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Backward error analysis of the shift-and-invert Arnoldi algorithm.

Christian Schröder1, Leo Taslaman2

  • 1Institut für Mathematik, MA 4-5, Technische Universität Berlin, Berlin, Germany.

Numerische Mathematik
|June 16, 2017
PubMed
Summary
This summary is machine-generated.

This study analyzes the inexact shift-and-invert Arnoldi algorithm, revealing backward stability conditions. The research provides insights into perturbation bounds and breakdown conditions for numerical linear algebra.

Keywords:
65F2565F5065G50

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Area of Science:

  • Numerical Analysis
  • Scientific Computing
  • Linear Algebra

Background:

  • The shift-and-invert Arnoldi algorithm is crucial for eigenvalue problems.
  • Inexact computations in linear system solves and orthonormalization steps can affect algorithm stability.
  • Non-orthonormality of the computed Krylov basis is a key challenge.

Purpose of the Study:

  • To perform a backward error analysis of the inexact shift-and-invert Arnoldi algorithm.
  • To investigate the impact of inexactness in linear system solutions and orthonormalization.
  • To establish conditions for backward stability and derive a breakdown condition.

Main Methods:

  • Backward error analysis of the inexact shift-and-invert Arnoldi algorithm.
  • Consideration of inexact linear system solves and orthonormalization steps.
  • Analysis of non-orthonormal Krylov bases and perturbation bounds.
  • Implicit restarts for relaxing stability conditions.
  • Analysis extended to the Hermitian case.

Main Results:

  • The computed basis and Hessenberg matrix satisfy an exact shift-and-invert Krylov relation for a perturbed matrix.
  • Bounds for the perturbation are derived.
  • The algorithm is backward stable if the condition number of the Hessenberg matrix is not too large.
  • Implicit restarts relax the condition number requirement.
  • Hermitian backward errors are considered.

Conclusions:

  • The inexact shift-and-invert Arnoldi algorithm demonstrates backward stability under specific conditions.
  • The analysis provides a foundation for understanding and improving the algorithm's robustness.
  • A sensible breakdown condition is derived from the analysis.