We consider the convergence of the algorithm GMRES of Saad and Schultz for solving linear equations Bx=b, where B ∈ ℂn×n is nonsingular and diagonalizable, and b ∈ ℂn. Our analysis explicitly includes the initial residual vector r0. We show that the GMRES residual norm satisfies a weighted polynomial least-squares problem on the spectrum of B, and that GMRES convergence reduces to an ideal GMRES problem on a rank-1 modification of the diagonal matrix of eigenvalues of B. Numerical experiments show that the new bounds can accurately describe GMRES convergence.
- convergence analysis
- iterative methods
- linear systems
- Krylov subspace methods
Titley-Peloquin, D., Pestana, J., & Wathen, A. J. (2014). GMRES convergence bounds that depend on the right-hand-side vector. IMA Journal of Numerical Analysis , 34(2), 462-479. https://doi.org/10.1093/imanum/drt025