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**Efficient preconditioning for sequences of parametric complex symmetric linear systems.**
*(English)*
Zbl 1066.65048

Summary: The solution of sequences of complex symmetric linear systems of the form \(A_jx_j=b_j\), \(j=0,\dots,s\), \(A_j=A+\alpha_jE_j\), \(A\) Hermitian, \(E_0,\dots,E_s\) complex diagonal matrices and \(\alpha_0,\dots,\alpha_s\) scalar complex parameters arise in a variety of challenging problems. This is the case of time dependent partial differential equations; 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, \(\text{Re} (\alpha_j)\geq 0\) and such that the diagonal entries of \(E_j\) \(j=0,\dots,s\) have nonnegative 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.

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.

### MSC:

65F35 | Numerical computation of matrix norms, conditioning, scaling |

65N22 | Numerical solution of discretized equations for boundary value problems involving PDEs |

65F10 | Iterative numerical methods for linear systems |

65F15 | Numerical computation of eigenvalues and eigenvectors of matrices |

35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |

35K05 | Heat equation |