# 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
```

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
```

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
```

## 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]
```

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]
```

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
...
```

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
```

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
```

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]
```

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]
```

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]
```

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]
```

## 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 # py3
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
```

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 # py3
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
```

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
```

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 # py3
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
```

## 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]
```

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]
```

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]
```

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]
```

## 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.