In this article, we will only consider sequences defined by a function whose domain is a subset of the set of all integers. Such sequences will be visualized, i.e. we will try to evaluate the first few (thousand) elements, using functional programming paradigm, where functions are more similar to the ones in math (in contrast to imperative style with side effects confusing to inexperenced coders). The idea is taken from subsection 3.5.2 of SICP and adapted to Python, which, compare to Scheme, is significantly more popular: Python is pre-installed on almost every modern Unix-like system, namely macOS, GNU/Linux and the *BSDs; and even at MIT, the new 6.01 in Python has recently replaced the legendary 6.001 (SICP).
One notable advantage of using Python is its huge standard library. For
example the "identity sequence" (sequence defined by the identity function) can
be imported directly from itertools:
>>> from itertools import count
>>> positive_integers = count(start=1)
>>> next(positive_integers)
1
>>> next(positive_integers)
2
>>> for _ in range(4): next(positive_integers)
...
3
4
5
6To open a Python emulator, simply lauch your terminal and run python. If that
is somehow still too struggling, navigate to
the interactive shell on Python.org.
Let's get it started with somethings everyone hates: recursively defined sequences, e.g. the famous Fibonacci (Fn = Fn-1 + Fn-2, F1 = 1, F0 = 0). Since Python doesn't support tail recursion, it's generally not a good idea to define anything recursively (which is, ironically, the only trivial functional solution in this case) but since we will only evaluate the first few terms (use the Tab key to indent the line when needed):
>>> def fibonacci(n, a=0, b=1):
... # To avoid making the code look complicated, n < 0 is not handled here.
... return a if n == 0 else fibonacci(n - 1, b, a + b)
...
>>> fibo_seq = (fibonacci(n) for n in count(start=0))
>>> for _ in range(7): next(fibo_seq)
...
0
1
1
2
3
5
8Note (click to expand)
Thefibo_seq above is just to demonstrate how
itertools.count can be use to create an infinite sequence defined
by a function. For better performance, this should be used instead
def fibonacci_sequence(a=0, b=1):
yield a
yield from fibonacci_sequence(b, a + b)It is noticable that the elements having been iterated through (using next)
will disappear forever in the void (oh no!), but that is the cost we are
willing to pay to save some memory, especially when we need to evaluate a
member of (arbitrarily) large index to estimate the sequence's limit. One case
in point is estimating a definite integral using
left Riemann sum:
>>> def integral(f, a, b):
... def leftRiemannSum(n):
... dx = (b-a) / n
... def x(i): return a + i*dx
... return sum(f(x(i)) for i in range(n)) * dx
... return leftRiemannSum
... The function integral(f, a, b) as defined above returns a function taking n
as an argument. As n approaches ∞, its result approaches the value of the
integral of f on [a, b]. For example, we are going to estimate π as the
area of a semicircle whose radius is sqrt(2):
>>> from math import sqrt
>>> def semicircle(x): return sqrt(abs(2 - x*x))
...
>>> pi = integral(semicircle, -sqrt(2), sqrt(2))
>>> pi_seq = (pi(n) for n in count(start=2))
>>> for _ in range(3): next(pi_seq) # the first few aren't quite close
...
2.000000029802323
2.514157464087051
2.7320508224700384At index around 1000, the result is somewhat acceptable:
3.1414873191059525
3.1414874770617427
3.1414876346231577
Since we are comfortable with sequence of sums, let's move on to sums of a
sequence, which are called series. For estimation, again, we are going to make
use of infinite sequences of partial sums, which are implemented as
itertools.accumulate by thoughtful Python developers.
Geometric and
p-series
can be defined as follow:
>>> from itertools import accumulate as partial_sums
>>> def geometric_series(r, a=1): return partial_sums(a*r**n for n in count(0))
...
