Functional Programming

Level 1 introduced Map, Apply, and pure functions. This lesson adds the other functions that let you replace almost every loop.

The examples in this level use symbols like x, f, a, and b without values. If you assigned them earlier, start a fresh kernel (Evaluation → Quit Kernel) or run Clear[x, f, a, b, c] first.

Fold: accumulate over a list

Fold[f, x, {a, b, c}]
Fold[Plus, 0, {1, 2, 3, 4}]
FoldList[Plus, 0, {1, 2, 3, 4}]
Fold[10 #1 + #2 &, 0, {4, 7, 2}]

f[f[f[x, a], b], c]
10
{0, 1, 3, 6, 10}
472

Fold[f, start, list] feeds each element into the running result. FoldList keeps every intermediate value. The last line builds a number from its digits.

Nest: repeat a function

Nest[f, x, 3]
NestList[2 # &, 1, 5]
NestWhile[#/2 &, 96, EvenQ]
NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, 6, # != 1 &]
FixedPoint[(# + 2/#)/2 &, 1.]

f[f[f[x]]]
{1, 2, 4, 8, 16, 32}
3
{6, 3, 10, 5, 16, 8, 4, 2, 1}
1.41421

NestWhile[f, x, test] keeps applying f while test is true.
The fourth line is the Collatz sequence starting at 6.
FixedPoint repeats until the result stops changing — here Newton's method converging to √2.

Working with several lists

MapThread[f, {{a, b}, {c, d}}]
MapIndexed[f, {a, b}]
Outer[Times, {1, 2}, {3, 4}]
Array[#^2 &, 4]
Through[{Min, Max}[{3, 1, 2}]]
(f @* g)[x]

{f[a, c], f[b, d]}
{f[a, {1}], f[b, {2}]}
{{3, 4}, {6, 8}}
{1, 4, 9, 16}
{1, 3}
f[g[x]]

MapThread walks several lists in parallel.
MapIndexed also passes each element's position.
Outer combines every element of one list with every element of another.
Through applies several functions to the same argument.
f @* g is composition: g first, then f.

Why bother?

Functional code says what to compute, not how to step through it. It is usually shorter, easier to test, and faster than loops that modify variables — and it's the style the rest of this course uses.

Practice

Use FoldList to compute the running balance of an account that starts at 100 and receives the changes {20, -50, 10, -5}.

Show solution

FoldList[Plus, 100, {20, -50, 10, -5}]

{100, 120, 70, 80, 75}