solving-equations.org
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* Solving Equations
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*** Outline
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***** Introduction
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******* What are we going to do?
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******* How are we going to do?
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******* Arsenal Required
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********* working knowledge of arrays
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*** Script
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    Welcome. 
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    In this tutorial we shall look at solving linear equations, obtaining
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    roots of polynomial and non-linear equations. In the process, we
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    shall look at defining functions as well. 
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    We would be using concepts related to arrays which we have covered
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    in a previous tutorial.
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    Let's begin with solving linear equations. 
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    {show a slide of the equations}
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    Consider the set of equations,
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    3x + 2y -z = 1, 2x-2y + 4z = -2, -x+ half y-z = 0.
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    We shall use the solve function, to solve the given system of linear
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    equations. Solve requires the coefficients and the constants to
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    be in the form of matrices of the form Ax = b to solve the system of linear equations. 
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    Lets start ipython -pylab interpreter.    
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    We begin by entering the coefficients and the constants as
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    matrices. 
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    In []: A = array([[3,2,-1], 
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                      [2,-2,4],
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                      [-1, 0.5, -1]])
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    A is a 3X3 matrix of the coefficients of x, y and z
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    In []: b = array([1, -2, 0])
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    Now, we can use the solve function to solve the given system. 
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    In []: x = solve(A, b)
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    Type x, to look at the solution obtained. 
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    Equation is of the form Ax = b, so we verify the solution by 
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    obtaining a matrix product of A and x, and comparing it with b. 
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    As we have covered earlier that we should use the dot function 
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    here, and not the * operator. 
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    In []: Ax = dot(A, x)
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    In []: Ax
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    The result Ax, doesn't look exactly like b, but if we carefully
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    observe, we will see that it is the same as b. To save ourself
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    all this trouble, we can use the allclose function. 
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    allclose checks if two matrices are close enough to each other
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    (with-in the specified tolerance level). Here we shall use the
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    default tolerance level of the function. 
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    In []: allclose(Ax, b)
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    The function returns True, which implies that the product of A &
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    x is very close to the value of b. This validates our solution x. 
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    Let's move to finding the roots of a polynomial. We shall use the
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    roots function for this.
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    The function requires an array of the coefficients of the
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    polynomial in the descending order of powers. 
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    Consider the polynomial x^2-5x+6 = 0
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    In []: coeffs = [1, -5, 6]
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    In []: roots(coeffs)
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    As we can see, roots returns the result in an array. 
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    It even works for polynomials with imaginary roots.
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    roots([1, 1, 1])
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    As you can see, the roots of that equation are of the form a + bj
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    What if I want the solution of non linear equations?
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    For that we use the fsolve function. In this tutorial, we shall use
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    the equation sin(x)+cos^2(x). fsolve is not part of the pylab
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    package which we imported at the beginning, so we will have to import
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    it. It is part of scipy package. Let's import it using.
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    In []: from scipy.optimize import fsolve
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    Now, let's look at the documentation of fsolve by typing fsolve?    
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    In []: fsolve?
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    As mentioned in documentation the first argument, func, is a python 
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    function that takes atleast one argument. So, we should now 
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    define a python function for the given mathematical expression
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    sin(x)+cos^2(x). 
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    The second argument, x0, is the initial estimate of the roots of
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    the function. Based on this initial guess, fsolve returns a root. 
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    Before, going ahead to get a root of the given expression, we
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    shall first learn how to define a function in python. 
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    Let's define a function called f, which returns values of the
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    given mathematical expression (sin(x)+cos^2(x)) for a each input. 
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    In []: def f(x):
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    ...        return sin(x)+cos(x)*cos(x)
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    ...
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    ...
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    hit the enter key to come out of function definition. 
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    def, is a key word in python that tells the interpreter that a
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    function definition is beginning. f, here, is the name of the
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    function and x is the lone argument of the function. The whole
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    definition of the function is done with in an indented block similar
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    to the loops and conditional statements we have used in our 
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    earlier tutorials. Our function f has just one line in it's 
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    definition. 
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    We can test our function, by calling it with an argument for
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    which the output value is known, say x = 0. We can see that
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    sin(x) + cos^2(x) has a value of 1, when x = 0. 
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    Let's check our function definition, by calling it with 0 as an
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    argument. 
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    In []: f(0)
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    We can see that the output is as expected. 
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    Now, that we have our function, we can use fsolve to obtain a root
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    of the expression sin(x)+cos^2(x). Recall that fsolve takes
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    another argument, the initial guess. Let's use 0 as our initial
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    guess. 
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    In []: fsolve(f, 0)
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    fsolve has returned a root of sin(x)+cos^2(x) that is close to 0. 
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    That brings us to the end of this tutorial. We have covered solution
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    of linear equations, finding roots of polynomials and non-linear
2
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    equations. We have also learnt how to define functions and call
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    them. 
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    Thank you!
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*** Notes