Homework 2: Higher Order Functions hw02.zip

Due by 11:59pm on Thursday, September 8

Required questions

Several doctests refer to these functions:

from operator import add, mul

square = lambda x: x * x

identity = lambda x: x

triple = lambda x: 3 * x

increment = lambda x: x + 1

Q1: Product

Write a function called product that returns term(1) * ... * term(n).

def product(n, term):
    """Return the product of the first n terms in a sequence.

    n: a positive integer
    term:  a function that takes one argument to produce the term

    >>> product(3, identity)  # 1 * 2 * 3
    6
    >>> product(5, identity)  # 1 * 2 * 3 * 4 * 5
    120
    >>> product(3, square)    # 1^2 * 2^2 * 3^2
    36
    >>> product(5, square)    # 1^2 * 2^2 * 3^2 * 4^2 * 5^2
    14400
    >>> product(3, increment) # (1+1) * (2+1) * (3+1)
    24
    >>> product(3, triple)    # 1*3 * 2*3 * 3*3
    162
    """
    sum=term(1)
    for i in range(n-1):
        sum*=term((i+2))
    return sum

Q2: Accumulate

Let’s take a look at how product is an instance of a more general function called accumulate, which we would like to implement:

def accumulate(merger, start, n, term):
    """Return the result of merging the first n terms in a sequence and start.
    The terms to be merged are term(1), term(2), ..., term(n). merger is a
    two-argument commutative function.

    >>> accumulate(add, 0, 5, identity)  # 0 + 1 + 2 + 3 + 4 + 5
    15
    >>> accumulate(add, 11, 5, identity) # 11 + 1 + 2 + 3 + 4 + 5
    26
    >>> accumulate(add, 11, 0, identity) # 11
    11
    >>> accumulate(add, 11, 3, square)   # 11 + 1^2 + 2^2 + 3^2
    25
    >>> accumulate(mul, 2, 3, square)    # 2 * 1^2 * 2^2 * 3^2
    72
    >>> # 2 + (1^2 + 1) + (2^2 + 1) + (3^2 + 1)
    >>> accumulate(lambda x, y: x + y + 1, 2, 3, square)
    19
    >>> # ((2 * 1^2 * 2) * 2^2 * 2) * 3^2 * 2
    >>> accumulate(lambda x, y: 2 * x * y, 2, 3, square)
    576
    >>> accumulate(lambda x, y: (x + y) % 17, 19, 20, square)
    16
    """
    "*** YOUR CODE HERE ***"
    sum = start
    for i in range(n):
        sum = merger(sum,term(i+1))
    return sum

accumulate has the following parameters:

term and n: the same parameters as in product
merger: a two-argument function that specifies how the current term is merged with the previously accumulated terms.
start: value at which to start the accumulation.

For example, the result of accumulate(add, 11, 3, square) is

1	11 + square(1) + square(2) + square(3) = 25

Note: You may assume that merger is commutative. That is, merger(a, b) == merger(b, a) for all a and b. However, you may not assume merger is chosen from a fixed function set and hard-code the solution.

After implementing accumulate, show how summation and product can both be defined as function calls to accumulate.

Important: You should have a single line of code (which should be a return statement) in each of your implementations for summation_using_accumulate and product_using_accumulate, which the syntax check will check for.

def summation_using_accumulate(n, term):
    """Returns the sum: term(0) + ... + term(n), using accumulate.

    >>> summation_using_accumulate(5, square)
    55
    >>> summation_using_accumulate(5, triple)
    45
    >>> # You aren't expected to understand the code of this test.
    >>> # Check that the bodies of the functions are just return statements.
    >>> # If this errors, make sure you have removed the "***YOUR CODE HERE***".
    >>> import inspect, ast
    >>> [type(x).__name__ for x in ast.parse(inspect.getsource(summation_using_accumulate)).body[0].body]
    ['Expr', 'Return']
    """
    return accumulate(add, term(0), n, term)

def product_using_accumulate(n, term):
    """Returns the product: term(1) * ... * term(n), using accumulate.

    >>> product_using_accumulate(4, square)
    576
    >>> product_using_accumulate(6, triple)
    524880
    >>> # You aren't expected to understand the code of this test.
    >>> # Check that the bodies of the functions are just return statements.
    >>> # If this errors, make sure you have removed the "***YOUR CODE HERE***".
    >>> import inspect, ast
    >>> [type(x).__name__ for x in ast.parse(inspect.getsource(product_using_accumulate)).body[0].body]
    ['Expr', 'Return']
    """
    return accumulate(mul, 1, n, term)