A non-stationary two-dimensional acceleration for the one-dimensional projection method [thesis]

Howard Dean Pyron
APPENDIX 3a converge or not for a given linear system depending on the spectral radius of M (22, p, 13). The Jacobl method results if A= D+ {L+ U) and the Gauss-Seldel method if A = (D + L) + U, where D is a diagonal matrix and L and U are strictly lower and upper triangular matrices, respectively. The Iteration matrix for the Jacobi method is M = -D~^(L + U) and for Gauss-Seidel is M = -(D + L)~^U. Jacobi is a totalstep method in that all components of the solution vector are changed at each
more » ... e changed at each iteration; while Gauss-Seidel is a single-step method in that only one component of the approximation to the solution vector Is changed at each iteration step. The third group is comprised of the relaxation methods, which were first suggested by Gauss (2, p. 146) and later by Southwell (20), Gauss believed that his method was particu larly effective for certain systems of linear equations which came from the normal matrix of a linear least squares fit. Although his method has never had the popularity which Gauss hoped it would have (2, p. 1^5)i It could be more popular for particular systems with the use of some of the newer interac tive display units. An advantage of the relaxation approach is that the number of iteration steps can be decreased by making the proper choice of the next residual element to eliminate at each step. However, there are no set rules or formulas to follow in making the choice; as a result, over-relaxation and under-relaxation approaches have been developed. Obtaining and j = 1, 2, 3, ..., n. 7*1 Now r^^^ -rj approaches zero as k-»oo, and [(Aj / Aj)a^ -a^] cannot be zero for a non-singular matrix, therefore we have lim (r^*^ -r^) = [(A^ / A, )a, -a J lim Ax^ = 0, k-»0Q J J 3 1 i J k-*oo J which proves the lemma.
doi:10.31274/rtd-180813-2350 fatcat:hqlnkieo5jbllopaaokqktil6m