*Due by 11:59pm on Tuesday, 10/7*

**Submission:** See
Lab 1 for
submission instructions. We have provided a hw4.py starter file for the questions below.

**Readings:** You might find the following references
useful:

- Question 1
- Question 2
- Question 3
- Question 4
- Question 5
- Question 6
- Question 7: Challenge Problem (optional)

**Acknowledgements.** This interval arithmetic example is based on
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))
```

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 ***"
```

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. Add an assert statement to Alyssa's code
to ensure that no such interval is used as a divisor:

```
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'
"""
"*** YOUR CODE HERE ***"
reciprocal_y = interval(1/upper_bound(y), 1/lower_bound(y))
return mul_interval(x, reciprocal_y)
```

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 ***"
```

After considerable work, Alyssa P. Hacker delivers her finished system. Several years later, after she has forgotten all about it, she gets a frenzied call from an irate user, Lem E. Tweakit. It seems that Lem has noticed that the formula for parallel resistors can be written in two algebraically equivalent ways:

`par1(r1, r2) = (r1 * r2) / (r1 + r2)`

or

`par2(r1, r2) = 1 / (1/r1 + 1/r2)`

He has written the following two programs, each of which computes the
`parallel_resistors`

formula differently::

```
def par1(r1, r2):
return div_interval(mul_interval(r1, r2), add_interval(r1, r2))
def par2(r1, r2):
one = interval(1, 1)
rep_r1 = div_interval(one, r1)
rep_r2 = div_interval(one, r2)
return div_interval(one, add_interval(rep_r1, rep_r2))
```

Lem complains that Alyssa's program gives different answers for the two ways of computing. This is a serious complaint.

Demonstrate that Lem is right. Investigate the behavior of the system
on a variety of arithmetic expressions. Make some intervals `a`

and
`b`

, and show that `par1`

and `par2`

can give different results. You
will get the most insight by using intervals whose width is a small
percentage of the center value:

```
# These two intervals give different results for parallel resistors:
"*** YOUR CODE HERE ***"
```

*Note*: No tests will be run on your solution to this problem.

Eva Lu Ator, another user, has also noticed the different intervals computed by different but algebraically equivalent expressions. She says that the problem is multiple references to the same interval.

The Multiple References Problem: a formula to compute with intervals using Alyssa's system will produce tighter error bounds if it can be written in such a form that no variable that represents an uncertain number is repeated.

Thus, she says, `par2`

is a better program for parallel resistances
than `par1`

. Is she right? Why? Write a function that returns an
explanation string:

```
def multiple_references_explanation():
return """The mulitple reference problem..."""
```

*Note*: No tests will be run on your solution to this problem.

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 ***"
```

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 ***"
```