# Homework 6

*Due by 9:00pm on Friday, 03/15/19*

## Instructions

Download hw06.zip. Inside the archive, you will find starter files for the questions in this homework, along with a copy of the OK autograder.

**Submission:** When you are done, submit with `python3 ok --submit`

. You may submit more than once before the deadline; only the final submission will be scored. Check that you have successfully submitted your code on okpy.org. See this article for more instructions on okpy and submitting assignments.

**Readings:** This homework relies on following references:

## Required Questions

### Intervals (data abstraction)

**Acknowledgements.** This interval arithmetic example is based on
a classic problem from Structure and Interpretation of Computer Programs,
Section 2.1.4.

**Introduction.** Alyssa P. Hacker is designing a system to help people
solve engineering problems. One feature she wants to provide in her
system is the ability to manipulate inexact quantities (such as
measured parameters of physical devices) with known precision, so that
when computations are done with such approximate quantities the results
will be numbers of known precision.

Alyssa's idea is to implement interval arithmetic as a set of arithmetic operations for combining "intervals" (objects that represent the range of possible values of an inexact quantity). The result of adding, subracting, multiplying, or dividing two intervals is itself an interval, representing the range of the result.

Alyssa postulates the existence of an abstract object called an "interval" that has two endpoints: a lower bound and an upper bound. She also presumes that, given the endpoints of an interval, she can construct the interval using the data constructor interval. Using the constructor and selectors, she defines the following operations:

```
def str_interval(x):
"""Return a string representation of interval x.
>>> str_interval(interval(-1, 2))
'-1 to 2'
"""
return '{0} to {1}'.format(lower_bound(x), upper_bound(x))
def add_interval(x, y):
"""Return an interval that contains the sum of any value in interval x and
any value in interval y.
>>> str_interval(add_interval(interval(-1, 2), interval(4, 8)))
'3 to 10'
"""
lower = lower_bound(x) + lower_bound(y)
upper = upper_bound(x) + upper_bound(y)
return interval(lower, upper)
def mul_interval(x, y):
"""Return the interval that contains the product of any value in x and any
value in y.
>>> str_interval(mul_interval(interval(-1, 2), interval(4, 8)))
'-8 to 16'
"""
p1 = lower_bound(x) * lower_bound(y)
p2 = lower_bound(x) * upper_bound(y)
p3 = upper_bound(x) * lower_bound(y)
p4 = upper_bound(x) * upper_bound(y)
return interval(min(p1, p2, p3, p4), max(p1, p2, p3, p4))
```

A constructor is something that creates whatever you want to make, and a selector gets the elements from the thing you made. For example, your constructor `interval`

will take in two numbers a and b. It will construct a two element list of them. Then your `lower_bound`

selector will return the smaller item in the list and the `upper_bound`

selector will return the bigger element in the list.

### Question 1

Alyssa's program is incomplete because she has not specified the implementation of the interval abstraction. Define the constructor and selectors in terms of two-element lists:

```
def interval(a, b):
"""Construct an interval from a to b."""
"*** YOUR CODE HERE ***"
def lower_bound(x):
"""Return the lower bound of interval x."""
"*** YOUR CODE HERE ***"
def upper_bound(x):
"""Return the upper bound of interval x."""
"*** YOUR CODE HERE ***"
```

Use OK to test your code:

```
python3 ok -q str_interval
python3 ok -q add_interval
python3 ok -q mul_interval
```

### Question 2

Alyssa implements division below, by multiplying by the reciprocal of
`y`

. Ben Bitdiddle, an expert systems programmer, looks over Alyssa's
shoulder and comments that it is not clear what it means to divide by
an interval that spans zero. Return `False`

if the interval being divided by contains zero.

```
def div_interval(x, y):
"""Return the interval that contains the quotient of any value in x divided by any value in y.
Division is implemented as the multiplication of x by the reciprocal of y.
>>> str_interval(div_interval(interval(-1, 2), interval(4, 8)))
'-0.25 to 0.5'
>>> div_interval(interval(4, 8), interval(-1, 2))
False
"""
"*** YOUR CODE HERE ***"
reciprocal_y = interval(1/upper_bound(y), 1/lower_bound(y))
return mul_interval(x, reciprocal_y)
```

