Recycling BiCG for families of shifted linear systems

Many problems in science and engineering fields require the solution of shifted linear systems. To solve these systems efficiently, the recycling BiCG algorithm in is extended in this paper. However, the shift-invariant property could no longer hold over the augmented Krylov subspace due to adding the recycling spaces. To remedy this situation, a strategy that constructs multiple recycling spaces is adopted and then a short-term recurrence for the solution update of the shifted system is derived when the seed system is solving. The new method not only improves the convergence but also has a potential to simultaneously compute approximate solutions for shifted linear systems using only as many matrix-vector multiplications as the one that a single system requires. In addition, some numerical experiments also confirm the efficiency of our method.