## Interval Computations, Test 1 for the course CS 5351/CS 4365, Fall 2010

Name: ___________________________________________________________

1. What is the main problem of interval computations? Why is this problem useful in practice?

2-3. Given the following number of people in each age bracket: [0, 50]: 1, [60, 80]: 2, [80, 90]: 1, compute the range of possible values of the variance.

4-7. We want to estimate the range of a function (1 - x) * (2 + x) on the interval [0,3]. Do the following:

• find the exact range by using calculus;
• estimate the range by applying naive (straightforward) interval computations to the original interval;
• use bisection and apply naive interval computations to both sub-intervals;
• use bisection and check monotonicity of the function on each of the sub-intervals, applying naive interval computations only if the function is not monotonic.

8-9. Give two numerical examples when an optimizing compiler helps improve the interval estimate, and a numerical example when an optimizing compiler makes the excess width worse. Hint: what are the Single Use Expressions (SUE), and how are they related to interval computations?

10. Use calculus to find the range of a function f(x1, x2) = (x1)2 - x1 * x2 +(1/2) * (x2)2 - x2 when x1 is in the interval [-1, 1], and x2 is in the interval [-2, 2]. For extra credit: use naive interval computations to estimate this range.