Volume 62, pp. 138-162, 2024.
A class of Petrov-Galerkin Krylov methods for algebraic Riccati equations
Christian Bertram and Heike Faßbender
Abstract
A class of (block) rational Krylov-subspace-based projection methods for solving the
large-scale continuous-time algebraic Riccati equation (CARE)
with a large, sparse , and and of full low rank is proposed.
The CARE is projected onto a block rational Krylov subspace spanned by blocks of the form for some shifts The considered projections do not need to be orthogonal and are built from the matrices appearing in the block rational Arnoldi decomposition associated to The resulting projected Riccati equation is solved for the small square Hermitian Then the Hermitian low-rank approximation to is set up where the columns of span
The residual norm can be computed efficiently via the norm of a readily available matrix. We suggest reducing the rank of the approximate solution even further by truncating small eigenvalues from This truncated approximate solution can be interpreted as the solution of the Riccati residual projected to a subspace of This gives us a way to efficiently evaluate the norm of the resulting residual. Numerical examples are presented.
Full Text (PDF) [1.1 MB],
BibTeX
, DOI: 10.1553/etna_vol62s138
Key words
algebraic Riccati equation, large-scale matrix equation, (block) rational Krylov subspace, projection method
AMS subject classifications
15A24, 65F15
Links to the cited ETNA articles
[2] |
Vol. 42 (2014), pp. 147-164 Peter Benner and Tobias Breiten:
Rational interpolation methods for symmetric Sylvester equations
|
[18] |
Vol. 33 (2008-2009), pp. 53-62 M. Heyouni and K. Jbilou:
An extended block Arnoldi algorithm for large-scale solutions of the continuous-time algebraic Riccati equation
|