Functional Programming for Mathematicians

Author: Minh Van Nguyen <nguyenminh2@gmail.com>

This tutorial discusses some techniques of functional programming that might be of interest to mathematicians or people who use Python for scientific computation. We start off with a brief overview of procedural and object-oriented programming, and then discuss functional programming techniques. Along the way, we briefly review Python’s built-in support for functional programming, including filter, lambda, map and reduce. The tutorial concludes with some resources on detailed information on functional programming using Python.

Styles of programming

Python supports several styles of programming. You could program in the procedural style by writing a program as a list of instructions. Say you want to implement addition and multiplication over the integers. A procedural program to do so would be as follows:

sage: def add_ZZ(a, b):
....:     return a + b
...
sage: def mult_ZZ(a, b):
....:     return a * b
...
sage: add_ZZ(2, 3)
5
sage: mult_ZZ(2, 3)
6

>>> from sage.all import *
>>> def add_ZZ(a, b):
...     return a + b
...
>>> def mult_ZZ(a, b):
...     return a * b
...
>>> add_ZZ(Integer(2), Integer(3))
5
>>> mult_ZZ(Integer(2), Integer(3))
6

The Python module operator defines several common arithmetic and comparison operators as named functions. Addition is defined in the built-in function operator.add and multiplication is defined in operator.mul. The above example can be worked through as follows:

sage: from operator import add
sage: from operator import mul
sage: add(2, 3)
5
sage: mul(2, 3)
6

>>> from sage.all import *
>>> from operator import add
>>> from operator import mul
>>> add(Integer(2), Integer(3))
5
>>> mul(Integer(2), Integer(3))
6

Another common style of programming is called object-oriented programming. Think of an object as code that encapsulates both data and functionalities. You could encapsulate integer addition and multiplication as in the following object-oriented implementation:

sage: class MyInteger:
....:     def __init__(self):
....:         self.cardinality = "infinite"
....:     def add(self, a, b):
....:         return a + b
....:     def mult(self, a, b):
....:         return a * b
...
sage: myZZ = MyInteger()
sage: myZZ.cardinality
'infinite'
sage: myZZ.add(2, 3)
5
sage: myZZ.mult(2, 3)
6

>>> from sage.all import *
>>> class MyInteger:
...     def __init__(self):
...         self.cardinality = "infinite"
...     def add(self, a, b):
...         return a + b
...     def mult(self, a, b):
...         return a * b
...
>>> myZZ = MyInteger()
>>> myZZ.cardinality
'infinite'
>>> myZZ.add(Integer(2), Integer(3))
5
>>> myZZ.mult(Integer(2), Integer(3))
6

Functional programming using map

Functional programming is yet another style of programming in which a program is decomposed into various functions. The Python built-in functions map, reduce and filter allow you to program in the functional style. Note that in Python 3 (as compared to Python 2), these functions have different behaviors, and reduce has been removed: if you want to use reduce in Python 3, you must import it from functools.

The function

map(func, seq1, seq2, ...)

takes a function func and one or more sequences, and apply func to elements of those sequences. In particular, you end up with a list like so:

[func(seq1[0], seq2[0], ...), func(seq1[1], seq2[1], ...), ...]

In many cases, using map allows you to express the logic of your program in a concise manner without using list comprehension. For example, say you have two lists of integers and you want to add them element-wise. A list comprehension to accomplish this would be as follows:

sage: A = [1, 2, 3, 4]
sage: B = [2, 3, 5, 7]
sage: [A[i] + B[i] for i in range(len(A))]
[3, 5, 8, 11]

>>> from sage.all import *
>>> A = [Integer(1), Integer(2), Integer(3), Integer(4)]
>>> B = [Integer(2), Integer(3), Integer(5), Integer(7)]
>>> [A[i] + B[i] for i in range(len(A))]
[3, 5, 8, 11]

Alternatively, you could use the Python built-in addition function operator.add together with map to achieve the same result:

sage: from operator import add
sage: A = [1, 2, 3, 4]
sage: B = [2, 3, 5, 7]
sage: list(map(add, A, B))
[3, 5, 8, 11]

>>> from sage.all import *
>>> from operator import add
>>> A = [Integer(1), Integer(2), Integer(3), Integer(4)]
>>> B = [Integer(2), Integer(3), Integer(5), Integer(7)]
>>> list(map(add, A, B))
[3, 5, 8, 11]

An advantage of map is that you do not need to explicitly define a for loop as was done in the above list comprehension.

