Abstract <p>A block extension of the generalized minimal residual method (GMRES) with a new block deflation technology is presented. Unlike currently known methods, a block can be deflated not only when it has degenerated but also when some residuals converge with the required accuracy or when the residuals become linearly dependent with a given accuracy. In addition, the method allows continuing the process when adding new right-hand sides. In this case, after the block deflation and adding new right-hand sides, the method retains a compact form and low complexity. This allows it to be used in the case when not all right-hand sides are known in advance. It also makes it possible to limit the maximum block size, thus balancing between performance and the final space dimension, i.e. the required memory. Numerical experiments confirm the high efficiency of the proposed method compared to the nonblock extension of GMRES and its naïve block generalization.</p>