Numerical Methods Course Notes Version 0.1 (UCSD Math 174, Fall ...

42 CHAPTER 3. SOLVING LINEAR SYSTEMS
Example Problem 3.9. Find conditions on ω which guarantee convergence of Richardson’s Iteration
for finding approximate iterative solutions to the system Ax = b, where
⎡
A = ⎣
6 1 1
2 4 0
1 2 6
⎤
⎡
⎦ , b = ⎣
12
0
6
⎤
⎦ .
Solution:
By Theorem 3.8, with Q the identity matrix, we have convergence if and only if
‖I − ωA‖ 2
< 1
We now use the fact that “eigenvalues commute with polynomials;” that is if f(x) is a polynomial
and λ is an eigenvalue of a matrix A, then f(λ) is an eigenvalue of the matrix f(A). In this case the
polynomial we consider is f(x) = x 0 −ωx 1 . Using octave or Matlab you will find that the eigenvalues
of A are approximately 7.7321, 4.2679, and 4. Thus the eigenvalues of I − ωA are approximately
1 − 7.7321ω, 1 − 4.2679ω, 1 − 4ω.
With some work it can be shown that all three of these values will be less than one in absolute
value if and only if
0 < ω < 3
7.7321 ≈ 0.388
(See also Exercise (3.10).)
Compare this to the results of Example Problem 3.4, where for this system, ω = 1 apparently did
not lead to convergence, while for Example Problem 3.5, with ω = 1/6, convergence was observed.
⊣
3.4.8 A Free Lunch?
The analysis leading to Theorem 3.8 leads to an interesting possible variant of the iterative scheme.
For simplicity we will only consider an alteration of Richardson’s Iteration. In the altered algorithm
we presuppose the existence, via some oracle, of a sequence of weightings, ω i , which we use in each
iterative update. Thus our algorithm becomes:
1. Select some initial iterate x (0) .
2. Given iterate x (k−1) , define
x (k) = (I − ω k A) x (k−1) + ω k b.
Following the analysis for Theorem 3.8, it can be shown that
e (k) = (I − ω k A) e (k−1)
where, again, e (k) = x (k) − x, with x the actual solution to the linear system. Expanding e (k−1)
similarly gives
e (k) = (I − ω k A) (Iω k−1 A) e (k−2)
( k∏
)
= I − ω i A e (0) .
i=1