Implicitly restarted global Krylov subspace methods for matrix equations AXB = C

The restarted global Krylov subspace methods are popular to solve matrix equations. Although these methods reduce storage costs, some important information is lost at the time of restart and this slows down the convergence. However, it is possible to keep some crucial information from the previous cycle and then apply them at the restart. To retain this information, it is necessary to update the starting block vector. For this purpose, we introduce the implicitly restarted global Arnoldi process that is based on the implicit double-shift QR iteration. Moreover, we develop the implicitly restarted global FOM and GMRES methods to speed up the convergence. In the mentioned methods, a starting block vector is selected so that the smallest eigenvalues in magnitude are deflated, and the corresponding approximate block Ritz vectors and harmonic Ritz vectors are augmented to the current Krylov subspace, respectively. It is demonstrated that the deflation of the smallest eigenvalues can be particularly imperative for global methods. Finally, the efficiency of these methods are appraised on two academic examples as well as on a case study in surface fitting application.