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-* Least Squares Fit
-*** Outline
-***** Introduction
-******* What do we want to do? Why?
-********* What's a least square fit?
-********* Why is it useful?
-******* How are we doing it?
-******* Arsenal Required
-********* working knowledge of arrays
-********* plotting
-********* file reading
-***** Procedure
-******* The equation (for a single point)
-******* It's matrix form
-******* Getting the required matrices
-******* getting the solution
-******* plotting
-*** Script
-    Welcome. 
-    
-    In this tutorial we shall look at obtaining the least squares fit
-    of a given data-set. For this purpose, we shall use the same
-    pendulum data that we used in the tutorial on plotting from files.
-
-    To be able to follow this tutorial comfortably, you should have a
-    working knowledge of arrays, plotting and file reading. 
-
-    A least squares fit curve is the curve for which the sum of the
-    squares of it's distance from the given set of points is
-    minimum. 
-
-    Previously, when we plotted the data from pendulum.txt we got a 
-    scatter plot of points as shown. 
-
-    In our example, we know that the length of the pendulum is
-    proportional to the square of the time-period. But when we plot
-    the data using lines we get a distorted line as shown. What
-    we expect ideally, is something like the redline in this graph. 
-    From the problem we know that L is directly proportional to T^2.
-    But experimental data invariably contains errors and hence does
-    not produce an ideal plot. The best fit curve for this data has 
-    to be a linear curve and this can be obtained by performing least
-    square fit on the data set. We shall use the lstsq function to
-    obtain the least squares fit curve. 
-
-    The equation of the line is of the form T^2 = mL+c. We have a set
-    of values for L and the corresponding T^2 values. Using this, we
-    wish to obtain the equation of the straight line. 
-
-    In matrix form the equation is represented as shown, 
-    Tsq = A.p where Tsq is an NX1 matrix, and A is an NX2 matrix as shown.
-    And p is a 2X1 matrix of the slope and Y-intercept. In order to 
-    obtain the least square fit curve we need to find the matrix p
-
-    Let's get started. As you can see, the file pendulum.txt
-    is on our Desktop and hence we navigate to the Desktop by typing 
-    cd Desktop. Let's now fire up IPython: ipython -pylab
-
-    We have already seen (in a previous tutorial), how to read a file
-    and obtain the data set using loadtxt(). Let's quickly get the required data
-    from our file. 
-
-    l, t = loadtxt('pendulum.txt', unpack=True)
-
-    loadtxt() directly stores the values in the pendulum.txt into arrays l and t
-    Let's now calculate the values of square of the time-period. 
-
-    tsq = t*t
-
-    Now we shall obtain A, in the desired form using some simple array
-    manipulation 
-
-    A = array([l, ones_like(l)])
-
-    As we have seen in a previous tutorial, ones_like() gives an array similar
-    in shape to the given array, in this case l, with all the elements as 1. 
-    Please note, this is how we create an array from an existing array.
-
-    Let's now look at the shape of A. 
-    A.shape
-    This is an 2X90 matrix. But we need a 90X2 matrix, so we shall transpose it.
-
-    A = A.T
-    
-    Type A, to confirm that we have obtained the desired array. 
-    A
-    Also note the shape of A. 
-    A.shape
-
-    We shall now use the lstsq function, to obtain the coefficients m
-    and c. lstsq returns a lot of things along with these
-    coefficients. We may look at the documentation of lstsq, for more
-    information by typing lstsq? 
-    result = lstsq(A,tsq)
-
-    We extract the required coefficients, which is the first element
-    in the list of things that lstsq returns, and store them into the variable coef. 
-    coef = result[0]
-
-    To obtain the plot of the line, we simply use the equation of the
-    line, we have noted before. T^2 = mL + c. 
-
-    Tline = coef[0]*l + coef[1]
-    plot(l, Tline, 'r')
-
-    Also, it would be nice to have a plot of the points. So, 
-    plot(l, tsq, 'o')
-
-    This brings us to the end of this tutorial. In this tutorial,
-    you've learnt how to obtain a least squares fit curve for a given
-    set of points using lstsq. There are other curve fitting functions
-    available in Pylab such as polyfit.
-
-    Hope you enjoyed it. Thanks. 
-
-*** Notes
-