Proceedings of the 2004 Asia and South Pacific Design Automation Conference
The block Lanczos method for linear systems with multiple right-hand sides
Applied Numerical Mathematics
Restarted block-GMRES with deflation of eigenvalues
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
Choice of initial guess in iterative solution of series of systems arising in fluid flow simulations
Journal of Computational Physics
International Journal of Computer Mathematics
Restarted block-GMRES with deflation of eigenvalues
Applied Numerical Mathematics - 6th IMACS International symposium on iterative methods in scientific computing
The Lanczos Method for Parameterized Symmetric Linear Systems with Multiple Right-Hand Sides
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A block IDR(s) method for nonsymmetric linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
Acceleration of randomized Kaczmarz method via the Johnson---Lindenstrauss Lemma
Numerical Algorithms
A new family of global methods for linear systems with multiple right-hand sides
Journal of Computational and Applied Mathematics
Accelerated GCRO-DR method for solving sequences of systems of linear equations
Journal of Computational and Applied Mathematics
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We analyze a class of Krylov projection methods but mainly concentrate on a specific conjugate gradient (CG) implementation by Smith, Peterson, and Mittra [IEEE Transactions on Antennas and Propogation, 37 (1989), pp. 1490--1493] to solve the linear system AX=B, where A is symmetric positive definite and B is a multiple of right-hand sides. This method 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 orthogonally onto the generated Krylov subspace to get the approximate solutions. The whole process is repeated with another unsolved system as a seed until all the systems are solved. We observe in practice a superconvergence behavior of the CG process of the seed system when compared with the usual CG process. We also observe that only a small number of restarts is required to solve all the systems if the right-hand sides are close to each other. These two features together make the method particularly effective. In this paper, we give theoretical proof to justify these observations. Furthermore, we combine the advantages of this method and the block CG method and propose a block extension of this single seed method.