The C-method is an exact method for analyzing gratings and rough surfaces. This method leads to large-size dense complex non-Hermitian eigenvalue. In this paper, we introduce a parallel QR algorithm that is specifically designed for the C-method. We define the ``early shift'' for the matrix according to the observed properties. We propose a combination of the ``early shift'', Wilkinson's shift and exceptional shift together to accelerate convergence. First, we use the ``early shift'' in order to have quick deflation of some eigenvalues. The multi-window bulge chain chasing and parallel aggressive early deflation are used. This approach ensures that most computations are performed in level 3 BLAS operations. The aggressive early deflation approach can detect deflation much quicker and accelerate convergence. Mixed MPI-OpenMP techniques are used for performing the codes to hybrid shared and distributed memory platforms. We validate our approach by comparison with experimental data for scattering patterns of two-dimensional rough surfaces.
2. Li, L. and J. Chandezon, "Improvement of the coordinate transformation method for surface-relief gratings with sharp edges," J. Opt. Soc. Am. A, 2247-2255, 1996.
3. Granet, G., "Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: Application to multilayer crossed gratings," J. Opt. Soc. Am. A, Vol. 15, 1121-1131, 1998.
4. Ait Braham, K., R. Dusseaux, and G. Granet, "Scattering of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces --- Study with the curvilinear coordinate method," Waves Random Complex Media, Vol. 18, 255-274, 2008.
5. Dusseaux, R., K. Ait Braham, and G. Granet, "Implementation and validation of the curvilinear coordinate method for the scattering of electromagnetic waves from two-dimensional dielectric random rough surfaces," Waves Random Complex Media, Vol. 18, 551-570, 2008.
6. Dusseaux, R., E. Vannier, O. Taconet, and G. Granet, "Study of backscatter signature for seedbed surface evolution under rainfall --- Influence of radar precision," Progress In Electromagnetics Research, Vol. 125, 415-437, 2012.
7. Elfouhaily, T. M. and C. A. Guerin, "A critical survey of approximate scattering wave theories from random rough surfaces," Waves in Random and Complex Media, Vol. 14, R1-10, 2004.
8. Bai, Z., J. W. Demmel, J. J. Dongarra, A. Ruhe, and H. van Der Vorst, Templates for the Solution of Algebraic Eigenvalue Problems. Software, Environments, and Tools, SIAM, 2000.
9. Golub, G. H. and F. Uhlig, "The QR algorithm: 50 years later its genesis by John Francis and Vera Kublanovskaya and subsequent developments," IMA J. Numer. Anal., Vol. 29, 467-485, 2009.
10. Golub, G. H. and C. F. Van Loan, Matrix Computations, Johns Hopkins University Press, Baltimore, 1996.
11. Braman, K., R. Byers, and R. Mathias, "The multi-shift QR algorithm. Part I: Maintaining well-focused shifts and level 3 performance," SIAM J. Matrix Anal. Appl., Vol. 23, 929-947, 2002.
12. Granat, R., B. Kagstrom, and D. Kressner, "A novel parallel QR algorithm for hybrid distributed memory HPC systems," SIAM J. Sci. Comput., Vol. 32, 2345-2378, 2010.
13. Braman, K., R. Byers, and R. Mathias, "The multi-shift QR algorithm. Part II: Aggressive early deflation," SIAM J. Matrix Anal. Appl., Vol. 23, 948-972, 2002.
14. MPI --- Messaging passing interface, See http://www.mcs.anl.gov/research/projects/mpi/.
15. OpenMP --- Open multi-processing, See http://openmp.org/wp/.
16. BLAS --- Basic linear algebra subprograms, See http://www.netlib.org/blas/.
17. ScaLAPACK --- Scalable linear algebra package, See http://www.netlib.org/scalapack/.
18. Kong, J. A., K. H. Ding, and C. O. Ao, Scattering of Electromagnetic Waves --- Numerical Simulations, Wiley-Interscience, New York, 2001.
19. Afifi, S. and R. Dusseaux, "Scattering by anisotropic rough layered 2D interfaces," IEEE Trans. Antennas Propag., Vol. 60, 5315-5328, 2012.
20. Berginc, G., "Small-slope approximation method: A further study of vector wave scattering from two-dimensional surfaces and comparison with experimental data," Progress In Electromagnetics Research, Vol. 37, 251-287, 2002.
21. Johnson, J. T., L. Tsang, R. T. Shin, K. Pak, C. H. Chan, A. Ishimaru, and Y. Kuga, "Backscattering enhancement of electromagnetic waves from two-dimensional perfectly conducting random rough surfaces: A comparison of Monte-Carlo simulations with experimental data," IEEE Trans. Antennas Propag., Vol. 44, 748-756, 1996.