st-scripts: comparison solving-equations.org

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-:3893bac8e424
+:2eff0ebac2dc
 ********* working knowledge of arrays
 *** Script
 Welcome.
-In this tutorial we shall look at solving linear equations, roots
+In this tutorial we shall look at solving linear equations, obtaining
-of polynomials and other non-linear equations. In the process, we
+roots of polynomial and other non-linear equations. In the process, we
-shall look at defining functions.
+shall look at defining functions as well.
 We would be using concepts related to arrays which we have covered
-in previous session
+in a previous tutorial.
 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.
-Then we begin by entering the coefficients and the constants as
+We begin by entering the coefficients and the constants as
 matrices.
 In []: A = array([[3,2,-1],
 [2,-2,4],
 [-1, 0.5, -1]])
 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
+The result Ax, doesn't look exactly like b, but if we carefully
-observe, you will see that it is the same as b. To save yourself
+observe, we will see that it is the same as b. To save ourself
-this trouble, you can use the allclose function.
+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, 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.
+roots function to calculate the roots of a polynomial.
 The function requires an array of the coefficients of the
 polynomial in the descending order of powers.
+Consider the polynomial x^2-5x+6
 In []: coeffs = [1, -5, 6]
 In []: roots(coeffs)
-As you can see, roots returns the result in an array.
+As we can see, roots returns the result in an array.
-# It even works for polynomials with imaginary solutions.
+It even works for polynomials with imaginary roots.
-# roots([1, 1, 1])
+roots([1, 1, 1])
-To find the roots of non linear equations, we use the fsolve
+What if I want the solution of non linear equations?
-function. We shall use the function sin(x)+cos^2(x) as our
+For that we use the fsolve function. We shall use the function
-function, in this tutorial. This function is not part of pylab
+sin(x)+cos^2(x) as our function, in this tutorial. This function
-package which we import during starting, so we will have to
+is not part of pylab package which we import at the beginning,
-import it. It is part of scipy package.
+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 using fsolve?
+Now, let's look at the documentation of fsolve by typing fsolve?
 In []: fsolve?
-As mentioned in docs the first argument, func, is a python
+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
 In []: def f(x):
 ...        return sin(x)+cos(x)*cos(x)
 ...
 ...
-hit return thrice for coming out of function definition.
+hit the enter key thrice for coming 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 same
-as for loops and conditional statements we have used in our
+as the loops and conditional statements we have used in our
-earlier sessions. Our function f has just one line in it's
+earlier tutorials. Our function f has just one line in it's
 definition.
-You can test your function, by calling it with an argument for
+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 []: 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 other
+of linear equations, finding roots of polynomials and non-linear
-non-linear
 equations. We have also learnt how to define functions and call
 them.
 Thank you!

changeset 81	2eff0ebac2dc
parent 79	3893bac8e424
child 82	c7abfeddc958