An extended GS method for dense linear systems
Journal of Computational and Applied Mathematics
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In this paper an iterative scheme of first degree is developed for the purpose of solving linear systems of boundary element equations of the form $Hx = c$ where $H$ is a dense square nonsingular matrix. The iterative scheme considered is $$ (D + (\Omega H)_{sl})x^{(k + 1)} = (D - (\Omega H)_{u})x^{(k)} + \Omega c, $$ \noindent where $(\Omega H)_u$ and $(\Omega H)_{s1}$ are defined as the upper triangular and strictly lower triangular terms of $\Omega H$, respectively. The parameter matrix $\Omega$ is selected to minimize the Frobenius norm $\|D - (\Omega H)_u\|_F$. Mathematical arguments and numerical experiments are presented to show that minimizing $\|D - (\Omega H)_u\|_F$ provides for faster convergence. Numerical tests are performed for systems of boundary element equations generated by three-dimensional potential and elastostatic problems. Computation times are determined and compared against those for Gaussian elimination and Gauss--Seidel iteration.