Next Chapter: Recursion and Recursive Functions

## Functions

### Syntax

The concept of a function is one of the most important in mathematics. A common usage of functions in computer languages is to implement mathematical functions. Such a function is computing one or more results, which are entirely determined by the parameters passed to it.

This is mathematics, but we are talking about programming and Python. So what is a function in programming? In the most general sense, a function is a structuring element in programming languages to group a bunch of statements so they can be utilized in a program more than once. The only way to accomplish this without functions would be to reuse code by copying it and adapting it to different contexts, which would be a bad idea. Redundant code - repeating code in this case - should be avoided! Using functions usually enhances the comprehensibility and quality of a program. It also lowers the cost for development and maintenance of the software.

Functions are known under various names in programming languages, e.g. as subroutines, routines, procedures, methods, or subprograms.

A function in Python is defined by a `def`

statement. The general syntax looks like this:

```
def function-name(Parameter list):
statements, i.e. the function body
```

The parameter list consists of none or more parameters. Parameters are called arguments, if the function is called. The function body consists of indented statements. The function body gets executed every time the function is called. We demonstrate this in the following picture:

The code from the picture can be seen in the following:

```
def f(x, y):
z = 2 * (x + y)
return z
print("Program starts!")
a = 3
res1 = f(a, 2+a)
print("Result of function call:", res1)
a = 4
b = 7
res2 = f(a, b)
print("Result of function call:", res2)
```

We call the function twice in the program. The function has two parameters, which are called `x`

and `y`

. This means that the function `f`

is expecting two value, or I should say "two objects". the first time, we call this function with `f(a, 2+a)`

. This means that `a`

goes to `x`

and the result of `2+a`

(5) 'goes to' the variable `y`

. The mechanism for assigning arguments to parameters is called **argument passing**. When we reach the `return`

statement, the object referenced by `z`

will be return, which means that it will be assigned to the variable `res1`

. After leaving the function `f`

, the variable `z`

and the parameters `x`

and `y`

will be deleted automatically.

The references to the objects can be seen in the next diagram:

The next Python code block contains an example of a function without a return statement. We use the `pass`

statement inside of this function. `pass`

is a null operation. This means that when it is executed, nothing happens. It is useful as a placeholder in situations when a statement is required syntactically, but no code needs to be executed:

A more useful function:

```
def fahrenheit(T_in_celsius):
""" returns the temperature in degrees Fahrenheit """
return (T_in_celsius * 9 / 5) + 32
for t in (22.6, 25.8, 27.3, 29.8):
print(t, ": ", fahrenheit(t))
```

### Default arguments in Python

When we define a Python function, we can set a default value to a parameter. If the function is called without the argument, this default value will be assigned to the parameter. This makes a parameter optional. To say it in other words: Default parameters are parameters, which don't have to be given, if the function is called. In this case, the default values are used.

We will demonstrate the operating principle of default parameters with a simple example. The following function `hello`

, - which isn't very useful, - greets a person. If no name is given, it will greet everybody:

```
def hello(name="everybody"):
""" Greets a person """
print("Hello " + name + "!")
hello("Peter")
hello()
```

### The Defaults Pitfall

In the previous section we learned about default parameters. Default parameters are quite simple, but quite often programmers new to Python encounter a horrible and completely unexpected surprise. This surprise arises from the way Python treats the default arguments and the effects steming from mutable objects.

Mutable objects are those which can be changed after creation. In Python, dictionaries are examples of mutable objects. Passing mutable lists or dictionaries as default arguments to a function can have unforeseen effects. Programmer who use lists or dictionaries as default arguments to a function, expect the program to create a new list or dictionary every time that the function is called. However, this is not what actually happens. Default values will not be created when a function is called. Default values are created exactly once, when the function is defined, i.e. at compile-time.

Let us look at the following Python function "spammer" which is capable of creating a "bag" full of spam:

```
def spammer(bag=[]):
bag.append("spam")
return bag
```

Calling this function once without an argument, returns the expected result:

```
spammer()
```

The surprise shows when we call the function again without an argument:

```
spammer()
```

Most programmers will have expected the same result as in the first call, i.e. `['spam']`

To understand what is going on, you have to know what happens when the functions is defined. The compiler creates an attribute `__defaults__`

:

```
def spammer(bag=[]):
bag.append("spam")
return bag
spammer.__defaults__
```

Whenever we will call the function, the parameter `bag`

will be assigned to the list object referenced by `spammer.__defaults__[0]`

:

```
for i in range(5):
print(spammer())
print("spammer.__defaults__", spammer.__defaults__)
```

