* Solving Equations
*** Outline
***** Introduction
******* What are we going to do?
******* How are we going to do?
******* Arsenal Required
********* working knowledge of arrays

*** Script
    Welcome. 
    
    In this tutorial we shall look at solving linear equations, roots
    of polynomials and other non-linear equations. In the process, we
    shall look at defining functions. 

    Let's begin with solving linear equations. 
    {show a slide of the equations}
    We shall use the solve function, to solve this system of linear
    equations.  Solve requires the coefficients and the constants to
    be in the form of matrices to solve the system of linear equations. 

    We begin by entering the coefficients and the constants as
    matrices. 

    In []: A = array([[3,2,-1],
                      [2,-2,4],                   
                      [-1, 0.5, -1]])
    In []: b = array([1, -2, 0])

    Now, we can use the solve function to solve the given system. 
    
    In []: x = solve(A, b)

    Type x, to look at the solution obtained. 

    Next, we verify the solution by obtaining a product of A and x,
    and comparing it with b. Note that we should use the dot function
    here, and not the * operator. 

    In []: Ax = dot(A, x)
    In []: Ax

    The result Ax, doesn't look exactly like b, but if you carefully
    observe, you will see that it is the same as b. To save yourself
    this trouble, you can use the allclose function. 

    allclose checks if two matrices are close enough to each other
    (with-in the specified tolerance level). Here we shall use the
    default tolerance level of the function. 

    In []: allclose(Ax, b)
    The function returns True, which implies that the product of A &
    x, and b are close enough. This validates our solution x. 

    Let's move to finding the roots of polynomials. We shall use the
    roots function to calculate the roots of the polynomial x^2-5x+6. 

    The function requires an array of the coefficients of the
    polynomial in the descending order of powers. 
    
    In []: coeffs = [1, -5, 6]
    In []: roots(coeffs)
    As you can see, roots returns the coefficients in an array. 

    To find the roots of any arbitrary function, we use the fsolve
    function. We shall use the function sin(x)+cos^2(x) as our
    function, in this tutorial. First, of all we import fsolve, since it
    is not already available to us. 

    In []: from scipy.optimize import fsolve

    Now, let's look at the arguments of fsolve using fsolve?
    
    In []: fsolve?

    The first argument, func, is a python function that takes atleast
    one argument. So, we should now define a python function for the
    given mathematical expression sin(x)+cos^2(x). 

    The second argument, x0, is the initial estimate of the roots of
    the function. Based on this initial guess, fsolve returns a root. 

    Before, going ahead to get a root of the given expression, we
    shall first learn how to define a function in python. 
    Let's define a function called f, which returns values of the
    given mathematical expression (sin(x)+cos^2(x)) for a each input. 

    In []: def f(x):
               return sin(x)+cos(x)*cos(x)
   
    def, is a key word in python that tells the interpreter that a
    function definition is beginning. f, here, is the name of the
    function and x is the lone argument of the function. The whole
    definition of the function is done with in an indented block. Our
    function f has just one line in it's definition. 

    You can test your function, by calling it with an argument for
    which the output value is know, say x = 0. We can see that
    sin(x) + cos^2(x) has a value of 1, when x = 0. 

    Let's check our function definition, by calling it with 0 as an
    argument. 
    In []: f(0)
    We can see that the output is as expected. 

    Now, that we have our function, we can use fsolve to obtain a root
    of the expression sin(x)+cos^2(x). Recall that fsolve takes
    another argument, the initial guess. Let's use 0 as our initial
    guess. 

    In []: fsolve(f, 0)
    fsolve has returned a root of sin(x)+cos^2(x) that is close to 0. 

    That brings us to the end of this tutorial on solving linear
    equations, finding roots of polynomials and other non-linear
    equations. We have also learnt how to define functions and call
    them. 

    Thank you!

*** Notes