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Truncated low‐rank methods for solving general linear matrix equations
Author(s) -
Kressner Daniel,
Sirković Petar
Publication year - 2015
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.1973
Subject(s) - mathematics , coefficient matrix , rank (graph theory) , system of linear equations , linear system , matrix (chemical analysis) , linear equation , projection (relational algebra) , numerical linear algebra , galerkin method , lyapunov function , mathematical analysis , finite element method , combinatorics , algorithm , nonlinear system , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material , thermodynamics
Summary This work is concerned with the numerical solution of large‐scale linear matrix equationsA 1 X B 1 T + ⋯ + A K X B K T = C . The most straightforward approach computes X ∈ R m × nfrom the solution of an m n × m n linear system, typically limiting the feasible values of m , n to a few hundreds at most. Our new approach exploits the fact that X can often be well approximated by a low‐rank matrix. It combines greedy low‐rank techniques with Galerkin projection and preconditioned gradients. In turn, only linear systems of size m × m and n × n need to be solved. Moreover, these linear systems inherit the sparsity of the coefficient matrices, which allows to address linear matrix equations as large as m = n = O (10 5 ). Numerical experiments demonstrate that the proposed methods perform well for generalized Lyapunov equations. Even for the case of standard Lyapunov equations, our methods can be advantageous, as we do not need to assume that C has low rank. Copyright © 2015 John Wiley & Sons, Ltd.