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The conditions for the convergence of the simultaneous over-relaxation method
are found via Kahan's Theorem (Equaton 3.10). Note that
Since
, (
are eigenvalues of H), and the spectral radius is defined by
,
![$\displaystyle \prod_i \lambda_i = det(H)=(1-w)^n \leq max\vert\lambda_i\vert^n \Rightarrow \rho(H)\geq\vert w-1\vert$](img50.png) |
(3.10) |
Now the Fundamental Theorem of Linear Iterative Methods (see final section) tells us we
have convergence if
, and since
,
Obviously
is Jacobi-GS, and
is foolish, so we consider
. It is now
our task to find which value in this region is most efficient.
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Timothy Jones
2006-02-24