Problem 1 (25 points). Consider the system Ax = b with A =
1 −1
−1 3
and b =
1
1
.
a) [by hand] Perform 2 Gauss-Seidel iterations starting with x0 =
0
0
. What are x1 and x2?
b) [by hand] Is the iteration guaranteed to converg
MAT 421 – MODULE 2 test – Summer 2019 – Welfert
c 2019 Arizona State University School of Mathematics & Statistics
Problem 2 (25 points). Consider the system Ax = b with A =
1 −1
−1 3
and b =
1
1
.
a) [by hand] Explain why the conjugate gradient (CG) algorithm converges in at most 2 iterations, regardless of the choice of x0.
b) [by Matlab] Solve the system using CG starting with x0 =
0
0
. What is x1? What is x2?
c) [by hand] Solve the system using CG starting with x0 =
1
2 −
√
2
. What is x1?
3
MAT 421 – MODULE 2 test – Summer 2019 – Welfert
c 2019 Arizona State University School of Mathematics & Statistics
Problem 3 (25 points). Let A =
1 4
−1 6
.
a) [by hand] Apply one iteration of inverse shifted iteration with shift µ = 6, starting with y =
0
1
. What
are x1 and λ1?
b) [by hand] Towards which value should the eigenvalue estimate converge? Justify.
c) [by Matlab] Implement the method in Matlab with TOL = 10−10
.
4
MAT 421 – MODULE 2 test – Summer 2019 – Welfert
c 2019 Arizona State University School of Mathematics & Statistics
Problem 4 (25 points). Consider the nonlinear system (
(a − 1)2 + b
2 = 1
a
2 + b
2 = 2
and set x :=
a
b
.
a) [by hand] Sketch the two sets of points corresponding to each equation. How many solutions does the
system have? What are the solutions?
b) [by hand] Apply 2 iterations of Newton’s method starting with x0 =
0
1
. What is x2?
c) [by Matlab] Verify convergence wth Matlab using TOL = 10−8
.
d) [hand or Matlab] What happens if x0 =
1
0
?
Sample Solution
