Recycling BiCG for families of shifted linear systems

Many problems in science and engineering fields require the solution of shifted linear systems. To solve such systems efficiently, the recycling BiCG (RBiCG) algorithm in [SIAM J. SCI. COMPUT, 34 (2012) 1925-1949] 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 to enforce the collinearity condition on the shifted system 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 solution of a single system requires. In addition, some numerical experiments also confirm the efficiency of our method.