Polynomial interpolation

The following table shows data $y_i$ obtained at locations $x_i$, $i=1,\dots,10$ during a biological experiment.

$x$	0	0.2	0.4	0.6	0.72	0.75	1	1.2	1.3	1.35
$y$	0.8	0.75	0.9	1.0	0.8	0.83	0.95	0.87	0.7	0.55

The problem is to estimate the value of $y$ at the location $x = 0.5$. In order to do this we shall first attempt to fit polynomials to the data.

$\phantom{0}$As the first step create a file called interpolate.m.

$\phantom{0}$Include code in this file to set up two row vectors, one called x , say, containing the locations $x_i$ and the other ( y ) the data points $y_i$.

The inbuilt MATLAB routines that carry out polynomial interpolation are polyfit.m and polyval.m. Recall from lectures that if we have $n$ data points then there is just one polynomial of degree $n-1$, i.e.

$$\bgroup\color{black} p_{n-1}(x) = c_0 +c_1 x +c_2 x^2 + \dots + c_{n-1} x^{n-1} \egroup$$

that will pass exactly through those data points. The MATLAB command

c9 = polyfit(x,y,9) ;

will return the coefficients $c_i$ of the ninth degree polynomial passing exactly through all ten data points. The coefficients are found in the vector c9. (Note that the coefficients are given in reverse order, e.g. the first element in c9 will be $c_9$.) The MATLAB command

yp9 = polyval(c9,xout) ;

evaluates $p_9(x)$ at all $x$-points contained in the vector xout and returns a vector of the results in yp9.

$\phantom{0}$ In interpolate.m set up a vector xout = min(x):0.01:max(x) and then include code to use the MATLAB routines above to calculate the approximating polynomial $p_9(x)$. Make sure to include some comments in your file explaining what the MATLAB commands are doing.

$\phantom{0}$Include code to plot out the original data y as a function of x (use red circles) and yp9 as a function of xout . The commands xlabel , ylabel and title should be used to annotate the graph.

Check that the approximating polynomial does indeed pass through the data points? From the graph, do you think that it is it a useful approximation near $x=0.5$?

$\phantom{0}$Calculate the estimated value of $y(0.5)$.
Code like y05=polyval(c9,0.5) will do the job.

$\phantom{0}$ Evaluate the interpolating polynomial at $x=0.5$.

As the ninth degree polynomial is clearly so poor, we can try to fit a lower degree polynomial instead. Thus take the two points nearest to $x=0.5$, and attempt a linear (first degree) polynomial fit:

c1 = polyfit(x(3:4),y(3:4),1)

Here x(3:4) e.g. means that we use only $x$-points 3 and 4, i.e. those bracketing $x=0.5$.

$\phantom{0}$Add extra code to your file interpolate.m to calculate $p_1(x)$ and plot it on your existing graph (use another linestyle like plot(xout,yp1,'r - -') for example, that will give you a red dashed line and remember to use the switch hold on).

Clearly this is more useful than the ninth degree fit. You can now also try a quadratic fit (take the three nearest $x$-points) or a cubic fit (four nearest $x$-points). What seems to be the best compromise?

$\phantom{0}$ Evaluate the interpolating quadratic polynomial at $x=0.5$.