Unformatted text preview: (k − 1) .
j =i This clearly suggests that the performance of the iteration can be aﬀected by
adjusting (or “relaxing”) the correction term—i.e., by replacing ck with ωck .
The resulting algorithm, xi (k ) = xi (k − 1) + ωck , is in fact (7.10.22), which
produces (7.10.21). Moreover, it was observed early on that Gauss–Seidel applied
to ﬁnite diﬀerence approximations for elliptic partial diﬀerential equations, such Copyright c 2000 SIAM Buy online from SIAM
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7.10 Diﬀerence Equations, Limits, and Summability
http://www.amazon.com/exec/obidos/ASIN/0898714540 625 as the one in Example 7.6.2 (p. 563), often produces successive corrections ck
that have the same sign, so it was reasoned that convergence might be accelerated
for these applications by increasing the magnitude of the correction factor at each
step (i.e., by setting ω > 1). Thus the technique became known as “successive
overrelaxation” rather than simply “successive relaxation.” It’s not hard to see
that ρ (Hω ) < 1 only if 0 < ω < 2 (Exercise 7.10.9), and it can be proven
that positive deﬁniteness of A is suﬃcient to guarantee ρ (Hω ) < 1 whenever
0 < ω < 2. But determining ω to minimize ρ (Hω ) is generally a diﬃcult task.
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Please report violations to [email protected] Nevertheless, there is one famous special case for which the optimal value
of ω can be explicitly given. If det (αD − L − U) = det αD − β L − β −1 U for
all real α and β = 0, and if the iteration matrix HJ for Jacobi’s method has
real eigenvalues with ρ (HJ ) < 1, then the eigenvalues λJ for HJ are related
to the eigenvalues λω of Hω by D
H (λω + ω − 1)2 = ω 2 λ2 λω .
J (7.10.23) From this it can be proven that the optimum value of ω for SOR is
ωopt = 2 and ρ Hωopt = ωopt − 1. IG
R 1 − ρ2 (HJ ) 1+ (7.10.24) Furthermore, setting ω = 1 in (7.10.23) yields ρ (HGS ) = ρ2 (HJ...
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