# Cross-interactive residual smoothing for global and block Lanczos-type solvers for linear systems with multiple right-hand sides

@article{Aihara2021CrossinteractiveRS, title={Cross-interactive residual smoothing for global and block Lanczos-type solvers for linear systems with multiple right-hand sides}, author={Kensuke Aihara and Akira Imakura and Keiichi Morikuni}, journal={ArXiv}, year={2021}, volume={abs/2106.00284} }

Global and block Krylov subspace methods are efficient iterative solvers for large sparse linear systems with multiple right-hand sides. However, global or block Lanczos-type solvers often exhibit large oscillations in the residual norms and may have a large residual gap relating to the loss of attainable accuracy of the approximations. Conventional residual smoothing schemes suppress the oscillations but do not aid in improving the attainable accuracy, whereas a novel residual smoothing scheme…

## Figures and Tables from this paper

## References

SHOWING 1-10 OF 26 REFERENCES

Residual Smoothing Techniques: Do They Improve the Limiting Accuracy of Iterative Solvers?

- Mathematics
- 1999

Many iterative methods for solving linear systems, in particular the biconjugate gradient (BiCG) method and its “squared” version CGS (or BiCGS), produce often residuals whose norms decrease far from…

Variants of residual smoothing with a small residual gap

- MathematicsBIT Numerical Mathematics
- 2019

Short-recurrence Krylov subspace methods, such as conjugate gradient squared-type methods, often exhibit large oscillations in the residual norms, leading to a large residual gap and a loss of…

OBLIQUE PROJECTION METHODS FOR LINEAR SYSTEMS WITH MULTIPLE RIGHT-HAND SIDES

- Mathematics
- 2005

In the present paper, we describe new Lanczos-based methods for solving nonsymmetric linear systems of equations with multiple right-hand sides. These methods are based on global oblique projections…

Generalized conjugate gradient squared

- Mathematics
- 1996

The Conjugate Gradient Squared (CGS) is an iterative method for solving nonsymmetric linear systems of equations. However, during the iteration large residual norms may appear, which may lead to…

Reliable updated residuals in hybrid Bi-CG methods

- Mathematics, Computer ScienceComputing
- 2006

This paper proposes a more restrictive strategy for accumulating groups of updates for updating the residual and the approximation, and it is shown that this may improve the accuracy significantly, while maintaining speed of convergence.

Accuracy Improvement of the Shifted Block BiCGGR Method for Linear Systems with Multiple Shifts and Multiple Right-Hand Sides

- Mathematics
- 2015

We consider solving linear systems with multiple shifts and multiple right-hand sides. In order to solve these linear systems efficiently, we develop the Shifted Block BiCGGR method. This method is…

A block version of BiCGSTAB for linear systems with multiple right-hand sides.

- Mathematics
- 2003

We present a new block method for solving large nonsymmetric linear systems of equations with multiple right-hand sides. We first give the matrix polynomial interpretation of the classical block…

Estimating the Attainable Accuracy of Recursively Computed Residual Methods

- Mathematics
- 1997

Many conjugate gradient-like methods for solving linear systems $Ax=b$ use recursion formulas for updating residual vectors instead of computing the residuals directly. For such methods it is shown…

Residual Smoothing Techniques for Iterative Methods

- Mathematics, Computer ScienceSIAM J. Sci. Comput.
- 1994

Certain relationships between the residuals and residual norms of the biconjugate gradient (BCG) and quasi-minimal residual (QMR) methods are noted, from which it follows that QMR can be obtained from BCG by a technique of this form.

Global FOM and GMRES algorithms for matrix equations

- Mathematics
- 1999

In the present paper, we present new methods for solving nonsymmetric linear systems of equations with multiple right-hand sides. These methods are based on global oblique and orthogonal projections…