1. (Due September 12) Use calculus to find the range of a function (4 + x) * (2 − x) on the interval [0, 6].

Solution to Homework 1

2. (Due September 13) Use calculus to find the range of a function (2 − x₁) * (4 + x₂) when x₁ is in the interval [0, 6] and x₂ is in the interval [−1, 6].

Solution to Homework 2

3. (Due September 18) Linearize the expression from Problem 1 around the interval's midpoint. Use the linearized expression to find the approximate value of the range of the original function, both with the actual derivative and with the result of numerical differentiation.

Solution to Homework 3

4. (Due September 18) Linearize the expression from Problem 2 around the intervals' midpoints. Use the linearized expression to find the approximate value of the range of the original function, both with the actual derivative and with the result of numerical differentiation.

Solution to Homework 4

5. (Due September 25) Write and test a program that estimates the range by using numerical differentiation, and a program that uses Cauchy techniques. Test it on an example of a simple function on a narrow box, e.g., for f(x₁, x₂) = x₁² − x₂² when x₁ is in [0.9, 1.1] and x₂ is in [0.8, 1.2]. For this example, you can find the exact range by using monotonicity.

In general, for each programming assignment, submit a softcopy of the program, and a softcopy of the testing results.

6. (Due September 25) Use straightforward interval computations to estimate the range of the functions from Problems 1 and 2.

Solution to Homework 6

7. (Due September 27) Write a program that implements all operations of interval arithmetic: addition, subtraction, multiplication, 1/x, and division. Submit printouts of the code and of the test results, be prepared to show how the program works if needed.

Comments:

• the range of 1/x is only bounded when the set of possible values of x does not contain 0; thus, the inverse interval 1/[a,b] is only defined when the interval [a,b] does not contain 0;
• the main need for interval arithmetic operations comes from naive interval computations, where the result of one operation are used as inputs for another interval arithmetic operation; in view of this, please program interval arithmetic operations so that they return the interval, not just print its endpoints; in an object-oriented programming language like Java, the most convenient way is to return an interval as an object;
• endpoints come from measurements; we saw many examples in class when endpoints are not integers; please make sure that your operations are defined for real numbers, not just for integers.

8. (Due October 2) Check and, if possible, use monotonicity to estimate the ranges of the functions from Problems 1 and 2.

Solution to Homework 8

9. (Due October 16) Use monotonicity and one iteration of bisection to estimate the range of the function from Problem 1 when x is in the interval [−4,0].

Solution to Homework 9

10. (Due October 16) Use monotonicity and one iteration of bisection to estimate the range of the function from Problem 2 when x₁ is in the interval [0,6] and x₂ is in the interval [−6,6].

Solution to Homework 10

11. (Due October 18) Use full interval algorithm -- monotonicity, centered form, and bisection (one iteration) -- to estimate the range of the function from Problem 1 when x is in the interval [−4,0].

Solution to Homework 11

12. (Due October 23) Use full interval algorithm -- monotonicity, centered form, and bisection (one iteration) -- to estimate the range of the function from Problem 2 when x₁ is in the interval [0,6] and x₂ is in the interval [−6,6].

Solution to Homework 12

13. (Due October 30) Write and test a program that implements the full interval algorithm. This program should compute an enclosure for the range of any given function on any given box by:

• first checking for monotonicity and then
• using the centered form -- either for the original function or (if the given function is monotonic) for the corresponding functions of one fewer variable.

  public static Interval partial(Interval[] inputs, int i){}

14. (Due November 13) Use the interval-based optimization algorithm to locate the maximum of the function f(x) = 2x + 4x² on the interval [−0.4, 0]. Divide this interval into two, then divide the remaining intervals into two again, etc. At each iteration, dismiss subintervals where maximum cannot be attained. Stop when you get intervals of width 0.1.

Solution to Homework 14

15. (Due November 15) Use the constraints method to solve the following two problems:

• find x₁ from the interval [0, 4] and x₂ from the interval [−2, 0], for which x₁ + x₂ = 0 and x₁ * x₂ = −1.0;
• find x₁ and x₂ from the same intervals for which x₁ + x₂ = 0 and x₁ * x₂ = −20.0.

Solution to Homework 15

16. (Due November 20) Write a method for simulating the propagation of random errors. We assume that:

• we have a function f(x₁, ...,x_n) -- given, as usual, as a method that takes an array x and transforms it into a real number;
• we also have the measurement results X₁, ..., X_n, and
• we have standard deviations d₁, ..., d_n.

In your program, first, you apply f to the measurement results, getting Y = f(X₁, ..., X_n). Then, you select the number of iterations N (e.g., N = 50).

After that, for k = 1, ..., N, you do the following:

• use the random number generator (RNG) to generate n numbers r₁, ..., r_n which are normally distributed with 0 mean and standard deviation 1;
  one way to generate each of these n random numbers r_i is to do the following:
  • 12 times run the RNG that generates numbers uniformly distributed on the interval [0, 1],
  • subtract 0.5 from each of 12 values and
  • add the resulting 12 differences;
• compute dy_k = f(x₁ + d₁ * r₁, ... x_n + d_n * r_n) − Y.

Test this program on some example. The resulting value should be close to the square root of the sum (c₁ * d₁)² + ... + (c_n * d_n)², where, as before, c_i is the partial derivative of f with respect to x_i.

17. (Due November 20) Write down:

• an expression for which an optimizing compiler improves the estimates provided by straightforward interval computations, and
• an expression for which an optimizing compiler worsens improves the estimates provided by straightforward interval computations.

Solution to Homework 17

18. (Due November 20) If we only use numbers with 2 digit after the decimal point, and we use rounding that preserves guaranteed bounds, what will be the result of multiplying the intervals [0.20, 0.69] and [0.39, 0.61]?

Solution to Homework 18

19. (Due November 22) If out of 10 experts, 9 think that the statement is true, what degree of confidence shall we assign to this statement?

Solution to Homework 19

20. (Due November 22) If an expert marked her confidence in a statement as 6 on a scale from 0 to 8, what degree of confidence shall we assign to this statement?

Solution to Homework 20

21. (Due November 22) If we have μ(1)=0.2 and μ(2)=0.8, what value shall we assign to μ(1.7)? Use linear interpolation.

Solution to Homework 21

22. (Due November 22) If the expert's degrees of confidence in two statements A and B are 0.7 and 0.9, and we use min as an "and"-operation and max as an "or"-operation, what degree of confidence shall we assign to A & B? To A \/ B? If a statement C has degree of confidence 0.8, what is our degree of belief in (A & B) \/ C?

Solution to Homework 22

23. (Due November 29) For a membership function μ(x) = 1 − |x|, what are the α-cuts corresponding to α = 0.6? to α = 0.4?

Solution to Homework 23

24. (Due November 29) Let us assume that the quantity x is described by the membership function μ(x) = max(1 − |1 − x|, 0) and the quantity y is described by the membership function μ(y) = max(1 − |− 1 − y|, 0). Use the values α = 0.25, 0.5, 0.75, and 1.0 to describe membership functions for z = x − y and t = x * y.

Solution to Homework 24

25. (Due November 29) Suppose that we have three alternatives, for which the gains are in the intervals [0, 100], [30, 60], and [80, 90].

• is there an alternative which is guaranteed to be optimal (i.e., for which the gain is the largest)?
• list all alternatives which can be optimal;
• which alternative(s) should we select if we use Hurwicz optimism-pessimism criterion with α = 0, 0.5, and 1.

Solution to Homework 25