3,255 research outputs found
Projection methods for large-scale T-Sylvester equations
The matrix Sylvester equation for congruence, or T-Sylvester equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning T-Sylvester equations is rather well understood and there are stable and e cient numerical algorithms which solve these equations for small- to medium-sized matrices. However, developing numerical algorithms for solving large-scale T-Sylvester equations still remains an open problem. In this paper, we present several projection algorithms based on di erent Krylov spaces for solving this problem when the right-hand side of the T-Sylvester equation is a low-rank matrix. The new algorithms have been extensively tested, and the reported numerical results show that they work very well in practice, o ering a clear guidance on which algorithm is the most convenient in each situation.This work has been supported by Ministerio de EconomĂa y Competitividad of Spain through grant MTM2012-32542.Publicad
Krylov methods for large-scale modern problems in numerical linear algebra
Large-scale problems have attracted much attention in the last decades since
they arise from different applications in several fields. Moreover, the matrices that
are involved in those problems are often sparse, this is, the majority of their entries
are zero. Around 40 years ago, the most common problems related to large-scale and
sparse matrices consisted in solving linear systems, finding eigenvalues and/or eigenvectors,
solving least square problems or computing singular value decompositions.
However, in the last years, large-scale and sparse problems of different natures have
appeared, motivating and challenging numerical linear algebra to develop effective
and efficient algorithms to solve them.
Common difficulties that appear during the development of algorithms for solving
modern large-scale problems are related to computational costs, storage issues and
CPU time, given the large size of the matrices, which indicate that direct methods
can not be used. This suggests that projection methods based on Krylov subspaces
are a good option to develop procedures for solving large-scale and sparse modern
problems.
In this PhD Thesis we develop novel and original algorithms for solving two
large-scale modern problems in numerical linear algebra: first, we introduce the
R-CORK method for solving rational eigenvalue problems and, second, we present
projection methods to compute the solution of T-Sylvester matrix equations, both
based on Krylov subspaces.
The R-CORK method is an extension of the compact rational Krylov method
(CORK) [104] introduced to solve a family of nonlinear eigenvalue problems that can
be expressed and linearized in certain particular ways and which include arbitrary
polynomial eigenvalue problems, but not arbitrary rational eigenvalue problems.
The R-CORK method exploits the structure of the linearized problem by representing
the Krylov vectors in a compact form in order to reduce the cost of storage,
resulting in a method with two levels of orthogonalization. The first level of orthogonalization
works with vectors of the same size as the original problem, and the
second level works with vectors of size much smaller than the original problem. Since
vectors of the size of the linearization are never stored or orthogonalized, R-CORK
is more efficient from the point of view of memory and orthogonalization costs than
the classical rational Krylov method applied to the linearization. Moreover, since
the R-CORK method is based on a classical rational Krylov method, the implementation
of implicit restarting is possible and we present an efficient way to do it, that
preserves the compact representation of the Krylov vectors.
We also introduce in this dissertation projection methods for solving the TSylvester
equation, which has recently attracted considerable attention as a consequence
of its close relation to palindromic eigenvalue problems and other applications.
The theory concerning T-Sylvester equations is rather well understood, and before the work in this thesis, there were stable and efficient numerical algorithms
to solve these matrix equations for small- to medium- sized matrices. However,
developing numerical algorithms for solving large-scale T-Sylvester equations was a
completely open problem. In this thesis, we introduce several projection methods
based on block Krylov subspaces and extended block Krylov subspaces for solving
the T-Sylvester equation when the right-hand side is a low-rank matrix. We also offer
an intuition on the expected convergence of the algorithm based on block Krylov
subspaces and a clear guidance on which algorithm is the most convenient to use in
each situation.
All the algorithms presented in this thesis have been extensively tested, and the
reported numerical results show that they perform satisfactorily in practice.Adicionalmente se recibió ayuda parcial de los proyectos de investigación: “Structured Numerical Linear Algebra: Matrix
Polynomials, Special Matrices, and Conditioning” (Ministerio de EconomĂa y Competitividad de España, NĂşmero
de proyecto: MTM2012-32542) y “Structured Numerical Linear Algebra for Constant, Polynomial and Rational Matrices” (Ministerio de EconomĂa y Competitividad de España,
NĂşmero de proyecto: MTM2015-65798-P), donde el investigador principal de ambos proyectos fue Froilán MartĂnez
Dopico.Programa Oficial de Doctorado en IngenierĂa MatemáticaPresidente: JosĂ© Mas MarĂ.- Secretario: Fernando de Terán Vergara.- Vocal: JosĂ© Enrique Román Molt
On the ADI method for the Sylvester Equation and the optimal- points
The ADI iteration is closely related to the rational Krylov projection
methods for constructing low rank approximations to the solution of Sylvester
equation. In this paper we show that the ADI and rational Krylov approximations
are in fact equivalent when a special choice of shifts are employed in both
methods. We will call these shifts pseudo H2-optimal shifts. These shifts are
also optimal in the sense that for the Lyapunov equation, they yield a residual
which is orthogonal to the rational Krylov projection subspace. Via several
examples, we show that the pseudo H2-optimal shifts consistently yield nearly
optimal low rank approximations to the solutions of the Lyapunov equations
From low-rank approximation to an efficient rational Krylov subspace method for the Lyapunov equation
We propose a new method for the approximate solution of the Lyapunov equation
with rank- right-hand side, which is based on extended rational Krylov
subspace approximation with adaptively computed shifts. The shift selection is
obtained from the connection between the Lyapunov equation, solution of systems
of linear ODEs and alternating least squares method for low-rank approximation.
The numerical experiments confirm the effectiveness of our approach.Comment: 17 pages, 1 figure
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