Now, you know and understand what is going on, but you may ask yourself how to overcome this problem. The solution consists in using the immutable value `None`

as the default. This way, the function can set dynamically at run-time bag to an empty list:

```
def spammer(bag=None):
if bag is None:
bag = []
bag.append("spam")
return bag
for i in range(5):
print(spammer())
print("spammer.__defaults__", spammer.__defaults__)
```

```
def hello(name="everybody"):
""" Greets a person """
print("Hello " + name + "!")
print("The docstring of the function hello: " + hello.__doc__)
```

```
def sumsub(a, b, c=0, d=0):
return a - b + c - d
print(sumsub(12, 4))
print(sumsub(42, 15, d=10))
```

Keyword parameters can only be those, which are not used as positional arguments. We can see the benefit in the example. If we hadn't had keyword parameters, the second call to function would have needed all four arguments, even though the c needs just the default value:

```
print(sumsub(42,15,0,10))
```

```
def no_return(x, y):
c = x + y
res = no_return(4, 5)
print(res)
```

If we start this little script, *None* will be printed, i.e. the special value *None* will be returned by a return-less function. *None* will also be returned, if we have just a return in a function without an expression:

```
def empty_return(x, y):
c = x + y
return
res = empty_return(4, 5)
print(res)
```

Otherwise the value of the expression following return will be returned. In the next example 9 will be printed:

```
def return_sum(x, y):
c = x + y
return c
res = return_sum(4, 5)
print(res)
```

Let's summarize this behavior: Function bodies can contain one or more return statement. They can be situated anywhere in the function body. A return statement ends the execution of the function call and "returns" the result, i.e. the value of the expression following the return keyword, to the caller. If the return statement is without an expression, the special value `None`

is returned. If there is no return statement in the function code, the function ends, when the control flow reaches the end of the function body and the value `None`

will be returned.

### Returning Multiple Values

A function can return exactly one value, or we should better say one object. An object can be a numerical value, like an integer or a float. But it can also be e.g. a list or a dictionary. So, if we have to return, for example, 3 integer values, we can return a list or a tuple with these three integer values. That is, we can indirectly return multiple values. The following example, which is calculating the Fibonacci boundary for a positive number, returns a 2-tuple. The first element is the Largest Fibonacci Number smaller than x and the second component is the Smallest Fibonacci Number larger than x. The return value is immediately stored via unpacking into the variables lub and sup:

```
def fib_intervall(x):
""" returns the largest fibonacci
number smaller than x and the lowest
fibonacci number higher than x"""
if x < 0:
return -1
(old,new) = (0,1)
while True:
if new < x:
(old,new) = (new,old+new)
else:
if new == x:
new = old+new
return (old, new)
while True:
x = int(input("Your number: "))
if x <= 0:
break
(lub, sup) = fib_intervall(x)
print("Largest Fibonacci Number smaller than x: " + str(lub))
print("Smallest Fibonacci Number larger than x: " + str(sup))
```

```
def f():
print(s)
s = "Python"
f()
```

```
def f():
s = "Perl"
print(s)
f()
```

```
s = "Python"
f()
print(s)
```

```
def f():
print(s)
s = "Perl"
print(s)
s = "Python"
f()
print(s)
```

```
s = "Python"
f()
print(s)
```

If we execute the previous script, we get the error message: UnboundLocalError: local variable 's' referenced before assignment

The variable s is ambigious in f(), i.e. in the first print in f() the global s could be used with the value "Python". After this we define a local variable s with the assignment s = "Perl"

```
def f():
global s
print(s)
s = "dog"
print(s)
s = "cat"
f()
print(s)
```

We made the variable s global inside of the script on the left side. Therefore anything we do to s inside of the function body of f is done to the global variable s outside of f.

### Arbitrary Number of Parameters

There are many situations in programming, in which the exact number of necessary parameters cannot be determined a-priori. An arbitrary parameter number can be accomplished in Python with so-called tuple references. An asterisk "*" is used in front of the last parameter name to denote it as a tuple reference. This asterisk shouldn't be mistaken for the C syntax, where this notation is connected with pointers. Example:

```
def arithmetic_mean(first, *values):
""" This function calculates the arithmetic mean of a non-empty
arbitrary number of numerical values """
return (first + sum(values)) / (1 + len(values))
print(arithmetic_mean(45,32,89,78))
print(arithmetic_mean(8989.8,78787.78,3453,78778.73))
print(arithmetic_mean(45,32))
print(arithmetic_mean(45))
```

This is great, but we have still have one problem. You may have a list of numerical values. Like, for example,

x = [3, 5, 9]

You cannot call it with

arithmetic_mean(x)

because "arithmetic_mean" can't cope with a list. Calling it with

arithmetic_mean(x[0], x[1], x[2])

is cumbersome and above all impossible inside of a program, because list can be of arbitrary length.

The solution is easy. We add a star in front of the x, when we call the function.

arithmetic_mean(*x)

This will "unpack" or singularize the list.