Define small functions using lambda

There are times when you want to write a short, one-liner function. You could re-write the above addition function as follows:

sage: def add_ZZ(a, b): return a + b
...

>>> from sage.all import *
>>> def add_ZZ(a, b): return a + b
...

Or you could use a lambda statement to do the same thing in a much clearer style. The above addition and multiplication functions could be written using lambda as follows:

sage: add_ZZ = lambda a, b: a + b
sage: mult_ZZ = lambda a, b: a * b
sage: add_ZZ(2, 3)
5
sage: mult_ZZ(2, 3)
6

>>> from sage.all import *
>>> add_ZZ = lambda a, b: a + b
>>> mult_ZZ = lambda a, b: a * b
>>> add_ZZ(Integer(2), Integer(3))
5
>>> mult_ZZ(Integer(2), Integer(3))
6

Things get more interesting once you combine map with the lambda statement. As an exercise, you might try to write a simple function that implements a constructive algorithm for the Chinese Remainder Theorem. You could use list comprehension together with map and lambda as shown below. Here, the parameter A is a list of integers and M is a list of moduli.

sage: def crt(A, M):
....:     Mprod = prod(M)
....:     Mdiv = list(map(lambda x: Integer(Mprod / x), M))
....:     X = list(map(inverse_mod, Mdiv, M))
....:     x = sum([A[i]*X[i]*Mdiv[i] for i in range(len(A))])
....:     return mod(x, Mprod).lift()
...
sage: A = [2, 3, 1]
sage: M = [3, 4, 5]
sage: x = crt(A, M); x
11
sage: mod(x, 3)
2
sage: mod(x, 4)
3
sage: mod(x, 5)
1

>>> from sage.all import *
>>> def crt(A, M):
...     Mprod = prod(M)
...     Mdiv = list(map(lambda x: Integer(Mprod / x), M))
...     X = list(map(inverse_mod, Mdiv, M))
...     x = sum([A[i]*X[i]*Mdiv[i] for i in range(len(A))])
...     return mod(x, Mprod).lift()
...
>>> A = [Integer(2), Integer(3), Integer(1)]
>>> M = [Integer(3), Integer(4), Integer(5)]
>>> x = crt(A, M); x
11
>>> mod(x, Integer(3))
2
>>> mod(x, Integer(4))
3
>>> mod(x, Integer(5))
1

To produce a random matrix over a ring, say \(\ZZ\), you could start by defining a matrix space and then obtain a random element of that matrix space:

sage: MS = MatrixSpace(ZZ, nrows=5, ncols=3)
sage: MS.random_element()  # random

[ 6  1  0]
[-1  5  0]
[-1  0  0]
[-5  0  1]
[ 1 -1 -3]

>>> from sage.all import *
>>> MS = MatrixSpace(ZZ, nrows=Integer(5), ncols=Integer(3))
>>> MS.random_element()  # random
<BLANKLINE>
[ 6  1  0]
[-1  5  0]
[-1  0  0]
[-5  0  1]
[ 1 -1 -3]

Or you could use the function random_matrix:

sage: random_matrix(ZZ, nrows=5, ncols=3)  # random

[  2 -50   0]
[ -1   0  -6]
[ -4  -1  -1]
[  1   1   3]
[  2  -1  -1]

