Math 164 - perform two iterations of gauss-sidel to

Question

Quiz Four

use no notes or calculators.

1. Let

A =	3	2
	2	3

and

b =	5
	5

Perform two iterations of Gauss-Sidel to approximate the solution of the system Ax = b, using the initial guess

v =	0
	0

What condition on A guarantees that the iterations will converge?

2.Let

A = 16	2	-1	0
	-1	2	-1
	0	-1	2

be our favourite matrix f : R -> R be defined by f ≡ 1 and b ∈ R³ be defined by b_i = f(i/4), i = 1,2,3. Solve Ay = b. What is the mathematical significance of the solution vector y? Can you "check your answer"?

3. Consider the elliptic linear PDE -Δu = f in Ω = (0,1) x (0,1) with the boundary condition u = 0 on aa Using the central second difference formulae, write down a typical equation in the approximating (n-1)² x (n -1)² system. How would you use this to approximate a solution to the PDE?