Experiments

1. Consider the function with values of
   1. Using both linear interpolation and cubic spline interpolation, determine an estimate of the values of

   2. Using 400 samples, plot a linear interpolate of from to .
   3. Using 400 samples, plot a cubic spline interpolate of from to . How do these figures compare? Which do you prefer?
   4. A cubic spline is a particular third degree polynomial approximation to . The command polyfit can also be used to approximate with a third degree polynomial. Plot a cubic spline approximation of on the same figure with a third degree polynomial approximation derived from polyfit for ranging between 0 and 4. How do these compare? Explain.
   5. Plot on the same figure a first, second, fourth, and tenth degree polynomial approximation to the data for
2. Curve fitting to reduce noise. Consider the sequences where is a sequence of Gaussian random variables with zero mean and standard deviation of .5.
   1. Plot on the same figure and . How do they compare?
   2. The following technique can be used for reducing the effect of the noise on . First, find an Nth order polynomial fit to which results in a ``close'' approximation to . To evaluate this approach, plot on the same figure this approximation with and . In addition, to see the resulting error, plot the difference between and the polynomial approximation.
   3. What was your choice for N? Why?
   4. How good has this technique worked for removing the noise from the cosine signal?