>>> from sage.all import *
>>> random_matrix(ZZ, nrows=Integer(5), ncols=Integer(3))  # random
<BLANKLINE>
[  2 -50   0]
[ -1   0  -6]
[ -4  -1  -1]
[  1   1   3]
[  2  -1  -1]

The next example uses map to construct a list of random integer matrices:

sage: rows = [randint(1, 10) for i in range(10)]
sage: cols = [randint(1, 10) for i in range(10)]
sage: rings = [ZZ]*10
sage: M = list(map(random_matrix, rings, rows, cols))
sage: M[0]  # random

[ -1  -3  -1 -37   1  -1  -4   5]
[  2   1   1   5   2   1  -2   1]
[ -1   0  -4   0  -2   1  -2   1]

>>> from sage.all import *
>>> rows = [randint(Integer(1), Integer(10)) for i in range(Integer(10))]
>>> cols = [randint(Integer(1), Integer(10)) for i in range(Integer(10))]
>>> rings = [ZZ]*Integer(10)
>>> M = list(map(random_matrix, rings, rows, cols))
>>> M[Integer(0)]  # random
<BLANKLINE>
[ -1  -3  -1 -37   1  -1  -4   5]
[  2   1   1   5   2   1  -2   1]
[ -1   0  -4   0  -2   1  -2   1]

If you want more control over the entries of your matrices than the random_matrix function permits, you could use lambda together with map as follows:

sage: rand_row = lambda n: [randint(1, 10) for i in range(n)]
sage: rand_mat = lambda nrows, ncols: [rand_row(ncols) for i in range(nrows)]
sage: matrix(rand_mat(5, 3))  # random

[ 2  9 10]
[ 8  8  9]
[ 6  7  6]
[ 9  2 10]
[ 2  6  2]
sage: rows = [randint(1, 10) for i in range(10)]
sage: cols = [randint(1, 10) for i in range(10)]
sage: M = list(map(rand_mat, rows, cols))
sage: M = list(map(matrix, M))
sage: M[0]  # random

[ 9  1  5  2 10 10  1]
[ 3  4  3  7  4  3  7]
[ 4  8  7  6  4  2 10]
[ 1  6  3  3  6  2  1]
[ 5  5  2  6  4  3  4]
[ 6  6  2  9  4  5  1]
[10  2  5  5  7 10  4]
[ 2  7  3  5 10  8  1]
[ 1  5  1  7  8  8  6]

>>> from sage.all import *
>>> rand_row = lambda n: [randint(Integer(1), Integer(10)) for i in range(n)]
>>> rand_mat = lambda nrows, ncols: [rand_row(ncols) for i in range(nrows)]
>>> matrix(rand_mat(Integer(5), Integer(3)))  # random
<BLANKLINE>
[ 2  9 10]
[ 8  8  9]
[ 6  7  6]
[ 9  2 10]
[ 2  6  2]
>>> rows = [randint(Integer(1), Integer(10)) for i in range(Integer(10))]
>>> cols = [randint(Integer(1), Integer(10)) for i in range(Integer(10))]
>>> M = list(map(rand_mat, rows, cols))
>>> M = list(map(matrix, M))
>>> M[Integer(0)]  # random
<BLANKLINE>
[ 9  1  5  2 10 10  1]
[ 3  4  3  7  4  3  7]
[ 4  8  7  6  4  2 10]
[ 1  6  3  3  6  2  1]
[ 5  5  2  6  4  3  4]
[ 6  6  2  9  4  5  1]
[10  2  5  5  7 10  4]
[ 2  7  3  5 10  8  1]
[ 1  5  1  7  8  8  6]

Reducing a sequence to a value

The function reduce takes a function of two arguments and apply it to a given sequence to reduce that sequence to a single value. The function sum is an example of a reduce function. The following sample code uses reduce and the built-in function operator.add to add together all integers in a given list. This is followed by using sum to accomplish the same task:

sage: from functools import reduce
sage: from operator import add
sage: L = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]
sage: reduce(add, L)
55
sage: sum(L)
55

