.
Moreover, we observe that some of the referee's comments (on "unphysical" solutions or "subjective [...] boundary conditions") appear to refer to algorithms other than DPT. We emphasize that our paper gives explicit conditions under which DPT provably converges to a complete set of eigenvectors, without any ambiguity or user-driven choices. Surely our algorithm cannot be criticized for the shortcomings of its competitors.
Response to specific comments
Comment 1
The proposed dynamic perturbation theory (D-PT) is not general. It is formulated only for matrices with non-degenerate (simple) eigenvalues.
This comment suggests that the exclusion of degenerate eigenvalues is a peculiar deficiency of our method. This is misleading in several ways:
- Degenerate eigenvalues present a challenge for any eigenvalue algorithm because they correspond to singularities in the complex \(\lambda\) plane. (In the language of numerical analysis, degenerate spectra have infinite condition number.) Moreover, many important results in matrix perturbation theory (e.g. the Bauer-Fike theorem) assume simple eigenvalues, as do many physics texts on quantum mechanical perturbation theory.
- Algorithms such as non-degenerate RS-PT (which the simple eigenvalues of the unperturbed matrix to be simple) are not generally seen as defective or "not general". Instead, it is usually understood that degenerate eigenvalues must be treated separately, for instance by diagonalizing the perturbation \(\Delta\) in the degenerate eigenspace (a procedure called "degenerate perturbation theory" in quantum mechanics). The same is true of our method.
- We prove the convergence of D-PT for any matrix \(M\ =\ D+ \Delta\) such that \(\Vert \theta\Vert \Vert \Delta\Vert < (3-2\sqrt{2})\). This is as "general" a result as one can hope for in the context of perturbation theory. A similar result for RS-PT (see e.g. \cite{Kato_1966}) is usually considered a "general" theorem of perturbation theory.
- From a strictly mathematical perspective, degenerate spectra are not generic: they form a nowhere dense, zero-measure subset of the space of square matrices. While this comment is moot in physics applications (where symmetries almost always induce degeneracies), it is relevant in other fields where perturbative eigenproblems play an important role, e.g.in molecular evolutionary theory (the application which motivated this work).
The reported illustrations are with highly specialized trivial real symmetric matrices (2 × 2 or 3 × 3) or with a still simple, small size 100 × 100 real non-symmetric matrix with random numbers (as the elements of M) distributed uniformly in the interval [-1,1].