Solution of dense systems of linear equations in the discrete-dipole approximation

The discrete-dipole approximation (DDA) is a method for calculating the scattering of light by an irregular particle. The DDA has been used, for example, in calculations of optical properties of cosmic dust. In this method the particle is approximated by interacting electromagnetic dipoles. Computationally the DDA method includes the solution of large dense systems of linear equations where the coefficient matrix is complex symmetric. In this work, the linear systems of equations are solved by various iterative methods. QMR was found to be the best iterative method in this application. It converged in only a few more iterations than the full generalized minimal residual (GMRES) method. When the discretization of the particle was refined, the number of iterations remained constant even without preconditioning. The matrix-vector product in the iterative methods can be computed with the fast Fourier transform or the fast multipole algorithm. These algorithms make it feasible to solve dense linear systems of hundreds of thousands of unknowns.