An implementation of the QMR method based on coupled two-term recurrences
SIAM Journal on Scientific Computing
Matrix computations (3rd ed.)
Estimating the Attainable Accuracy of Recursively Computed Residual Methods
SIAM Journal on Matrix Analysis and Applications
Restarted GMRES for Shifted Linear Systems
SIAM Journal on Scientific Computing
Stability of Conjugate Gradient and Lanczos Methods for Linear Least Squares Problems
SIAM Journal on Matrix Analysis and Applications
Fast CG-Based Methods for Tikhonov--Phillips Regularization
SIAM Journal on Scientific Computing
A Symmetric Band Lanczos Process Based on Coupled Recurrences and Some Applications
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Iterative numerical methods for sampling from high dimensional Gaussian distributions
Statistics and Computing
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We consider the solution of the linear system (ATA + σI)Xσ = ATb, for various real values of σ. This family of shifted systems arises, for example, in Tikhonov regularization and computations in lattice quantum chromodynamics. For each single shift σ this system can be solved using the conjugate gradient method for least squares problems (CGLS). In literature various implementations of the, so-called, multishift CGLS methods have been proposed. These methods are mathematically equivalent to applying the CGLS method to each shifted system separately but they solve all systems simultaneously and require only two matrix-vector products (one by A and one by AT) and two inner products per iteration step. Unfortunately, numerical experiments show that, due to roundoff errors, in some cases these implementations of the multishift CGLS method can only attain an accuracy that depends on the square of condition number of the matrix A. In this paper we will argue that, in the multishift CGLS method, the impact on the attainable accuracy of rounding errors in the Lanczos part of the method is independent of the effect of roundoff errors made in the construction of the iterates. By making suitable design choices for both parts, we derive a new (and efficient) implementation that tries to remove the limitation of previous proposals. A partial roundoff error analysis and various numerical experiments show promising results.