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, 20]: 1, [20, 30]: 2, [30, 50]: 1, compute the range of possible values of the variance.

4. 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.

5-6. Use calculus to find the range of a function
f(x_{1}, x_{2}) = (x_{1})^{2} -
x_{1} * x_{2} +(1/2) * (x_{2})^{2}
- x_{2} when x_{1} is in the interval [-1, 1], and
x_{2} is in the interval [-3, 3].
Use naive interval computations to estimate this range.

7. Assuming that the computer uses 2 decimal digits,
compute the range of a - b * c, where a is in the interval

[0.8, 1.2], b is in the interval [0.7, 1.1], and c is
in the interval [2.2, 3.3], with appropriate round-offs.

8-10. Estimate the range of function (1 + x) * (2 - x) on the interval [1.0, 3.0] by using the following methods:

- use the centered form;
- use the mean valued form;
- use affine arithmetic.

11. A decision maker has three alternatives:

- for
the first alternative a
_{1}, the range of possible benefits is [100.0, 200.0]; - for
the second alternative a
_{2}, the range of possible benefits is [0.0, 300.0]; - for
the third alternative a
_{3}, the range of possible benefits is [-50.0, 300.0].

- a pessimist, for whom α = 0;
- an optimist, for whom α = 1;
- a decision maker for whom α = 0.5.

12. Use the interval-based optimization algorithm
to locate the maximum of the function
f(x) = x - x^{2} on the interval [0.2, 1.0]. Divide this
interval into two, then divide the remaining intervals into two
again, etc. Stop when you can locate the maximum with
accuracy 0.1.

13-14. Use the constraints method to solve the following two problems:

- find x
_{1}and x_{2}, both from the interval [-10, 10], that satisfy the system of equations x_{1}- x_{2}= 0 and

x_{1}* (1 - x_{2}) = 0.16; - find x
_{1}and x_{2}, both from the interval [-10, 10], that satisfy the system of equations x_{1}- x_{2}= 0 and

x_{1}* (1 - x_{2}) = 1000.

15. Describe, in as many details as you can, one of class project presentations (different from your own).