Solving Singular Generalized Eigenvalue Problems by a Rank-Completing Perturbation

Abstract

Generalized eigenvalue problems involving a singular pencil are very challenging to solve, both with respect to accuracy and efficiency. The existing package Guptri is very elegant but may be time-demanding, even for small and medium-sized matrices. We propose a simple method to compute the eigenvalues of singular pencils, based on one perturbation of the original problem of a certain specific rank. For many problems, the method is both fast and robust. This approach may be seen as a welcome alternative to staircase methods.

One-particle Green's functions from the quantum equation of motion algorithm

“…We further improve the results by averaging over 13 realizations of the matrix elements estimation performed with 8192 shots, giving an effective number of total shots of approximately 10 4 . We also notice that the general problem of minimizing the effect of noise in GEPs is well known in the literature [40],…”

Solving generalized eigenvalue problems by ordinary differential equations on a quantum computer

“…When false(A,Bfalse) is regular, this matrix pair always has n eigenvalues, including normal∞.The above two unusual properties appear when B is singular. Also, when B is singular, the GEP is well known to be ill-conditioned [19,31]. So it is hard to solve even for a quantum computer; see the lower bound analysis in §5.…”

“…For a classical computer, the GEPs involving singular matrices are challenging to solve from the viewpoint of stability and computational complexity [19,31]. For instance, in [20, section 7.7], three simple examples are listed to illustrate this by showing that when the matrices are singular, the eigenvalues can be empty, finite or the whole space.…”

“…For instance, it may happen that the GEP has infinitely many eigenvalues or λ=normal∞ is an eigenvalue (see §2 for more details). Also, when B is singular, the GEP is well known to be ill-conditioned [19]. So it is hard to solve even for a quantum computer.…”

Solving singular generalized eigenvalue problems. Part II: projection and augmentation

“…Introduction. In this paper, which is a sequel to Part I [9], we further study the computation of eigenvalues of singular matrix pencils. Whereas in [9] a method based on a rank-completing perturbation has been introduced (i.e., an update by a pencil of a rank that is precisely sufficient to render the updated pencil regular), we propose in this paper a scheme based on rank projection (i.e., a projection of the pencil onto a subspace of maximal possible size such that the projected pencil is generically regular), as well as an augmentation method, as two alternative techniques.…”