Conditioning and backward errors of eigenvalues of homogeneous matrix polynomials under Möbius transformations
This work provides theoretical guarantees for the numerical reliability of Möbius transformations in structured PEP algorithms, addressing a gap in understanding their effect on error metrics.
Möbius transformations can change the conditioning and backward errors of eigenvalues in polynomial eigenvalue problems (PEPs). The authors show that with a homogeneous formulation, well-conditioned Möbius transformations approximately preserve these quantities under certain perturbation norms, but not under others.
Möbius transformations have been used in numerical algorithms for computing eigenvalues and invariant subspaces of structured generalized and polynomial eigenvalue problems (PEPs). These transformations convert problems with certain structures arising in applications into problems with other structures and whose eigenvalues and invariant subspaces are easily related to the ones of the original problem. Thus, an algorithm that is efficient and stable for some particular structure can be used for solving efficiently another type of structured problem via an adequate Möbius transformation. A key question in this context is whether these transformations may change significantly the conditioning of the problem and the backward errors of the computed solutions, since, in that case, their use may lead to unreliable results. We present the first general study on the effect of Möbius transformations on the eigenvalue condition numbers and backward errors of approximate eigenpairs of PEPs. By using the homogeneous formulation of PEPs, we are able to obtain two clear and simple results. First, we show that, if the matrix inducing the Möbius transformation is well conditioned, then such transformation approximately preserves the eigenvalue condition numbers and backward errors when they are defined with respect to perturbations of the matrix polynomial which are small relative to the norm of the polynomial. However, if the perturbations in each coefficient of the matrix polynomial are small relative to the norm of that coefficient, then the corresponding eigenvalue condition numbers and backward errors are preserved approximately by the Möbius transformations induced by well-conditioned matrices only if a penalty factor, depending on those coefficients, is moderate. It is important to note that these simple results are no longer true if a non-homogeneous formulation is used.