Software tutorial/Functions as objects

Background

From this point onward in the course we will often have to deal with an arbitrary function, such as finding the function's zeros, or integrating the function between two points. It is helpful if we can write MATLAB or Python code that can operate on any function, not just a specific function, \(f(x)\).

For example, if we write a Python function to find the zero of \(f(x) = 3x - \frac{2}{x}\), then we would like to send that function, let's call it my_function, into another Python function that will find the zero of any function, \(f(x)\).

def my_function(x):
    """
    Returns the value of f(x) = 3x - 2/x, at the given x
    """
    return 3*x - 2/x

lower = 0.1
upper = 3.0
root = bisection(my_function, lower, upper)

Python (or MATLAB) will see that my_function isn't an ordinary variable, it is another function that will return the value of \(f(x)\) when given a value \(x\). This means that when the code inside the bisection routine wants to evaluate \(f(x)\), it can do so. If you want to change which function is being operated on, then you just call bisection, but change the first input to point to a different function.

MATLAB details: inline and anonymous functions

f = inline('1/x - (x-1)')
fzero(f,1)
roots(...)

Python details: functions are objects

In Python, everything is an object. All four of these variables, my_string, my_integer, my_float and my_function are objects.

my_string = 'abc'
my_integer = 123
my_float = 45.6789
def my_function(x):
    return 3*x - 2/x

You can use the type(...) function to determine a variable's type:

>>> type(my_string)    # string object
<type 'str'>   
>>> type(my_integer)   # integer object
<type 'int'>
>>> type(my_float)     # floating point object
<type 'float'>
>>> type(my_function)  # function object
<type 'function'>

Find zeros of a nonlinear function

• scipy.optimize.brentq
• scipy.optimize.newton
• brenth,
• ridder,
• bisect,
• newton
• fsolve
• fixed_point