Convolution

## Current implementation of iterative solver in DDSCAT

DDSCAT numerical solution is based on iterative solution involving matrix times vector multiplication proposed by Goodman et al. (1990) . The basic idea of this algorithm, in case of one dimensional problem, is succintly described in
http://www.cs.utk.edu/~dongarra/etemplates/node384.html

The method is based on calculating matrix times vector by extending problem to larger one.

## Other proposals

Barrowes et al. (2001) (PDF)  proposed a new FFT-based method to expedite matrix-vector multiplies by reducing calculations to a one dimensional case. They describe their method as having a similar purpose to the FFT-based method of Goodman et al , but more general in implementation due to its ability to multiply arbitrary blocked Toeplitz matrices times a vector. This approach is similar to that of Lee (1986) .

Flatau (2004) looked at algorithm of 1d case(PDF) . His proposal was based on writting down matrix times vector multiplication in terms of the symmetric and skew-symmetric matrices. Extension to more dimensions are possible, see  and http://www.liv.ac.uk/maths/ETC/mpta/

Test Fortran codes related to this paper are available.

Bibliography
1. Barrowes B. E., Teixeira F. L., and Kong J. A., Fast algorithm for Matrix-vector multiply of asymmetric multilevel block-Toeplitz matrices in 3-D scattering. Microwave and Optical Technology Letters, 31, 28-32, 2001.
2. Flatau, P. J. Fast solvers for one dimensional light scattering in the discrete dipole approximation, Optics Express , 12 , 3149-3155, 2004.
3. Goodman, J.J., B.T. Draine, and P.J. Flatau. Application of fast- Fourier-transform techniques to the discrete-dipole approximation. Optics Letters , 16 :1198-1200, 1991.
4. Lee, D., Fast multiplication of a recursive block Toeplitz matrix by a vector and its application, Journal of complexity, 2, 295-305, 1986.
5. Ng M. K., Circulant and skew-circulant splitting methods for Toeplitz systems, 159, 101–108, 2003.