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Article summary:
1. Frommer and Glassner developed a variant of the restarted GMRES method for shifted linear systems with only one matrix-vector multiplication per iteration.
2. A variant of the restarted GMRES augmented with some approximate eigenvectors for the shifted systems is presented, which can lead to faster convergence at little extra expense.
3. Numerical experiments show its efficiency.
Article analysis:
The article is generally reliable and trustworthy, as it provides evidence from numerical experiments to support its claims. The authors provide a clear explanation of their proposed method and how it differs from existing methods, making it easy to understand and follow their reasoning. Furthermore, the authors provide an overview of related work in the field, which helps to contextualize their research and demonstrate that they are aware of other approaches that have been taken in this area.
However, there are some potential biases in the article that should be noted. For example, the authors do not explore any counterarguments or alternative approaches to solving shifted linear systems, which could be beneficial for readers who may be looking for more comprehensive coverage on this topic. Additionally, while the authors provide evidence from numerical experiments to support their claims, they do not discuss any potential risks associated with their proposed method or any limitations that may arise when using it in practice. This could lead readers to overestimate the effectiveness of this approach without considering all possible outcomes or implications.