A capacitance solver for incremental variation-aware extraction
Proceedings of the 2008 IEEE/ACM International Conference on Computer-Aided Design
Adaptive conjugate gradient DFEs for wideband MIMO systems
IEEE Transactions on Signal Processing
The Lanczos Method for Parameterized Symmetric Linear Systems with Multiple Right-Hand Sides
SIAM Journal on Matrix Analysis and Applications
Low-Rank Tensor Krylov Subspace Methods for Parametrized Linear Systems
SIAM Journal on Matrix Analysis and Applications
Accelerated GCRO-DR method for solving sequences of systems of linear equations
Journal of Computational and Applied Mathematics
Hi-index | 0.01 |
In this paper, we consider using conjugate gradient (CG) methods for solving multiple linear systems $A^{(i)} x^{(i)} = b^{(i)},$ for $1 \le i \le s,$ where the coefficient matrices $A^{(i)}$ and the right-hand sides $b^{(i)}$ are different in general.\ In particular, we focus on the seed projection method which generates a Krylov subspace from a set of direction vectors obtained by solving one of the systems, called the seed system, by the CG method and then projects the residuals of other systems onto the generated Krylov subspace to get the approximate solutions.\ The whole process is repeated until all the systems are solved.\ Most papers in the literature [T.\ F.\ Chan and W.\ L.\ Wan, {\it SIAM J.\ Sci.\ Comput.}, 18 (1997), pp.\ 1698--1721; B.\ Parlett {\it Linear Algebra Appl.}, 29 (1980), pp.\ 323--346; Y.\ Saad, {\it Math.\ Comp.}, 48 (1987), pp.\ 651--662; V.\ Simoncini and E.\ Gallopoulos, {\it SIAM J.\ Sci.\ Comput.}, 16 (1995), pp.\ 917--933; C.\ Smith, A.\ Peterson, and R.\ Mittra, {\it IEEE Trans.\ Antennas and Propagation}, 37 (1989), pp. 1490--1493] considered only the case where the coefficient matrices $A^{(i)}$ are the same but the right-hand sides are different.\ We extend and analyze the method to solve multiple linear systems with varying coefficient matrices and right-hand sides. A theoretical error bound is given for the approximation obtained from a projection process onto a Krylov subspace generated from solving a previous linear system. Finally, numerical results for multiple linear systems arising from image restorations and recursive least squares computations are reported to illustrate the effectiveness of the method.