# Functions¶

import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline


## What’s wrong with this code?¶

max_iter = 100
h, w = 400, 400

img = np.zeros((h, w)).astype('int')
for i, real in enumerate(np.linspace(-1.5, 0.5, w)):
for j, imag in enumerate(np.linspace(-1, 1, h)):
c = complex(real, imag)
z = 0 + 0j
for k in range(max_iter):
z = z*z + c
if abs(z) > 2:
break
img[j, i] = k

plt.grid(False)
plt.imshow(img, cmap=plt.cm.jet)
pass

• hard to understand
• uses global variables
• not re-usable except by copy and paste

## Refactoring to use functions¶

def mandel(c, z=0, max_iter=100):
for k in range(max_iter):
z = z*z + c
if abs(z) > 2:
return k
return k

def mandelbrot(w, h, xl=-1.5, xu=0.5, yl=-1, yu=1):
img = np.zeros((h, w)).astype('int')
for i, real in enumerate(np.linspace(xl, xu, w)):
for j, imag in enumerate(np.linspace(yl, yu, h)):
c = complex(real, imag)
img[j, i] = mandel(c)
return img

img = mandelbrot(w=400, h=400)
plt.grid(False)
plt.imshow(img, cmap=plt.cm.jet)
pass


## Function is re-usable¶

img = mandelbrot(w=400, h=400, xl=-0.75, xu=-0.73, yl=0.1, yu=0.12)
plt.grid(False)
plt.imshow(img, cmap=plt.cm.jet)
pass


### Anonymous functions (lambdas)¶

def square(x):
return x*x

square(3)

9

square2 = lambda x: x*x

square2(3)

9


### First class functions¶

# functions can be treated the same way as (say) an integer


## Functions can be passed in as arguments¶

def grad(x, f, h=0.01):
return (f(x+h) - f(x-h))/(2*h)

def f(x):
return 3*x**2 + 5*x + 3

grad(0, f)

5.000000000000004


## Functions can also be returned by functions¶

import time

def timer(f):
def g(*args, **kwargs):
start = time.time()
result = f(*args, **kwargs)
elapsed = time.time() - start
return result, elapsed
return g

def f(n=1000000):
s = sum([x*x for x in range(n)])
return s

timed_func = timer(f)

timed_func()

(333332833333500000, 0.18930506706237793)


## Decorators¶

@timer
def g(n=1000000):
s = sum([x*x for x in range(n)])
return s

g()

(333332833333500000, 0.192213773727417)


### Map, filter, reduce¶

map(lambda x: x*x, [1,2,3,4])

<map at 0x10996d5c0>

list(map(lambda x: x*x, [1,2,3,4]))

[1, 4, 9, 16]

list(filter(lambda x: x%2==0, [1,2,3,4]))

[2, 4]

from functools import reduce

reduce(lambda x, y: x*y, [1,2,3,4], 10)

240


### List comprehension¶

[x*x for x in [1,2,3,4]]

[1, 4, 9, 16]

[x for x in [1,2,3,4] if x%2 == 0]

[2, 4]


### Set and dictionary comprehension¶

{i%3 for i in range(10)}

{0, 1, 2}

{i: i%3 for i in range(10)}

{0: 0, 1: 1, 2: 2, 3: 0, 4: 1, 5: 2, 6: 0, 7: 1, 8: 2, 9: 0}


### Generator expressions¶

(i**2 for i in range(10,15))

<generator object <genexpr> at 0x109e5ca40>

for x in (i**2 for i in range(10,15)):
print(x)

100
121
144
169
196


#### Generator expressions¶

Generator expressions return a potentially infinite stream, but one at a time thus sparing memory. They are ubiquitous in Python 3, allowing us to handle arbitrarily large data sets.

# Note that count can generate an infinite stream
def count(i=0):
while True:
yield i
i += 1

c = count()
next(c)

0

next(c)

1

next(c)

2

list(zip('abcde', count(10)))

[('a', 10), ('b', 11), ('c', 12), ('d', 13), ('e', 14)]

for i in count():
print(i)
if i >= 10:
break

0
1
2
3
4
5
6
7
8
9
10

def palindrome_numbers(n):
yield from range(1, n+1)
yield from range(n, 0, -1)

list(palindrome_numbers(5))

[1, 2, 3, 4, 5, 5, 4, 3, 2, 1]


### Itertools¶

import itertools as it

for i in it.islice(count(), 5, 10):
print(i)

5
6
7
8
9

for i in it.takewhile(lambda i: i< 5, count()):
print(i)

0
1
2
3
4

import operator as op

[i for i in it.starmap(op.add, [(1,2), (2,3), (3,4)])]

[3, 5, 7]

fruits = ['appple', 'banana', 'cherry', 'durain', 'eggplant',  'fig']

for k, group in it.groupby(sorted(fruits, key=len), len):
print(k, list(group))

3 ['fig']
6 ['appple', 'banana', 'cherry', 'durain']
8 ['eggplant']


## Functools¶

import functools as fn

rng1 = fn.partial(np.random.normal, 2, .3)
rng2 = fn.partial(np.random.normal, 10, 1)

rng1(10)

array([ 2.13849718,  1.5807533 ,  1.92939089,  2.32091577,  1.75429334,
2.39892103,  2.13631947,  1.90810476,  1.54398362,  2.22273936])

rng2(10)

array([  9.46427924,  10.75766948,   9.79962611,  10.46099347,
10.44005324,   9.69270764,   8.788236  ,  10.32903729,
8.98723117,   9.97326292])

fn.reduce(op.add, rng2(10))

95.284222396568097


### Modules¶

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt

from pandas import DataFrame, Series
import scipy.stats as ss

DataFrame(ss.beta(2,5).rvs((3,4)), columns=['a', 'b', 'c', 'd'])

a b c d
0 0.128479 0.285557 0.380817 0.223367
1 0.391506 0.282001 0.231474 0.196180
2 0.567670 0.122379 0.278288 0.151692

## Where does Python search for modules?¶

import sys
sys.path

['',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages/StreamLib-1.0.1-py3.5.egg',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages/julia-0.1.1-py3.5.egg',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages/pybind11-1.9.dev0-py3.5.egg',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages/scikits.odes-2.3.0.dev0-py3.5-macosx-10.6-x86_64.egg',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages/ReFlowRESTClient-0.4-py3.5.egg',
'/Users/cliburn/spark/python',
'/Users/cliburn/git-teach/sta-663-2017-public/notebook',
'/Users/cliburn/anaconda2/envs/p3/lib/python35.zip',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/plat-darwin',
'/Users/cliburn/.local/lib/python3.5/site-packages',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages/Sphinx-1.4.1-py3.5.egg',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages/aeosa',
'/Users/cliburn/anaconda2/envs/p3/lib/python3.5/site-packages/IPython/extensions',
'/Users/cliburn/.ipython']


## Creating your own module¶

%%file my_module.py

PI = 3.14

def my_f(x):
return PI*x

Overwriting my_module.py

import my_module as mm

mm.PI

3.14

mm.my_f(2)

6.28

from my_module import PI

PI * 2 * 2

12.56


Note: Modules can also be nested within each other - e.g. numpy.random to creaate a package. We will explore how to create packages in a later session.