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Quiz Four
use no notes or calculators.
1. Let
and
Perform two iterations of Gauss-Sidel to approximate the solution of the system Ax = b, using the initial guess
What condition on A guarantees that the iterations will converge?
2.Let
be our favourite matrix f : R -> R be defined by f ≡ 1 and b ∈ R3 be defined by bi = 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?
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