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MAT 421 - MODULE 2 test – Summer 2019 – Welfert

c 2019 Arizona State University School of Mathematics & Statistics
MAT 421 MODULE 2 test
Summer 2019 NAME:

1. Check due date and time on Canvas.
2. This exam consists of 4 problems.
3. Access to all Canvas course material, personal courses notes, or Matlab is authorized.
4. Provide complete and well-organized answers. Explain how you get your answer. Circle your final
   answer. Show your work!
5. Write directly on a copy of this exam. Include additional pages as needed.
6. Upload a PDF copy of your completed and signed exam (including the front page) to Canvas by the
   due date/time. Make sure your scan is clear and legible.
7. HONORS statement: I certify that I have received no help from anyone during this test.
   Signature:
   1
   MAT 421 - MODULE 2 test – Summer 2019 – Welfert

c 2019 Arizona State University School of Mathematics & Statistics
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 converge? Justify.
c) [by Matlab] Verify by Matlab. What is x? How many iterations are needed to converge for TOL =
10-6
?
2
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 -
v
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 ?1?
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

?
5

Sample Solution