Analyzing the convergence factor of residual inverse iteration

• Published in: BIT (Nordisk Tidskrift for Informationsbehandling) Vol. 51; no. 4; pp. 937 - 957
• Main Authors: Jarlebring, Elias, Michiels, Wim
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-12-2011; Springer
• Subjects: Computational Mathematics and Numerical Analysis; Exact sciences and technology; Mathematics; Mathematics and Statistics; Nonlinear algebraic and transcendental equations; Numeric Computing; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Residual inverse iteration; 65F15; Convergence factors; Nonlinear eigenvalue problems; 65H17; 15A18; Transcendental equation; Eigenvector; Numerical linear algebra; Eigenvalue; Iteration; Nonlinear problems; Convergence; Non linear equation; Numerical analysis; Algebraic equation; Eigenvalue problem; EQUATIONS; SYSTEMS; STATES; NONLINEAR EIGENVALUE PROBLEMS; ARNOLDI METHOD
• ISSN: 0006-3835; 1572-9125; 1572-9125
• DOI: 10.1007/s10543-011-0336-2

We will establish here a formula for the convergence factor of the method called
residual inverse iteration
, which is a method for nonlinear eigenvalue problems and a generalization of the well-known
inverse iteration
. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector
w
k
and we can use the formula for the convergence factor to analyze how it depends on the choice of
w
k
. We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.