Backward error analysis for linearizations in heavily damped quadratic eigenvalue problem

Abstract

Summary Heavily damped quadratic eigenvalue problem (QEP) is a special type of QEPs. It has a large gap between small and large eigenvalues in absolute value. One common way for solving QEP is to linearize the original problem via linearizations. Previous work on the accuracy of eigenpairs of not heavily damped QEP focuses on analyzing the backward error of eigenpairs relative to linearizations. The objective of this paper is to explain why different linearizations lead to different errors when computing small and large eigenpairs. To obtain this goal, we bound the backward error of eigenpairs relative to the linearization methods. Using these bounds, we build upper bounds of growth factors for the backward error. We present results of numerical experiments that support the predictions of the proposed methods.

The tropical scaling for the polynomial eigenvalue problem solved by a contour integral method

“…We now discuss why the numerical solution of R(𝜆) in the SS-RR method requires special attention. The standard way of solving small to medium size dense R(𝜆) is via L(𝜆) (6), the approximate eigenpairs of which are solved by the QZ algorithm. Then, the eigenpairs (𝜆, y) of R(𝜆) are obtained from the eigenpairs of L(𝜆).…”

“…Our test problems are taken from a variety of sources, some of which are available in the collection NLEVP. 15 The linearization L(𝜆) is constructed from (6). The eigenpairs of the linearizations are computed using MATLAB's eig function and the eigenpairs of P(𝜆) are recovered by Theorem 1.…”

“…A key question to answer is how accurate the approximate eigenpairs of the original eigenproblem are computed by the SS‐RR method. In the pioneering work, 6 the upper bounds are constructed for the backward errors of the approximate eigenpairs of the PEP relative to those of the approximate eigenpairs of linearizations. These bounds are limited for one special case of PEP, such as the heavily damped quadratic eigenvalue problem (QEP).…”

“…In Section 2, we show the proposed method (tropically scaled SS-RR method) in detail. After introducing a few key definitions at the beginning of Section 3, we construct one-sided factorization for the projected PEP and the companion linearization (6). Based on the one-sided factorization, we establish the bound for backward errors of the computed eigenpairs of the original PEP relative to those of the approximate eigenpairs of the companion linearization.…”

“…6 and A 0 , A 1 , A 2 ∈ C 1000×1000 . For the projected mod − sleeper problem,||R 0 || 2 ≈ 1.3 × 10 1 , ||R 1 || 2 ≈ 1.7 × 10 7 , and ||R 2 || 2 ≈ 1.0 × 10 0 .…”