BiCGStab(ℓ) for Families of Shifted Linear Systems

We consider a seed system Ax = b together with a shifted linear system of the form

We develop modifications of the BiCGStab(ℓ) method which allow to solve the seed and the shifted system at the expense of just the matrix-vector multiplications needed to solve Ax = b via BiCGStab(ℓ). On the shifted system, these modifications do not perform the corresponding BiCGStab(ℓ)-method, but we show, that in the case that A is positive real and σ ≥ 0, the resulting method is still a well-smoothed variant of BiCG. Numerical examples from an application arising in quantum chromodynamics are given to illustrate the efficiency of the method developed.