* Solving Equations

*** Outline

***** Introduction

******* What are we going to do?

******* How are we going to do?

******* Arsenal Required

********* working knowledge of arrays

*** Script

    Welcome. 

    We would be using concepts related to arrays which we have covered

    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. 

    Lets start ipython -pylab interpreter.    

    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. 

    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

    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

    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)

    In []: from scipy.optimize import fsolve

    In []: fsolve?

    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 same

    definition. 

    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

    equations. We have also learnt how to define functions and call

    them. 

    Thank you!

*** Notes