Ruhe's rational Krylov method is a popular tool for approximating eigenvalues of a given matrix, though its convergence behavior is far from being fully understood. Under fairly general assumptions we characterize in an asymptotic sense which eigenvalues of a Hermitian matrix are approximated by rational Ritz values and how fast this approximation takes place. Our main tool is a constrained extremal problem from logarithmic potential theory, where an additional external field is required for taking into account the poles of the underlying rational Krylov space. Several examples illustrate our analytic results.

MSC codes

  1. 15A18
  2. 31A05
  3. 31A15
  4. 65F15


  1. rational Krylov
  2. Ritz values
  3. orthogonal rational functions
  4. logarithmic potential theory

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Information & Authors


Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 1740 - 1774
ISSN (online): 1095-7162


Submitted: 9 April 2009
Accepted: 12 January 2010
Published online: 17 March 2010

MSC codes

  1. 15A18
  2. 31A05
  3. 31A15
  4. 65F15


  1. rational Krylov
  2. Ritz values
  3. orthogonal rational functions
  4. logarithmic potential theory



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