Math 1080: Homework #5 solution

Description

Problem 1:
Find the absolute and relative condition number for the following problems. Comment on the values of x for
which the problem would be considered well-conditioned or ill-conditioned.
a) 𝑓(𝑥) = (ln 𝑥)
2
b) 𝑓(𝑥) = ‖𝑥‖2 = √∑ 𝑥𝑖
𝑛 2
𝑖=1
c) 𝑓(𝐴) = [𝑡𝑟𝑎𝑐𝑒(𝐴) det(𝐴)] for 2×2 matrix A
Use ‖. ‖∞ norm in the formula for the condition number and treat the input as a vector of dimension 4, so your
Jacobian becomes a 2 x 4 matrix.
d) 𝑓(𝑥, 𝑦) = [
𝑥𝑦 𝑥
2
𝑦
2 𝑥𝑦]
Use ‖. ‖1 norm in the formula for the condition number and treat the output as a vector of dimension 4, so your
Jacobian becomes a 4 x 2 matrix.
Problem 2:
Determine whether the following algorithms are backward stable, stable, or unstable:
a) Computation of
2 2
f ( x, y ) = x − y
as 𝑓̃(𝑥, 𝑦) = [𝑓𝑙(𝑥) ⊗ 𝑓𝑙(𝑥)] ⊖ [𝑓𝑙(𝑦) ⊗ 𝑓𝑙(𝑦)]
b) Computation of
2 2
f ( x, y ) = x − y
as 𝑓̃(𝑥, 𝑦) = [𝑓𝑙(𝑥) ⊕ 𝑓𝑙(𝑦)] ⊗ [𝑓𝑙(𝑥) ⊖ 𝑓𝑙(𝑦)]
c) Computation of 𝑓(𝑥) = 1⁄(1 + 𝑥) as 𝑓̃(𝑥, 𝑦) = 1 ⊘ [1 ⊕ 𝑓𝑙(𝑥)]
Problem 3:
Determine the accuracy of the algorithms in Problem 2. Which of the algorithms a) and b) is more accurate?