Talk:Successive over-relaxation
From Wikipedia, the free encyclopedia
The algorighm section has a k loop and also a superscript k. I don't think these k's are related, but I'm not sure. If they are not related, I think we should choose a different variable for the loop.Epachamo 00:26, 3 March 2006 (UTC)
- They are related: φ(k) denotes the guess for the solution in the kth iteration. Of course, in a real implementation, one wouldn't keep all the previous φ in memory, but only the current guess. -- Jitse Niesen (talk) 00:37, 3 March 2006 (UTC)
-
- That makes sense. Thanks! Epachamo 05:39, 3 March 2006 (UTC)
[edit] for future reference
in case anyone gets up the gumption to add a section on convergence analysis, rewriting the iteration:
- (D + ωL)φ(k + 1) = ( − ωU + (1 − ω)D)φ(k) + ωb
in the corrector form:
- φ(k + 1) = φ(k) + ω(D + ωL) − 1r(k)
is useful. (the residual is r(k) = b − Aφ(k).) since the error and residual are related by Ae = r, we get:
- e(k + 1) = (I − ω(D + ωL) − 1A)e(k).
also, perhaps SSOR should be made into a redirect that points here, and a section should be added. that is, SSOR is just two applications of different flavors of SOR one right after the other. one, in the "forward" direction, is as above; the other, in the "backward" direction, is as above but with U in place of L. the two iterations of SOR can be combined in a multiplicative (the usual) or additive fasion. even though SSOR is usually slower than SOR (each with its own optimal damping ω; see, for instance, Young's "Iterative solution of large linear systems," p462, 1971), there is the advantage that SSOR is a symmetric procedure and allows it to be used as a preconditioner in methods that respect symmetry. Lunch 05:52, 15 September 2006 (UTC)
[edit] Rewrite sentence?
Quote: As in the Gauss–Seidel method, it is not necessary to solve a linear system in order to implement the iteration (∗); indeed, given that the goal is to solve the linear system Aφ = b in the first place, it would be silly to use an iterative method in which a linear system must be solved in each step.
(*) has no "real" linear system, since the matrix on the left side is a (non-strict) lower triangular matrix, so the usual stepwise substitution can be used - this sentence implies something different. Change? -- anonymous
- I think it's quite clear, but that doesn't mean it can't be improved. Go for it. -- Jitse Niesen (talk) 01:27, 2 February 2007 (UTC)