Use OK to test your code:

`python3 ok -q div_interval`

### Question 3

Using reasoning analogous to Alyssa's, define a subtraction function for intervals:

```
def sub_interval(x, y):
"""Return the interval that contains the difference between any value in x
and any value in y.
>>> str_interval(sub_interval(interval(-1, 2), interval(4, 8)))
'-9 to -2'
"""
"*** YOUR CODE HERE ***"
```

Use OK to test your code:

`python3 ok -q sub_interval`

### Question 4

After debugging her program, Alyssa shows it to a potential user, who
complains that her program solves the wrong problem. He wants a program
that can deal with numbers represented as a center value and an
additive tolerance; for example, he wants to work with intervals such
as `3.5 +/- 0.15`

rather than `3.35`

to `3.65`

. Alyssa returns to her
desk and fixes this problem by supplying an alternate constructor and
alternate selectors in terms of the existing ones:

```
def make_center_width(c, w):
"""Construct an interval from center and width."""
return interval(c - w, c + w)
def center(x):
"""Return the center of interval x."""
return (upper_bound(x) + lower_bound(x)) / 2
def width(x):
"""Return the width of interval x."""
return (upper_bound(x) - lower_bound(x)) / 2
```

Unfortunately, most of Alyssa's users are engineers. Real engineering situations usually involve measurements with only a small uncertainty, measured as the ratio of the width of the interval to the midpoint of the interval. Engineers usually specify percentage tolerances on the parameters of devices.

Define a constructor `make_center_percent`

that takes a center and a
percentage tolerance and produces the desired interval. You must also
define a selector percent that produces the percentage tolerance for a
given interval. The center selector is the same as the one shown
above:

```
def make_center_percent(c, p):
"""Construct an interval from center and percentage tolerance.
>>> str_interval(make_center_percent(2, 50))
'1.0 to 3.0'
"""
"*** YOUR CODE HERE ***"
def percent(x):
"""Return the percentage tolerance of interval x.
>>> percent(interval(1, 3))
50.0
"""
"*** YOUR CODE HERE ***"
```

Use OK to test your code:

`python3 ok -q make_center_percent`

### Question 5

Write a function `quadratic`

that returns the interval of all values
`f(t)`

such that `t`

is in the argument interval `x`

and `f(t)`

is a
quadratic function:

`f(t) = a*t*t + b*t + c`

Make sure that your implementation returns the smallest such interval, one that does not suffer from the multiple references problem.

*Hint*: the derivative `f'(t) = 2*a*t + b`

, and so the extreme
point of the quadratic is `-b/(2*a)`

:

```
def quadratic(x, a, b, c):
"""Return the interval that is the range of the quadratic defined by
coefficients a, b, and c, for domain interval x.
>>> str_interval(quadratic(interval(0, 2), -2, 3, -1))
'-3 to 0.125'
>>> str_interval(quadratic(interval(1, 3), 2, -3, 1))
'0 to 10'
"""
"*** YOUR CODE HERE ***"
```

Use OK to test your code:

`python3 ok -q quadratic`

## Extra Questions

Extra questions are not worth extra credit and are entirely optional. They are designed to challenge you to think creatively!

### Question 6

Write a function polynomial that takes an interval `x`

and a list of
coefficients `c`

, and returns the interval containing all values of
`f(t)`

for `t`

in interval `x`

, where:

`f(t) = c[k-1] * pow(t, k-1) + c[k-2] * pow(t, k-2) + ... + c[0] * 1`

Like quadratic, your polynomial function should return the smallest such interval, one that does not suffer from the multiple references problem.

*Hint*: You can approximate this result. Try using Newton's
method.

```
def polynomial(x, c):
"""Return the interval that is the range of the polynomial defined by
coefficients c, for domain interval x.
>>> str_interval(polynomial(interval(0, 2), [-1, 3, -2]))
'-3 to 0.125'
>>> str_interval(polynomial(interval(1, 3), [1, -3, 2]))
'0 to 10'
>>> str_interval(polynomial(interval(0.5, 2.25), [10, 24, -6, -8, 3]))
'18.0 to 23.0'
"""
"*** YOUR CODE HERE ***"
```

Use OK to test your code:

`python3 ok -q polynomial`