# A weighted global GMRES algorithm with deflation for solving large Sylvester matrix equations

## TL;DR

This paper introduces a novel weighted global GMRES algorithm with deflation for efficiently solving large Sylvester matrix equations, combining residual-based weighting and deflated restarting strategies.

## Contribution

It proposes a new weighted global GMRES method with deflation for large Sylvester equations, improving efficiency without changing inner products, and provides theoretical and numerical validation.

## Key findings

- The new algorithm effectively solves large Sylvester equations.
- It reduces computational costs compared to existing methods.
- Numerical examples demonstrate improved performance and accuracy.

## Abstract

The solution of large scale Sylvester matrix equation plays an important role in control and large scientific computations. A popular approach is to use the global GMRES algorithm. In this work, we first consider the global GMRES algorithm with weighting strategy, and propose some new schemes based on residual to update the weighting matrix. Due to the growth of memory requirements and computational cost, it is necessary to restart the algorithm efficiently. The deflation strategy is popular for the solution of large linear systems and large eigenvalue problems, to the best of our knowledge, little work is done on applying deflation to the global GMRES algorithm for large Sylvester matrix equations. We then consider how to combine the weighting strategy with deflated restarting, and propose a weighted global GMRES algorithm with deflation for solving large Sylvester matrix equations. Theoretical analysis is given to show why the new algorithm works effectively. Further, unlike the weighted GMRES-DR presented in [{\sc M. Embree, R. B. Morgan and H. V. Nguyen}, {\em Weighted inner products for GMRES and GMRES-DR}, (2017), arXiv:1607.00255v2], we show that in our new algorithm, there is no need to change the inner product with respect to diagonal matrix to that with non-diagonal matrix, and our scheme is much cheaper. Numerical examples illustrate the numerical behavior of the proposed algorithms.