Numerical recipes

866 Chapter 19. Partial Differential Equations
The Gauss-Seidel method, equation (19.5.6), corresponds to the matrix decomposition
(L + D) · x (r) = −U · x (r−1) + b (19.5.13)
The fact that L is on the left-hand side of the equation follows from the updating
in place, as you can easily check if you write out (19.5.13) in components. One
can show [1-3] that the spectral radius is just the square of the spectral radius of the
Jacobi method. For our model problem, therefore,
ρs � 1 − π2
J 2
r � pJ 2 ln 10
π 2
� 1 2
pJ
4
(19.5.14)
(19.5.15)
The factor of two improvement in the number of iterations over the Jacobi method
still leaves the method impractical.
Successive Overrelaxation (SOR)
We get a better algorithm — one that was the standard algorithm until the 1970s
—ifwemakeanovercorrection to the value of x (r) at the rth stage of Gauss-Seidel
iteration, thus anticipating future corrections. Solve (19.5.13) for x (r) , add and
subtract x (r−1) on the right-hand side, and hence write the Gauss-Seidel method as
x (r) = x (r−1) − (L + D) −1 · [(L + D + U) · x (r−1) − b] (19.5.16)
The term in square brackets is just the residual vector ξ (r−1) ,so
Now overcorrect, defining
x (r) = x (r−1) − (L + D) −1 · ξ (r−1)
x (r) = x (r−1) − ω(L + D) −1 · ξ (r−1)
(19.5.17)
(19.5.18)
Here ω is called the overrelaxation parameter, and the method is called successive
overrelaxation (SOR).
The following theorems can be proved [1-3]:
• The method is convergent only for 0