A stabilized QMR version of block BICG

Block BICG (BBICG) is an appealing method for solving AX = B with A epsilon R(nxn) and X, B epsilon R(nxs). Because of its short-term recurrence form, memory allocation and computational cost do not depend on additional parameters. Unfortunately, loss of orthogonality prevents convergence in many cases. We present a new version of the algorithm that generates blocks of vectors that are vector-wise A-biorthogonal; moreover, a near-breakdown safeguard strategy inside the block stabilizes the computation of the coefficients. In order to smooth the possibly erratic behavior of the residual norm curve, the approximate solution is determined using a block QMR procedure. The new method considerably improves the robustness of the original algorithm, showing very good performance on dense or preconditioned matrices over both BBICG and the single right-hand side solver coupled two-term QMR method applied on each system.