* 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, obtaining
    roots of polynomial and non-linear equations. In the process, we
    shall look at defining functions as well. 

    We would be using concepts related to arrays which we have covered
    in a previous tutorial.

    Let's begin with solving linear equations. 
    {show a slide of the equations}
    Consider the set of equations,
    3x + 2y -z = 1, 2x-2y + 4z = -2, -x+ half y-z = 0.
    We shall use the solve function, to solve the given system of linear
    equations. Solve requires the coefficients and the constants to
    be in the form of matrices of the form Ax = b to solve the system of linear equations. 

    Lets start ipython -pylab interpreter.    
    We begin by entering the coefficients and the constants as
    matrices. 

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

    A is a 3X3 matrix of the coefficients of x, y and z

    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. 

    Equation is of the form Ax = b, so we verify the solution by 
    obtaining a matrix product of A and x, and comparing it with b. 
    As we have covered earlier 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 we carefully
    observe, we will see that it is the same as b. To save ourself
    all this trouble, we 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 is very close to the value of b. This validates our solution x. 

    Let's move to finding the roots of a polynomial. We shall use the
    roots function for this.

    The function requires an array of the coefficients of the
    polynomial in the descending order of powers. 
    Consider the polynomial x^2-5x+6 = 0
    
    In []: coeffs = [1, -5, 6]
    In []: roots(coeffs)
    As we can see, roots returns the result in an array. 
    It even works for polynomials with imaginary roots.
    roots([1, 1, 1])
    As you can see, the roots of that equation are of the form a + bj

    What if I want the solution of non linear equations?
    For that we use the fsolve function. In this tutorial, we shall use
    the equation sin(x)+cos^2(x). fsolve is not part of the pylab
    package which we imported at the beginning, so we will have to import
    it. It is part of scipy package. Let's import it using.

    In []: from scipy.optimize import fsolve

    Now, let's look at the documentation of fsolve by typing fsolve?    
    
    In []: fsolve?

    As mentioned in documentation 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)
    ...
    ...
    hit the enter key to come out of function definition. 
   
    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 similar
    to the loops and conditional statements we have used in our 
    earlier tutorials. Our function f has just one line in it's 
    definition. 

    We can test our function, by calling it with an argument for
    which the output value is known, 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. We have covered solution
    of linear equations, finding roots of polynomials and non-linear
    equations. We have also learnt how to define functions and call
    them. 

    Thank you!

*** Notes