--- a/solving-equations.org Mon Sep 13 18:35:56 2010 +0530
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,142 +0,0 @@
-* 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