>>> from sage.all import *
>>> from functools import reduce
>>> from operator import add
>>> L = [Integer(1), Integer(2), Integer(3), Integer(4), Integer(5), Integer(6), Integer(7), Integer(8), Integer(9), Integer(10)]
>>> reduce(add, L)
55
>>> sum(L)
55

In the following sample code, we consider a vector as a list of real numbers. The dot product is then implemented using the functions operator.add and operator.mul, in conjunction with the built-in Python functions reduce and map. We then show how sum and map could be combined to produce the same result.

sage: from functools import reduce
sage: from operator import add
sage: from operator import mul
sage: U = [1, 2, 3]
sage: V = [2, 3, 5]
sage: reduce(add, map(mul, U, V))
23
sage: sum(map(mul, U, V))
23

>>> from sage.all import *
>>> from functools import reduce
>>> from operator import add
>>> from operator import mul
>>> U = [Integer(1), Integer(2), Integer(3)]
>>> V = [Integer(2), Integer(3), Integer(5)]
>>> reduce(add, map(mul, U, V))
23
>>> sum(map(mul, U, V))
23

Or you could use Sage’s built-in support for the dot product:

sage: u = vector([1, 2, 3])
sage: v = vector([2, 3, 5])
sage: u.dot_product(v)
23

>>> from sage.all import *
>>> u = vector([Integer(1), Integer(2), Integer(3)])
>>> v = vector([Integer(2), Integer(3), Integer(5)])
>>> u.dot_product(v)
23

Here is an implementation of the Chinese Remainder Theorem without using sum as was done previously. The version below uses operator.add and defines mul3 to multiply three numbers instead of two.

sage: from functools import reduce
sage: def crt(A, M):
....:     from operator import add
....:     Mprod = prod(M)
....:     Mdiv = list(map(lambda x: Integer(Mprod / x), M))
....:     X = map(inverse_mod, Mdiv, M)
....:     mul3 = lambda a, b, c: a * b * c
....:     x = reduce(add, map(mul3, A, X, Mdiv))
....:     return mod(x, Mprod).lift()
...
sage: A = [2, 3, 1]
sage: M = [3, 4, 5]
sage: x = crt(A, M); x
11

>>> from sage.all import *
>>> from functools import reduce
>>> def crt(A, M):
...     from operator import add
...     Mprod = prod(M)
...     Mdiv = list(map(lambda x: Integer(Mprod / x), M))
...     X = map(inverse_mod, Mdiv, M)
...     mul3 = lambda a, b, c: a * b * c
...     x = reduce(add, map(mul3, A, X, Mdiv))
...     return mod(x, Mprod).lift()
...
>>> A = [Integer(2), Integer(3), Integer(1)]
>>> M = [Integer(3), Integer(4), Integer(5)]
>>> x = crt(A, M); x
11

Filtering with filter

The Python built-in function filter takes a function of one argument and a sequence. It then returns a list of all those items from the given sequence such that any item in the new list results in the given function returning True. In a sense, you are filtering out all items that satisfy some condition(s) defined in the given function. For example, you could use filter to filter out all primes between 1 and 50, inclusive.

sage: list(filter(is_prime, [1..50]))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]

>>> from sage.all import *
>>> list(filter(is_prime, (ellipsis_range(Integer(1),Ellipsis,Integer(50)))))
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47]

For a given positive integer \(n\), the Euler phi function counts the number of integers \(a\), with \(1 \leq a \leq n\), such that \(\gcd(a, n) = 1\). You could use list comprehension to obtain all such \(a\)’s when \(n = 20\):

sage: [k for k in range(1, 21) if gcd(k, 20) == 1]
[1, 3, 7, 9, 11, 13, 17, 19]

>>> from sage.all import *
>>> [k for k in range(Integer(1), Integer(21)) if gcd(k, Integer(20)) == Integer(1)]
[1, 3, 7, 9, 11, 13, 17, 19]

