Krylov Subspace Methods for Complex Non-Hermitian Linear Systems

摘要

Krylov subspace methods are considered for the solution of large sparse linearsystems Ax = b with complex non-Hermitian coefficient matrices. Such linear systems arise in important applications, such an inverse scattering, numerical solution of time dependent Schroedinger equations, underwater acoustics, eddy current computations, numerical computations in quantum chromodynamics, and numerical conformal mapping. Typically the resulting coefficient matrices A exhibit special structures, such as complex symmetry, or they are shifted Hermitian matrices. A Krylov subspace approach is described with iterates defined by a quasi-minimal residual property, the QMR method, for solving general complex non-Hermitian linear systems. Then, special Krylov subspace methods designed for the two families of complex symmetric respectively shifted Hermitian linear systems. Some results are also included concerning the obvious approach to general complex linear systems by solving equivalent real linear systems for the real and imaginary parts of x. Finally, numerical experiments for linear systems arising from the complex Helmholtz equation are reported.