User Defined Functions

User defined functions make for neater and more efficient programming.

We have already made use of several library functions in the math, scipy and numpy libraries.



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import numpy as np
import scipy.constants as constants
print('Pi = ', constants.pi)









    



Pi =  3.141592653589793



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h = float(input("Enter the height of the tower (in metres): "))
t = float(input("Enter the time interval (in seconds): "))
s = constants.g*t**2/2
print("The height of the ball is",h-s,"meters")









    



Enter the height of the tower (in metres): 10
Enter the time interval (in seconds): 1
The height of the ball is 5.096675 meters

Link to What's in Scipy.constants: https://docs.scipy.org/doc/scipy/reference/constants.html

Library Functions in Maths

(and numpy)



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x = 4**0.5
print(x)
x = np.sqrt(4)
print(x)

User Defined Functions

Here we'll practice writing our own functions.

Functions start with

def name(input):

and must end with a statement to return the value calculated

return x

To run a function your code would look like this:

import numpy as np

def name(input)

FUNCTION CODE HERE

return D

y=int(input("Enter y:"))

 D = name(y)

 print(D)

First - write a function to calculate n factorial. Reminder:

$n! = \pi^n_{k=1} k$



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def factorial(n):
    f = 1.0
    for k in range(1,n+1):
        f *= k
    return f

print("This programme calculates n!")
n = int(input("Enter n:"))
a = factorial(10)
print("n! = ", a)









    



This programme calculates n!
Enter n:10
n! =  3628800.0

Finding distance to the origin in cylindrical co-ordinates:



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from math import sqrt, cos, sin

def distance(r,theta,z):
    x = r*cos(theta)
    y = r*sin(theta)
    d = sqrt(x**2+y**2+z**2)
    return d

D = distance(2.0,0.1,1.5)
print(D)

2.5

Another Example: Prime Factors and Prime Numbers

Reminder: prime factors are the numbers which divide another number exactly.

Factors of the integer n can be found by dividing by all integers from 2 up to n and checking to see which remainders are zero.

Remainder in python calculated using

n % k



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def factors(n):
    factorlist=[]
    k = 2
    while k<=n:
        while n%k==0:
            factorlist.append(n)
            n //= k
        k += 1
    return factorlist

list=factors(12)
print(list)









    



[12, 6, 3]



In [7]:

    
print(factors(17556))









    



[17556, 8778, 4389, 1463, 209, 19]



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print(factors(23))

The reason these are useful are for things like the below, where you want to make the same calculation many times. This finds all the prime numbers (only divided by 1 and themselves) from 2 to 100.



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for n in range(2,100):
    if len(factors(n))==1:
        print(n)

More Extended Example

Exercise 2.11 in Newman - binomial coefficients.



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