HW 31 Discussion

* The improved accuracy of the values in the table did not seem to affect
  the accuracy of the integral approximations.
* We have 7 digits in the data, so the best we can do is a 6 or 7 digits (or
  a relative error of 10^(-6), 10^(-7))
* Recall the error estimate from the notes (page 7).  The absolute error between 
  the y values in the table and the y values obtained from the cubic spline 
  are less than
        h^4 * max | f''''(x) |/24
* For this problem, h is the difference in the x-coordinates in the table.  
  This is 
        h = 2.727272e-01
  
  The exact function is in the hw31fun.m file.  Computing and plotting the 
  4th derivative of this reveals that the maximum of this is about 52.7.  

  The error in the y values obtained from the spline is guaranteed to be less 
  than
       (2.727272e-01)^4*52.7/24 =  0.01216
  
  This is not great and this is reflected in the calculations for HW 31.
  
  IMPORTANT!!!!
  ************************************************************************
  The spacing of the x-coordinates is what is limiting the accuracy of our
  approximations.  The conclusion is if you need more accuracy than this,
  then you need more raw data.
  ************************************************************************
  
  This is one of the reasons that computers dedicated to various scientific
  simulations keep getting bigger and bigger.  Computer models that incorporate more
  effects (say including pressure, temperature, dew point, pollutants, chemical
  reactions, and the mass of clouds in a weather simulation) will be more accurate than a 
  model that just includes pressure, temperature and dew point.  
  
  A more complex model requires more variables and more equations which leads to the
  need for more memory and storage space.  More accuracy in the computed solutions ultimately 
  requires finer spacing in the grid points.  This is not bad for one-dimensional problems.  It
  starts to become a concern for two dimensions and is one of the primary factors of concern
  for three-dimesional problems.    

* Results
    Trapezoidal rule applied to raw data
      relative error = -2.364762546621578e-02
    Create spline, grow the table and apply trapezoidal rule
      relative error = 1.445224131550613e-04
    Apply Simpson's rule to the grown table
      relative error =  1.357486298435793e-03
    Integrate the spline
      relative error =  1.354507674463709e-03       

* Suprisingly, the trapezoidal rule applied to the grown table is most accurate
  (by 1 order of magnitde over the Simpson's and spline integration)
* Simpson's and spline integration errors are almost identical.