Efficient preconditioning for sequences of parametric complex symmetric linear systems

Solution of sequences of complex symmetric linear systems of the form A(j)x(j)=b(j), j=0,...,s, A(j)=A+alpha E-j(j), A Hermitian, E-0,...,E-s complex diagonal matrices and alpha(0),...,alpha(s) scalar complex parameters arise in a variety of challenging problems. This is the case of time dependent PDEs; lattice gauge computations in quantum chromodynamics; the Helmholtz equation; shift-and-invert and Jacobi-Davidson algorithms for large-scale eigenvalue calculations; problems in control theory and many others. If A is symmetric and has real entries then A(j) is complex symmetric. The case A Hermitian positive semidefinite, Re(alpha(j))>= 0 and such that the diagonal entries of E-j, j=0,...,s have non negative real part is considered here. Some strategies based on the update of incomplete factorizations of the matrix A and A(-1) are introduced and analyzed. The numerical solution of sequences of algebraic linear systems from the discretization of the real and complex Helmholtz equation and of the diffusion equation in a rectangle illustrate the performance of the proposed approaches.