4.6 Article

On the Backward Error Incurred by the Compact Rational Krylov Linearization

Journal

JOURNAL OF SCIENTIFIC COMPUTING
Volume 89, Issue 1, Pages -

Publisher

SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10915-021-01625-6

Keywords

Backward error; Polynomial eigenvalue problem; Linearization; Taylor basis; Newton basis; Lagrange interpolation; Orthogonal polynomials; Chebyshev polynomials

Funding

  1. National Natural Science Foundation of China [12001262, 61963028, 11961048]
  2. Natural Science Foundation of Jiangxi Province [20181ACB20001]
  3. Double-Thousand Plan of Jiangxi Province [jxsq2019101008]
  4. Anhui Initiative in Quantum Information Technologies [AHY150200]

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This study investigates the backward errors of computed eigenpairs using the compact rational Krylov linearization method for solving PEP or REP problems. One-sided factorizations are constructed to relate the eigenpairs of the linearization and those of the original problems, providing upper bounds for the backward error of approximate eigenpairs. Numerical experiments show successful reduction of actual backward errors by scaling, with errors well predicted by the established bounds.
One of the most successful methods for solving a polynomial (PEP) or rational eigenvalue problem (REP) is to recast it, by linearization, as an equivalent but larger generalized eigenvalue problem which can be solved by standard eigensolvers. In this work, we investigate the backward errors of the computed eigenpairs incurred by the application of the well-received compact rational Krylov (CORK) linearization. Our treatment is unified for the PEPs or REPs expressed in various commonly used bases, including Taylor, Newton, Lagrange, orthogonal, and rational basis functions. We construct one-sided factorizations that relate the eigenpairs of the CORK linearization and those of the PEPs or REPs. With these factorizations, we establish upper bounds for the backward error of an approximate eigenpair of the PEPs or REPs relative to the backward error of the corresponding eigenpair of the CORK linearization. These bounds suggest a scaling strategy to improve the accuracy of the computed eigenpairs. We show, by numerical experiments, that the actual backward errors can be successfully reduced by scaling and the errors, before and after scaling, are both well predicted by the bounds.

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