GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Algebraic multigrid theory: The symmetric case
Applied Mathematics and Computation - Second Copper Mountain conference on Multigrid methods Copper Mountain, Colorado
Deflation of conjugate gradients with applications to boundary value problems
SIAM Journal on Numerical Analysis
Preconditioned conjugate gradients for solving singular systems
Journal of Computational and Applied Mathematics - Special issue on iterative methods for the solution of linear systems
Implicit application of polynomial filters in a k-step Arnoldi method
SIAM Journal on Matrix Analysis and Applications
An iterative method for nonsymmetric systems with multiple right-hand sides
SIAM Journal on Scientific Computing
Matrix computations (3rd ed.)
Analysis of Projection Methods for Solving Linear Systems with Multiple Right-Hand Sides
SIAM Journal on Scientific Computing
Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods
SIAM Journal on Scientific Computing
The symmetric eigenvalue problem
The symmetric eigenvalue problem
The Geometry of Algorithms with Orthogonality Constraints
SIAM Journal on Matrix Analysis and Applications
A Deflated Version of the Conjugate Gradient Algorithm
SIAM Journal on Scientific Computing
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
GMRES with Deflated Restarting
SIAM Journal on Scientific Computing
Thick-Restart Lanczos Method for Large Symmetric Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
The block Lanczos method for linear systems with multiple right-hand sides
Applied Numerical Mathematics
A Comparative Study of Iterative Solvers Exploiting Spectral Information for SPD Systems
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Incremental spectral preconditioners for sequences of linear systems
Applied Numerical Mathematics
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We present a new algorithm that computes eigenvalues and eigenvectors of a Hermitian positive definite matrix while solving a linear system of equations with conjugate gradient (CG). Traditionally, all the CG iteration vectors could be saved and recombined through the eigenvectors of the tridiagonal projection matrix, which is equivalent theoretically to unrestarted Lanczos. Our algorithm capitalizes on the iteration vectors produced by CG to update only a small window of vectors that approximate the eigenvectors. While this window is restarted in a locally optimal way, the CG algorithm for the linear system is unaffected. Yet, in all our experiments, if the window has more than a properly chosen but small number of vectors, it converges to the required eigenvectors at a rate identical to unrestarted Lanczos. After the solution of the linear system, eigenvectors that have not accurately converged can be improved in an incremental fashion by solving additional linear systems. In this case, eigenvectors identified in earlier systems can be used to deflate, and thus accelerate, the convergence of subsequent systems. We have used this algorithm with excellent results in lattice quantum chromodynamics applications, where hundreds of right-hand sides may be needed. Specifically, about 70 eigenvectors are obtained to full accuracy after solving 24 right-hand sides. Deflating these from the large number of subsequent right-hand sides removes the dreaded critical slowdown, where the conditioning of the matrix increases as the quark mass reaches a critical value. Our experiments show almost a constant number of iterations for our method, regardless of quark mass, and speedups of 8 over original CG for light quark masses.