Edu projectors if mnn bnr crn is any full rank

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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 Buy from 7.10 Difference Equations, Limits, and Summability 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. 83 It is illegal to print, duplicate, or distribute this material 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 E T 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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