A functional approach is to use lambda to define a function that determines whether or not a given integer is relatively prime to 20. Then you could use filter instead of list comprehension to obtain all the required \(a\)’s.

sage: is_coprime = lambda k: gcd(k, 20) == 1
sage: list(filter(is_coprime, range(1, 21)))
[1, 3, 7, 9, 11, 13, 17, 19]

>>> from sage.all import *
>>> is_coprime = lambda k: gcd(k, Integer(20)) == Integer(1)
>>> list(filter(is_coprime, range(Integer(1), Integer(21))))
[1, 3, 7, 9, 11, 13, 17, 19]

The function primroots defined below returns all primitive roots modulo a given positive prime integer \(p\). It uses filter to obtain a list of integers between \(1\) and \(p - 1\), inclusive, each integer in the list being relatively prime to the order of the multiplicative group \((\ZZ/p\ZZ)^{\ast}\).

sage: def primroots(p):
....:     g = primitive_root(p)
....:     znorder = p - 1
....:     is_coprime = lambda x: gcd(x, znorder) == 1
....:     good_odd_integers = filter(is_coprime, [1..p-1, step=2])
....:     all_primroots = [power_mod(g, k, p) for k in good_odd_integers]
....:     all_primroots.sort()
....:     return all_primroots
...
sage: primroots(3)
[2]
sage: primroots(5)
[2, 3]
sage: primroots(7)
[3, 5]
sage: primroots(11)
[2, 6, 7, 8]
sage: primroots(13)
[2, 6, 7, 11]
sage: primroots(17)
[3, 5, 6, 7, 10, 11, 12, 14]
sage: primroots(23)
[5, 7, 10, 11, 14, 15, 17, 19, 20, 21]
sage: primroots(29)
[2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27]
sage: primroots(31)
[3, 11, 12, 13, 17, 21, 22, 24]

>>> from sage.all import *
>>> def primroots(p):
...     g = primitive_root(p)
...     znorder = p - Integer(1)
...     is_coprime = lambda x: gcd(x, znorder) == Integer(1)
...     good_odd_integers = filter(is_coprime, (ellipsis_range(Integer(1),Ellipsis,p-Integer(1), step=Integer(2))))
...     all_primroots = [power_mod(g, k, p) for k in good_odd_integers]
...     all_primroots.sort()
...     return all_primroots
...
>>> primroots(Integer(3))
[2]
>>> primroots(Integer(5))
[2, 3]
>>> primroots(Integer(7))
[3, 5]
>>> primroots(Integer(11))
[2, 6, 7, 8]
>>> primroots(Integer(13))
[2, 6, 7, 11]
>>> primroots(Integer(17))
[3, 5, 6, 7, 10, 11, 12, 14]
>>> primroots(Integer(23))
[5, 7, 10, 11, 14, 15, 17, 19, 20, 21]
>>> primroots(Integer(29))
[2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27]
>>> primroots(Integer(31))
[3, 11, 12, 13, 17, 21, 22, 24]

Further resources

This has been a rather short tutorial to functional programming with Python. The Python standard documentation has a list of built-in functions, many of which are useful in functional programming. For example, you might want to read up on all, any, max, min, and zip. The Python module operator has numerous built-in arithmetic and comparison operators, each operator being implemented as a function whose name reflects its intended purpose. For arithmetic and comparison operations, it is recommended that you consult the operator module to determine if there is a built-in function that satisfies your requirement. The module itertools has numerous built-in functions to efficiently process sequences of items.

Another useful resource for functional programming in Python is the Functional Programming HOWTO by A. M. Kuchling. Steven F. Lott’s book Building Skills in Python has a chapter on Functional Programming using Collections. See also the chapter Functional Programming from Mark Pilgrim’s book Dive Into Python.

You might also want to consider experimenting with Haskell for expressing mathematical concepts. For an example of Haskell in expressing mathematical algorithms, see J. Gibbons’ article Unbounded Spigot Algorithms for the Digits of Pi in the American Mathematical Monthly.