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

3.4. ITERATIVE SOLUTIONS 45
is the matrix
⎡
⎢
⎣
1 0 0 · · · 0
−a 2 1 0 · · · 0
−a 3 0 1 · · · 0
. . .
. .. .
−a n 0 0 · · · 1
(Hint: Multiply them together.)
(3.7) Under the strategy of scaled partial pivoting, which row of the following matrix will be the
first pivot row?
⎡
⎤
10 17 −10 0.1 0.9
−3 3 −3 0.3 −4
⎢ 0.3 0.1 0.01 −1 0.5
⎥
⎣ 2 3 4 −3 5 ⎦
10 100 1 0.1 0
(3.8) Let A be a symmetric positive definite n × n matrix with n distinct eigenvalues. Letting
y (0) = b/ ‖b‖ 2
, consider the iteration
y (k+1) =
Ay(k) ∥ Ay (k) ∥ ∥
2
.
(a) What is ∥ ∥ y
(k) ∥ ∥
2
?
(b) Show that y (k) = A k b/ ∥ ∥ A k b ∥ ∥
2
.
(c) Show that as k → ∞, y (k) converges to the (normalized) eigenvector associated with the
largest eigenvalue of A.
(3.9) Consider the equation
⎡
⎣
1 3 5
−2 2 4
4 −3 −4
⎤
⎦ x = ⎣
Letting x (0) = [1 1 0] ⊤ , find the iterate x (1) by one step of Richardson’s Method. And by
one step of Jacobi Iteration. And by Gauss Seidel.
(3.10) Let A be a symmetric n × n matrix with eigenvalues in the interval [α, β], with 0 < α ≤ β,
and α + β ≠ 0. Consider Richardson’s Iteration
⎡
⎤
⎥
⎦
−5
−6
10
x (k+1) = (I − ωA) x (k) + ωb.
Recall that e (k+1) = (I − ωA) e (k) .
(a) Show that the eigenvalues of I − ωA are in the interval [1 − ωβ, 1 − ωα].
(b) Prove that
max {|λ| : 1 − ωβ ≤ λ ≤ 1 − ωα}
is minimized when we choose ω such that 1 − ωβ = − (1 − ωα) . (Hint: It may help to
look at the graph of something versus ω.)
(c) Show that this relationship is satisfied by ω = 2/ (α + β).
(d) For this choice of ω show that the spectral radius of I − ωA is
|α − β|
|α + β| .
⎤
⎦