Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: This work is concerned with the numerical solution of large-scale linear matrix equations A1XB1T++AKXBKT=C. The most straightforward approach computes XRmxn from the solution of an mn x mn 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 ll 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 x m and n x 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 ll for generalized Lyapunov equations. Even for the case of standard Lyapunov equations, our methods can be advantageous, as  do not need to assume that C has low rank. Copyright (c) 2015 John Wiley & Sons, Ltd.
['Daniel Kressner', 'Petar Sirković']
I am writing a research paper on a idea Truncated low‐rank methods for solving general linear matrix equations
[{"from": "human", "value": "I am writing a research paper on a idea Truncated low\u2010rank methods for solving general linear matrix equations"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: This work is concerned with the numerical solution of large-scale linear matrix equations A1XB1T++AKXBKT=C. The most straightforward approach computes XRmxn from the solution of an mn x mn 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 ll 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 x m and n x 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 ll for generalized Lyapunov equations. Even for the case of standard Lyapunov equations, our methods can be advantageous, as  do not need to assume that C has low rank. Copyright (c) 2015 John Wiley & Sons, Ltd."}]