Deflated and augmented global Krylov subspace methods for the matrix equations

• The main point of this paper is to solve AX=BAX=B, where X,B∈Rm×sX,B∈Rm×s, for the first time by the presented methods.
• Presenting deflated and augmented global OR, MR, MINRES, and GMRES type methods for solving matrix equations.
• Bring a class of algorithms (Def-Aug-Global algorithm).
• Reducing the number of iterations and CPU time in 5 examples in comparison with the global CG, MINRES and GMRES methods.
• This work for interested people in solving matrix equation in all areas of science can be helpful.

Global Krylov subspace methods are among the most efficient algorithms to solve matrix equation AX=BAX=B. Deflation and augmentation techniques are used to accelerate the convergence of Krylov subspace methods. There are two different approaches for deflated and augmented methods: an augmentation space is applied explicitly in every step, or the global method is used for solving a projected problem and then a correction step is applied at the end. In this paper, we present a framework of deflation and augmentation approaches for accelerating the convergence of the global methods for the solution of nonsingular linear matrix equations AX=BAX=B. Then, we define deflated and augmented global algorithms. Also, we analyze the deflated and augmented global minimal residual and global orthogonal residual methods. Finally, we present numerical examples to illustrate the effectiveness of different versions of the new algorithms.