A practical example: We have a list

my_list = [('a', 232), ('b', 343), ('c', 543), ('d', 23)]

We want to turn this list into the following list:

[('a', 'b', 'c', 'd'), (232, 343, 543, 23)]

This can be done by using the *-operator and the zip function in the following way:

list(zip(*my_list))

```
def f(**kwargs):
print(kwargs)
```

```
f()
```

```
f(de="German",en="English",fr="French")
```

One use case is the following:

```
def f(a, b, x, y):
print(a, b, x, y)
d = {'a':'append', 'b':'block','x':'extract','y':'yes'}
f(**d)
```

### Exercises with Functions

#### Exercise 1

Write a function which takes a text and a dictionary to decrypt or encrypt the given text.

#### Exercise 2

Write a function txt2morse, which translates a text to morse code, i.e. the function returns a string with the morse code.

Write another function morse2txt which translates a string in Morse code into a „normal“ string.

The Morse character are separated by spaces. Words by three spaces.

#### Exercise 3

Perhaps the first algorithm used for approximating $\sqrt{S}$ is known as the "Babylonian method", named after the Babylonians, or "Hero's method", named after the first-century Greek mathematician Hero of Alexandria who gave the first explicit description of the method.

If a nmber $x_n$ is close to the square root of $a$ then $$x_{n+1} = \frac{1}{2}(x_n + \frac{a}{x_n})$$ will be a better approximation.

Write a program to calculate the square root of a number by using the Babylonian method.

Exercise 4)

Write a function which calculates the position of the n-th occurence of a string sub in another string s. If sub doesn't occur in s, -1 shall be returned.

```
import string
from random import sample
alphabet = string.ascii_letters
permutated_alphabet = sample(alphabet, len(alphabet))
encrypt_dict = dict(zip(alphabet, permutated_alphabet))
decrypt_dict = dict(zip(permutated_alphabet, alphabet))
def encrypt(text, edict):
""" Every character of the text 'text'
is mapped to the value of edict. Characters
which are not keys of edict will not change"""
res = ""
for char in text:
res = res + edict.get(char, char)
return res
# Donald Trump: 5:19 PM, September 9 2014
txt = """Windmills are the greatest
threat in the US to both bald
and golden eagles. Media claims
fictional ‘global warming’ is worse."""
ctext = encrypt(txt, encrypt_dict)
print(ctext + "\n")
print(encrypt(ctext, decrypt_dict))
```

```
latin2morse_dict = {'A':'.-', 'B':'-...', 'C':'-.-.', 'D':'-..',
'E':'.', 'F':'..-.', 'G':'--.','H':'....',
'I':'..', 'J':'.---', 'K':'-.-', 'L':'.-..',
'M':'--', 'N':'-.', 'O':'---', 'P':'.--.',
'Q':'--.-', 'R':'.-.', 'S':'...', 'T':'-',
'U':'..-', 'V':'...-', 'W':'.--', 'X':'-..-',
'Y':'-.--', 'Z':'--..', '1':'.----', '2':'...--',
'3':'...--', '4':'....-', '5':'.....', '6':'-....',
'7':'--...', '8':'---..', '9':'----.', '0':'-----',
',':'--..--', '.':'.-.-.-', '?':'..--..', ';':'-.-.-',
':':'---...', '/':'-..-.', '-':'-....-', '\'':'.----.',
'(':'-.--.-', ')':'-.--.-', '[':'-.--.-', ']':'-.--.-',
'{':'-.--.-', '}':'-.--.-', '_':'..--.-'}
# reversing the dictionary:
morse2latin_dict = dict(zip(latin2morse_dict.values(),
latin2morse_dict.keys()))
print(morse2latin_dict)
```

```
def txt2morse(txt, alphabet):
morse_code = ""
for char in txt.upper():
if char == " ":
morse_code += " "
else:
morse_code += alphabet[char] + " "
return morse_code
def morse2txt(txt, alphabet):
res = ""
mwords = txt.split(" ")
for mword in mwords:
for mchar in mword.split():
res += alphabet[mchar]
res += " "
return res
mstring = txt2morse("So what?", latin2morse_dict)
print(mstring)
print(morse2txt(mstring, morse2latin_dict))
```

```
def heron(a, eps=0.000000001):
""" Approximate the square root of a"""
previous = 0
new = 1
while abs(new - previous) > eps:
previous = new
new = (previous + a/previous) / 2
return new
print(heron(2))
print(heron(2, 0.001))
```

Solution to exercise 4:

```
def findnth(s, sub, n):
num = 0
start = -1
while num < n:
start = s.find(sub, start+1)
if start == -1:
break
num += 1
return start
s = "abc xyz abc jkjkjk abc lkjkjlkj abc jlj"
print(findnth(s,"abc", 3))
```

Next Chapter: Recursion and Recursive Functions