Abstract
We discuss the numerical solution of successive linear systems of equations \(Ax=b_i, \ i=1,2, \ldots ,m\), by iterative methods based on recycling Krylov subspaces. We propose various recycling algorithms which are based on the generalized conjugate residual (GCR) method. The recycling algorithms reuse the descent vectors computed while solving the previous linear systems \(Ax=b_j, \ j=1,2, \ldots, i-1\), such that a lot of computational work can be saved when solving the current system Ax = b i . The proposed algorithms are robust for solving sequences of linear systems arising in circuit simulation. Sequences of linear systems need to be solved, e.g., in model order reduction (MOR) for systems with many terminals. Numerical experiments illustrate the efficiency and robustness of the proposed method.
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Notes
- 1.
TITAN was then developed by Qimonda AG, Neubiberg (Germany); after insolvency of Qimonda in 2009, TITAN is now owned by Infineon Technologies AG, Neubiberg.
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Acknowledgements
This research was supported by the Alexander von Humboldt-Foundation and by the Research Network SyreNe—System Reduction for Nanoscale IC Design, funded by the German Federal Ministry of Education and Science (BMBF), grant no. 03BEPAE1. Responsibility for the contents of this publication rests with the authors. We would like to thank Timo Reis (TU Berlin) for helpful discussions on the positive realness conditions for matrices arising from RLC circuits.
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Benner, P., Feng, L. (2011). Recycling Krylov Subspaces for Solving Linear Systems with Successively Changing Right-hand Sides Arising in Model Reduction. In: Benner, P., Hinze, M., ter Maten, E. (eds) Model Reduction for Circuit Simulation. Lecture Notes in Electrical Engineering, vol 74. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0089-5_6
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