>>> def p_series(p): return partial_sums(1 / n**p for n in count(1))
... We can then use these to determine whether the series is convergent or divergent. For instance, the fact that p-series with p = 2 converges to π2/6 ≈ 1.6449340668482264 can be verified via
>>> s = p_series(p=2)
>>> for _ in range(11): next(s)
...
1.0
1.25
1.3611111111111112
1.4236111111111112
1.4636111111111112
1.4913888888888889
1.511797052154195
1.527422052154195
1.5397677311665408
1.5497677311665408
1.558032193976458We can observe that it takes quite a lot of steps to get the precision we would generally expect (s11 is only precise to the first decimal place; second decimal places: s101; third: s2304). Luckily, many techniques for series acceleration are available. Shanks transformation, for instance, can be implemented as follow:
>>> from itertools import islice, tee
>>> def shanks(seq):
... return map(lambda x, y, z: (x*z - y*y) / (x + z - y*2),
... *(islice(t, i, None) for i, t in enumerate(tee(seq, 3))))
... In the code above, lambda x, y, z: (x*z - y*y) / (x + z - y*2) denotes the
anonymous function (x, y, z) ↦ (xz-y2)/(x+z-2y) and
map is a higher order function applying that function to respective elements
of subsequences starting from index 1, 2, 3 of seq. On Python 2, one should
import imap from itertools to get the same
lazy behavior of map on
Python 3.
>>> s = shanks(p_series(2))
>>> for _ in range(10): next(s)
...
1.4500000000000002
1.503968253968257
1.53472222222223
1.5545202020202133
1.5683119658120213
1.57846371882088
1.5862455815659202
1.5923993101138652
1.5973867787856946
1.6015104548459742The result was quite satisfying, yet we can do one step futher by continuously applying the transformation to the sequence:
>>> def compose(transform, seq):
... yield next(seq)
... yield from compose(transform, transform(seq))
...
>>> s = compose(shanks, p_series(2))
>>> for _ in range(10): next(s)
...
1.0
1.503968253968257
1.5999812811165188
1.6284732442271674
1.6384666832276524
1.642311342667821
1.6425249569252578
1.640277484549416
1.6415443295058203
1.642038043478661Shanks transformation works on every sequence (not just sequences of partial sums). Back to previous example of using left Riemann sum to compute definite integral:
>>> pi_seq = compose(shanks, map(pi, count(2)))
>>> for _ in range(10): next(pi_seq)
...
2.000000029802323
2.978391111182236
3.105916845397819
3.1323116570377185
3.1389379264270736
3.140788413965646
3.140921512857936
3.1400282163913436
3.1400874774021816
3.1407097229603256
>>> next(islice(pi_seq, 300, None))
3.1415061302492413Now having series defined, let's see if we can learn anything about power series. Sequence of partial sums of power series ∑cn(x-a)n can be defined as
>>> from operator import mul
>>> def power_series(c, start=0, a=0):
>>> return lambda x: partial_sums(map(mul, c, (x**n for n in count(start))))
... We can use this to compute functions that can be written as Taylor series:
>>> from math import factorial
>>> def exp(x): return power_series(1/factorial(n) for n in count(0))(x)
...
>>> def cos(x):
... c = ((1 - n%2) * (1 - n%4) / factorial(n) for n in count(0))
... return power_series(c)(x)
...
>>> def sin(x):
... c = (n%2 * (2 - n%4) / factorial(n) for n in count(1))
... return power_series(c, start=1)(x)
... Amazing! Let's test 'em!
>>> e = compose(shanks, exp(1)) # this should converges to 2.718281828459045
>>> for _ in range(4): next(e)
...
1.0
2.749999999999996
2.718276515152136
2.718281825486623Impressive, huh? For sine and cosine, series acceleration is not even necessary:
>>> from math import pi as PI
>>> s = sin(PI/6)
>>> for _ in range(5): next(s)
...
0.5235987755982988
0.5235987755982988
0.49967417939436376
0.49967417939436376
0.5000021325887924
>>> next(islice(cos(PI/3), 8, None))
0.500000433432915