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The Riccati algorithm for eigenvalues and invariant subspaces of matrices with inexpensive action

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
358
Identifiers
DOI: 10.1016/s0024-3795(02)00392-0
Keywords
  • Riccati
  • Jacobi–Davidson
  • Ritz–Galerkin
  • Krylov
  • Invariant Subspace
  • Orthogonal Corrections
  • Stability
Disciplines
  • Computer Science
  • Mathematics

Abstract

Abstract We present new algorithms for the numerical approximation of eigenvalues and invariant subspaces of matrices with cheap action (for example, large but sparse). The methods work with inexact solutions of generalized algebraic Riccati equations. The simpler ones are variants of Subspace Iteration and Block Rayleigh Quotient Iteration in which updates orthogonal to current approximations are computed. Subspace acceleration leads to more sophisticated algorithms. Starting with a Block Jacobi Davidson algorithm, we move towards an algorithm that incorporates Galerkin projection of the non-linear Riccati equation directly, extending ideas of Hu and Reichel in the context of Sylvester equations. Numerical experiments show that this leads to very a competitive algorithm, which we will call the Riccati method, after J.F. Riccati (1676–1754).

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