1. (Due September 9) Given the following number of people in each age bracket: [0, 60]: 1, [70, 80]: 2, [80, 100]: 1, compute the range of possible values of the variance.

2. (Due September 14) Write a program that, given n intervals (as in the privacy case), computes the range of possible values of the variance.

3. (Due September 16) 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.

4. (Due September 21) We want to estimate 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].

• find the exact range by using calculus;
• estimate the range by applying naive (straightforward) interval computations to the original intervals;
• use bisection and apply naive interval computations to both sub-boxes.

5. (Due September 23) To estimate 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], we bisected the original box into four sub-boxes, with x₁ in [-1, 0] and [0, 1], and x₂ in [-2, 0] and [0, 2]. In class, we used monotonicity analysis to simplify and improve the estimation of the range for one of the 4 sub-boxes, when x₁ is in [0, 1], and x₂ is in [-2, 0]. Perform a similar analysis for the three remaining sub-boxes. Reminder: when there is no monotonicity, we have no other choice but to use naive interval computations.

6. (Due October 5) Propose a topic for a project that you want to do for this class.

7. (Due October 12) Assuming that the computer uses 2 decimal digits, compute the range of a + b * c, where a is in the interval [-1.1, -0.9], b is in the interval [0.010, 0.013], and c is in the interval [11, 15], with appropriate round-offs.

8. (Due October 12) Use both naive interval computations and centered form to estimate the range of the expression (x₁)² - x₁ * x₂ + (x₂)² when x₁ is in the interval [0.2, 1], and x₂ is in the interval [0, 0.6].

9. (Due October 14) Use the mean value form to estimate the range of the expression (x₁)² + x₁ * x₂ + (x₂)² when x₁ is in the interval [0.2, 1], and x₂ is in the interval [0, 0.6].

10. (Due October 19) Use the interval-based optimization algorithm that we learned in class to locate the maximum of the function f(x) = x + x² on the interval [-0.8, 0]. Divide this interval into two, then divide the remaining intervals into two again, etc. Stop when you get intervals of width 0.1.

10a. (Due October 26) Use the interval-based optimization algorithm that we learned in class to locate the maximum of the function f(x) = x² - x + 1 on the interval [-0.6, 1]. Divide this interval into two, then divide the remaining intervals into two again, etc. Stop when you get intervals of width 0.1.

11. (Due October 21)

• Implement four basic operations of interval arithmetic.
• Write a method that computes an enclosure for the range of a given function on a given box by using the Mean Value Form. Assume that a function is already implemented as a method f(x) with an array input, and that we have also already implemented a method that computes the enclosures for partial derivatives. For example, in Java, that would mean that we have a method

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

  that, given a box (described as an array of intervals) and an index i, estimates the range of the partial derivative of f over x _i. Test your method on the example of a function and intervals that we had in Problem 9.

12. (Due October 26) Use the interval-based optimization algorithm that we learned in class to locate the maximum of the function f(x) = 2x₁² - 2x₁ * x₂ + x₂² + x₁ on the box [-1, 1] X [-1, 1]. Two bisection cycles are enough.

13. (Due November 2) Write a method that uses the algorithm that we learned in class to locate the maximum of an arbitrary function on a given box. Just like in the previous programming assignment, Assume that a function is already implemented as a method f(x) with an array input, and that we have also already implemented a method that computes the enclosures for partial derivatives.

14. (Due October 28) Use the constraints method that we had in class to solve the following two problems:

• find x₁ and x₂, both from the interval [-10, 10], that satisfy the system of equations x₁ + x₂ = 0 and x₁ * (1 + x₂) = 0.21;
• find x₁ and x₂, both from the interval [-10, 10], that satisfy the system of equations x₁ + x₂ = 0 and x₁ * (1 + x₂) = 0.3.

16. (Due November 9) A decision maker has three alternatives:

• for the first alternative a₁, the range of possible benefits is [1.0, 2.0];
• for the second alternative a₂, the range of possible benefits is [0.0, 3.0];
• for the third alternative a₃, the range of possible benefits is [-1.0, 3.0].

• a pessimist, for whom α = 0;
• an optimist, for whom α = 1;
• a decision maker for whom α = 0.5;
• a decision maker for whom α = 0.3.

17. (Due November 11) In the affine arithmetic, instead of intervals, we have generalized intervals. Each generalized interval consists of the following three fields:

• a number a₀ (describing the results of computations with midpoints);
• an array a₁, ..., a_n (describing the coefficients at Δa_i), and
• an interval [a] (describing the effect of quadratic and higher order terms).

• a₀ = X_i;
• a_i = -1 and a_j = 0 for all j =/= i; and
• [a] = [0, 0].

• a method that transforms each input interval into the corresponding generalized interval;
• a method for adding generalized intervals, i.e., a method that takes as inputs generalized inputs corresponding to two quantities a and b, and produces a generalized interval corresponding to c = a + b;
• a method for subtracting generalized intervals, i.e., a method that takes as inputs generalized inputs corresponding to two quantities a and b, and produces a generalized interval corresponding to c = a - b;
• a method for multiplying generalized intervals, i.e., a method that takes as inputs generalized inputs corresponding to two quantities a and b, and produces a generalized interval corresponding to c = a * b;
• a method that transforms the generalized interval (a₀, (a₁, ..., a_n), [a]) corresponding to the final result into the final interval estimate a₀ + a₁ * [-Δ₁, Δ₁] + ... + a_n * [-Δ_n, Δ_n] + [a].