a) Determine the LU factorization of A, keeping 3 significant (decimal) digits in all calculations.
b) Describe the numerical issue in computing the coefficient U22.
c) What is the condition number κ∞(A) of A?
d) How many digits of accuracy could we trust in the solution x of Ax = b using the above factorization?
Would pivoting help?
2
MAT 421 – MODULE 1 test – Summer 2019 – Welfert

c 2019 Arizona State University School of Mathematics & Statistics
Problem 2 (25 points). For fixed x, consider the quantity
Q(x, h) = sin(x + h) − sin(x)
h
.
a) What is the limit (x) of Q(x, h) as h → 0? b) What is the optimal value of h > 0 to obtain the best accuracy in Q(x, h) on a computer with machineepsilon ε? What is the accuracy obtained (in terms of ε)? c) Plot |(x) − Q(x, h)| vs. h (in log-log axes) for 10−16 ≤ h ≤ 1 and i) x = π; ii) x = π/2 (you can plot
them on the same graph). Verify b).
3
MAT 421 – MODULE 1 test – Summer 2019 – Welfert

c 2019 Arizona State University School of Mathematics & Statistics
Problem 3 (25 points). A matrix A has thin SVD A =



√
1
3
√
1
2
√
1
3
− √
1
2
√
1
3
0



√
6 0
0 2
“
√
1
2
− √
1
2
√
1
2
√
1
2

T

a) [by hand] Verify that the given product is indeed a SVD. What is A explicitly?
b) What is the best rank-one approximation of A?
4
MAT 421 – MODULE 1 test – Summer 2019 – Welfert

c 2019 Arizona State University School of Mathematics & Statistics
Problem 4 (25 points). Consider the 4 points (0, 1), (2, 1), (2, 3), (4, 3).
a) Write the (overdetermined) linear system Ax = b arising from the linear regression problem (i.e., fit a
straight line).
b) [hand+Matlab] Determine a QR factorization of the system matrix A.
c) Use the factorization to solve the linear regression (least-squares) problem.
d) Sketch the regression line. Determine the coefficient of determination R2
.
5